Maple commands
for MAT2500: Calc 3created 2002 by bob jantzen; version 15-sep-2018 [Villanova University]
<execute worksheet with toolbar icon !!! to restore output>This worksheet summarizes the commands that are helpful in using MAPLE to as a supporting tool in this course. In a perfect world, students could refer to examples here when necessary to do assigned homework problems, but the reality is that no one has time to look at this worksheet. Fortunately the right click context sensitive menus plus palette insertion of Clickable Calculus enables elementary users to not have to deal very much with Maple commands. Although many of these commands may not be useful in this course, it is useful for instructors to at least be aware of them in case some might prove useful to the activities they choose to do in guiding student learning.Placing a "restart" command at the beginning of a new problem erases all previous assignments MAPLE has made in that session, but then packages must be reloaded if needed.If you are new to Maple, consult the tips for using the Maple interface for new users on the web page:
http://www.villanova.edu/artsci/mathematics/resources/maple/using/mapletips.htmwhere you can also view the worksheets for MAT1500 and MAT1505.
If you execute this worksheet all at once with the icon "!!!" on the toolbar, a few popup Tutor windows must be closed (click on the "Close" button X) as they open in order to allow Maple to continue to the end of the worksheet.load special packagesUsually for Calc 3 we need to load some MAPLE packages with extra commands the most used of which may be listed by replacing the colon by a semicolon (list not shown here):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIjpGJ0YvRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi1GNjYuUSJ+RidGL0YyRjlGPEY+RkBGQkZERkYvRklRJjAuMGVtRicvRkxGUS9GM1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1GQDYtUSJ+RidGPUZDRkZGSEZKRkxGTkZQL0ZTUSYwLjBlbUYnL0ZWRmVuRj0=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI6RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZURj0=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 is needed to display multiple plots together and for spacecurve [animate is fun, and implicitplots, contourplots and fieldplots are useful, while arrow is needed to visualize vectors].
Student[LinearAlgebra] is needed to deal with matrices and vectors as entered by the Matrix palette, although now just right click context sensitive menus give immediate access to the necessary commands, while the menu Tools, Load Package enables direct loading of all the TutorsLinearAlgebra is only needed to access rows or columns from a matrix easily.
Student[VectorCalculus] is needed for the vector calculus commands. Omitting the BasisFormat command below leads to i, j, k unit vector notation except these vectors are called 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. Vectors are displayed as column matrices in Maple output with this command present. This allows integration of vector functions.Student[MultivariateCalculus] has some useful visual tools for multivariable calculus concepts.example worksheets and tasksRemoving the number sign "#" from the beginning of each line and executing the line will bring up many useful example worksheets. They are suppressed here so that when the entire worksheet is executed, 10 extra windows are not opened.#?examples/indexThere are useful example worksheets under the topics of Calculus and Algebra:#?examples/Calculus1Visualization#?examples/MultivariateCalculus#?examples/StudentVectorCalculus#?examples/VectorCalculus#?examples/LA_Linear_Solve#?examples/LinearAlgebraComputation#?examples/LinearAlgebraVisualization1#?examples/LinearAlgebraVisualization2Too much of a good thing, eh? Overwhelming, but better more than less...
And for old timers who used to use the package "linalg" where a lot of short abbreviations existed, there is:#?examples/LinearAlgebraMigrationFrom the Tools Menu, Tasks, Browse, you can see a list of Calculus worksheet templates for common activities in Multivariable or Vector Calculus and Linear Algebra.basic MAPLE through single variable calculus MAT1500-1505The command list for MAT1500 summarizes the basic tools one needs for working with functions symbolically, numerically and graphically, for the topics of calculus through limits and derivatives:
http://www.villanova.edu/artsci/assets/documents/mathematics/mapleresources/cmdlist1.mwThe command list for MAT1505 deals instead with integration, differential equations, and infinite series:http://www.villanova.edu/artsci/assets/documents/mathematics/mapleresources/cmdlist2.mw
Once you get the hang of it you can enter most mathematical expressions from the palettes as long as you respect standard mathematical notation and it will do what you expect. Context dependent right click menus can help you do almost all simple operations needed for this course.In 2d math using the palettes, you already see the symbol on input and the output gives the value: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 you load the Student Vector Calculus package, suppress an arrow function definition with a colon since it messes up the output: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 you really want the symbol for an expression on the LHS of an equality so that the expression is labeled by its symbol, you can place right quotes around an expression and it will not by replaced by the expression for which it stands: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 calc 1 worksheet explains many of the useful shortcuts for entering mathematical expressions. Usually spaces between variables imply multiplication except between hard numbers like 2 and 4 where the asterisk is necessary. The right arrow key is needed to continue to the right from exponents and denominators as expressions are built up. Shift enter will give you a new line within an input region, and a semicolon is necessary to separate more than one input command within such a region.
More extensive Maple interface tips are given on the page:
http://www1.villanova.edu/villanova/artsci/mathematics/resources-and-opportunities/maple/mapletips.htmvectors: dot and crossproduct geometry with standard MAPLEWe use triangle brackets for vectors in MAPLE but the output shows them as vertical column matrix vectors: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General scalar multiplication works as expected: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The dot product symbol is just the centered bold dot from the bottom of the Common Symbols palette:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZtaWRkb3Q7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkwtRiw2JVEiYkYnRi9GMkY5The crossproduct can be inserted from the bottom of the Common Symbols palette:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmdGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlMtRiw2JVEiYkYnRi9GMkY5One can also make a 3D plot of the two vectors and their cross product (the first vector is red, the second vector is blue, third is the cross product in purple)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: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 unit vectors: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEmYV9oYXRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSkmbWlkZG90O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJmJfaGF0RidGL0YyLUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMjc3Nzc3OGVtRidGOQ==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiZGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0Y4LyUqbGluZWJyZWFrR1EobmV3bGluZUYnRisvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGQUYrRkE=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiZGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0Y4LyUqbGluZWJyZWFrR1EobmV3bGluZUYnRisvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGQUYrRkE=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiZGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0Y4LyUqbGluZWJyZWFrR1EobmV3bGluZUYnRisvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGQUYrRkE=This is the angle between the directions.
One can also better visualize the directions of the two vectors and of their cross product using the corresponding unit vectors (the first vector is red, the second vector is blue, the third is the cross product in purple)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Maple also has a visualization command for vector addition: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projection\302\267scalar projection/component of b along a:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZtaWRkb3Q7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkwtRiw2JVEmYV9oYXRGJ0YvRjJGOQ==vector projection/component of b along a: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vector projection/component of b orthogonal to a: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 as a check that these sum to the original vector: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjYvJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGKy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGK0Y/There is a cute visualization command for the projection operation: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 scalar 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parametrized curves in the planeConsider the parametrized curve in the plane, with 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the square brackets around the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= range inputs of the plot command: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Notice that the axes do not intersect at the origin, since MAPLE only shows the exact window which contains all the plotted points. To get the origin in the picture, we need an option to change the (horizontal and) vertical range: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This starts at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=: 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 and terminates at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4=: 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.REMARK: putting the right bracket after the second function shows LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= versus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= instead of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= versus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (we are plotting a list of functions):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 need to tell MAPLE which curve has which color to identify them in the same order:So LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= decreases and then increases while LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is always increasing uniformly. This is also useful information about the parametrized curve.If you simply type in a pair of expressions in a variable and right click on the output, you can select Plots, PlotBuilder, 2d parametric plot to plot the parametrized curve. You may type in 1 or more such expressions and select Plots, 2d plot to plot them together.
You may ignore the sections below:extra: eliminating the parameterTo see that this is a piece of a parabola we can eliminate the parameter easily by hand. MAPLE can also do this: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The first set in the output is the relation used to eliminate the parameter, while the second set is the left hand side of the equation which results, with 0 right hand side not given. In this case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: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We can only plot functions with the horizontal axis as the independent variable, otherwise we must use a parametrized curve instead. Notice that in plotting functions there are no square brackets, just the expression and the range of the independent variable and the optional range of the dependent variable. Here the y axis is horizontal and the x axis is vertical, NOT what we normally do: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extra: animatecurve (VCR)Why not trace out the curve instead of just looking at its final path? Easy.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUYsNiNRIUYnRj0=You can click on animatecurve and go to the HELP menu for its help and examples.Animations play like VCR's once you click on the plot area to bring it alive; then look to the last tool bar line to see the controls: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 might try this on the cycloid for two revolutions 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: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But only after assigning a definite value to the radius, otherwise MAPLE won't have definite points to plot:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMUYnRjlGOQ==After you are done with this particular assigned value, you might want to unassign it so you can use it as a variable name later on.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 functions and spacecurves
(parametrized curves in space)We can use MAPLE vector expressions or MAPLE vector functions. Expressions must be evaluated using another command rather than just using standard function notation for evaluation (to be explained below):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3D parametrized curves (curves in space or "space curves") are plotted with the spacecurve command: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However, one must click on the plot and then click on the boxed axes tool bar icon for this to be useful. Right clicking on the plot allows you to select AXES = BOXED, which can also be done with an optional argument: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Without numbers and axes, the curve cannot be related to ranges of numerical values.The MAPLE commands for calculus operations like limits, differentiation and integration can be applied to vectors by applying them simultaneously to all components:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNipGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LVEkbGltRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JVEidEYnLyUnaXRhbGljR0ZKL0Y7USdpdGFsaWNGJy1GNzYtUScmcmFycjtGJ0Y6Rj1GQEZCRkRGRi9GSUY/RksvRk5RLDAuMjIyMjIyMmVtRicvRlFGW28tSSNtbkdGJDYkUSIzRidGOkY6LyUsYWNjZW50dW5kZXJHRj9GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC40ZW1GJy8lJmRlcHRoR0Zoby8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUYjNiYtRiw2JVEiUkYnRlhGWi1GNzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y6Rj1GQEZCRkRGRkZpbkZLRk0vRlFGTy1JKG1mZW5jZWRHRiQ2JC1GIzYkRlVGOkY6RjpGK0Y6RitGOkYrRjo=[Note that this was really a one-sided limit since t can only approach 3 from the right here due to the positive input requirement for the ln function, but Maple does not complain.]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One can do one-sided limits as in the 1d case: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ASIDE for instructors:
single inputs (functions) versus multiple inputs (vectors) and the map commandASIDE: Without redefinition by loading the above Student packages, many 1d Maple commands like Limit, Diff, Int, etc won't except more than a single expression as their input. The maple "Multiple APplication" command "map" applies the "limit" command to each expression in the list of inputs R, with the remaining inputs for "limit" command following R, but this is unnecessary with the Student[VectorCalculus] package 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 with (which will generate an error if the VectorCalculus package is not loaded):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEmTGltaXRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2LC1GLDYlUSJSRidGL0YyLUY2NiQtRiM2JC1GLDYlUSJ0RidGL0YyL0YzUSdub3JtYWxGJ0ZELUkjbW9HRiQ2LVEiLEYnRkQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGQS1GRzYtUSI9RidGREZKL0ZORkxGT0ZRRlNGVUZXL0ZaUSwwLjI3Nzc3NzhlbUYnL0ZnbkZeby1JI21uR0YkNiRRIjBGJ0ZERkYtRiw2JVEmcmlnaHRGJ0YvRjIvJStleGVjdXRhYmxlR0ZMRkRGREZpbi1GLDYlUSZsaW1pdEYnRi9GMkY1RmdvRkQ=However, notice how one sees here the symbolic operation applied to each component individually instead of to the whole vector. Notice also that since you are using 2d math, you don't even know about the underlying Maple commands so you can safely ignore this. It is instructor stuff.Sometimes we need to increase the number of points plotted to get a smoother picture of a curve: 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visualizing lines and planesparametrized lines and planesWe can visualize parametrized lines with spacecurve, and planes with a parametrized 3d plot generalizing the parametrized curves plotting of spacecurve. We start with the position vectors of two points and construct the line segment from the first point to the second point . Then we take a third point to construct a plane through it with the orientation determined by the first two position vectors (their cross product is a normal to the plane). The individual plots are not very useful but displaying them together we see all the geometry.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEmTHBsb3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEmUHBsb3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==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 the connecting line segment from the tip of the red arrow to the tip of the blue arrow. Notice that the "unit parallelogram" of the plane has edges equal to those two vectors and the unit normal is perpendicular to them. The "parameter grid" on the plane repeats this basic unit parallelogram "tiling" the whole plane. Rotating this plot must be done to see an appropriate viewing angle.Maple has its own visualization command for this requiring the reference point and a normal as inputs: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 planesNext we look at a pair of planes specified by their nonparametrized 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Solving both for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, we can plot them as function graphs: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVElcGx0MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiw2JVEiJUYnRjZGOUZALUY9Ni1RIjpGJ0ZARkJGRUZHRklGS0ZNRk9GUUZURkBGK0ZAThe line of intersection crosses the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 plane at: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 is a trick for getting Maple to use this result in the position vector of the point: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but we can simply retype the result for this point: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while the direction of the line of intersection is the cross product of the two 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so the line of intersection 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrRi9GK0YrRi9GKy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=Yup. But the line is way too long for the parts of the planes displayed. By trial and error, we can shorten it up a bit.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrRi9GK0YrRi9GKy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=JSFHMaple has a visualization command for planes and their normals: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 the Student[LinearAlgebra] package has a cute visualizer for the intersection of two planes: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Or even three planes, which in general intersect in a point: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calculus operations with vector functions of a single variable Calculus operations applied to vector functions are applied simultaneously to all component functions. as a MAPLE vector expressionFirst we just assign a letter to the list of expressions:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2JkYrLUkobWZlbmNlZEdGJDYmLUYjNictRiw2JVEidEYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVUSwwLjMzMzMzMzNlbUYnLUYjNihGKy1GIzYmRistSSVtc3VwR0YkNiVGZ24tSSNtbkdGJDYkUSIyRidGQC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0ZARmpuLUYjNiZGKy1GZ282JUZnbi1Gam82JFEiM0YnRkBGXXBGK0ZARitGQEYrRkBGQC8lJW9wZW5HUTMmTGVmdEFuZ2xlQnJhY2tldDtGJy8lJmNsb3NlR1E0JlJpZ2h0QW5nbGVCcmFja2V0O0YnRitGQEYrRkBGK0ZARitGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNihGKy1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LVEkbGltRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JVEidEYnLyUnaXRhbGljR0ZKL0Y7USdpdGFsaWNGJy1GNzYtUScmcmFycjtGJ0Y6Rj1GQEZCRkRGRi9GSUY/RksvRk5RLDAuMTExMTExMWVtRicvRlFGTy1JI21uR0YkNiRRIjBGJ0Y6RjovJSxhY2NlbnR1bmRlckdGP0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmhvLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYsNiVRIkZGJ0ZYRlpGOkYrRjpGK0Y6LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KkYrLUkmbWZyYWNHRiQ2KC1JI21vR0YkNi1RMCZEaWZmZXJlbnRpYWxEO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUYjNidGK0Y0LUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjJlbUYnLyUmZGVwdGhHRlYvJSpsaW5lYnJlYWtHUSVhdXRvRictRiw2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJ0Y4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zmby8lKWJldmVsbGVkR1EmZmFsc2VGJy1GNTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y4L0Y8RltwL0Y/RltwL0ZBRltwL0ZDRltwL0ZFRltwL0ZHRltwL0ZJRltwRkpGTS1GUjYmRlQvRlhRJjAuNGVtRidGWkZmbkYrLUYsNiVRIkZGJ0Zcb0Zfb0YrRjhGK0Y4LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNihGKy1GIzYmLUYsNiVRInRGJ0Y2RjktRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yNzc3Nzc4ZW1GJy9GVUZeby1JI21uR0YkNiRRIjFGJ0ZARkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUYjNiQtRiw2JVEiJUYnRjZGOUZARitGQEZARkBGK0ZARitGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNihGKy1GIzYoLUkjbW9HRiQ2LVEmJmludDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRiw2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y6USdpdGFsaWNGJy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0Zobi8lKmxpbmVicmVha0dRJWF1dG9GJy1GNjYtUTAmRGlmZmVyZW50aWFsRDtGJ0Y5RjxGP0ZBRkNGRUZHRkkvRkxGUEZOLUYsNiVRInRGJ0ZURldGOS1GNjYtUSIrRidGOUY8Rj9GQUZDRkVGR0ZJL0ZMUSwwLjIyMjIyMjJlbUYnL0ZPRlxwLUkobWZlbmNlZEdGJDYmLUYjNictRiw2JVEjQzFGJ0ZURlctRjY2LVEiLEYnRjlGPC9GQEZWRkFGQ0ZFRkdGSUZkby9GT1EsMC4zMzMzMzMzZW1GJy1GIzYmLUYsNiVRI0MyRidGVEZXRmZwLUYsNiVRI0MzRidGVEZXRjlGK0Y5RjkvJSVvcGVuR1EzJkxlZnRBbmdsZUJyYWNrZXQ7RicvJSZjbG9zZUdRNCZSaWdodEFuZ2xlQnJhY2tldDtGJ0YrRjlGK0Y5RitGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNilGKy1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnLUkjbW5HRiQ2JFEiMEYnRjotRlM2JFEiMUYnRjovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHRlUtRiw2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y7USdpdGFsaWNGJy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0Zlby8lKmxpbmVicmVha0dRJWF1dG9GJy1GNzYtUTAmRGlmZmVyZW50aWFsRDtGJ0Y6Rj1GQEZCRkRGRkZIRkovRk1GUUZPLUYsNiVRInRGJ0Zbb0Zeb0Y6RitGOkYrRjo=as a MAPLE vector function (preferred if working with derivatives): prime and D notationNext we make a MAPLE vector function 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 we use physics notation for the position of a particle as a function of time, we get position, velocity, acceleration, and "jerk" but must use the D notation for the derivative applied to a function name when not evaluating it on an input 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The right quotes prevent Maple from evaluating these symbolic expressions to their explicit values, so we get their symbols on the left hand side for nicer output. Of course we can also just use the prime derivative notation instead of defining separate function names for the velocity, acceleration and jerk: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 the tangent line (optional)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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNkY5RkBGQEYrRkBGK0ZARitGQA==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 equations of the tangent line: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Plot each separately (not very useful without boxed axes) and display together: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiVGK0YrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRjNGK0YzIf we just show the part of the tangent line for the parameter interval 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 which starts at the point of tangency and terminates at the tip of the tangent vector, we see the tangent vector with its correct length attached to the curve at the point of tangency (by using colons we can suppress the individual plots, but it is useful to see them first before by using semicolons and then changing them to colons after it is clear that no problems have occurred: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEmVGxpbmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==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One can also put in the arrow representing the tangent vector itself with the arrow command: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 Student[VectorCalculus] package has a cute tangent vector visualizer. From the Help page, here is an example are a few options with it: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 the geometry of space curvesspace curve tangent, normal and binormalYou can copy the following command out of the tutor if you want to tweek it, but otherwise it inserts the plot or animation into the worksheet. You choose animation or plot by clicking on the right button in the tutor. The following commands are copied and pasted from the Tutor window, but the right quotes are unnecessary, and the 1d notation can easily be changed to 2d notation.[Available from the Tools Menu, Tutors, VectorCalculus, SpaceCurves.]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animation:You click on the plot and then use the VCR controls on the tool bar to play it.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpVE5CRnJhbWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ri1GNjYmLUYjNi0tRiw2JVEkY29zRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUY2NiQtRiM2JC1GLDYlUSJ0RidGL0YyRkNGQy1JI21vR0YkNi1RIixGJ0ZDLyUmZmVuY2VHRkIvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRk02LVEifkYnRkNGUC9GU0ZCRlRGVkZYRlpGZm5GaG4vRlxvRmpuLUYsNiVRJHNpbkYnRkFGQ0ZFRkxGXm9GPi1GNjYkLUYjNiYtSSNtbkdGJDYkUSIyRidGQy1GTTYtUSIqRidGQ0ZQRmFvRlRGVkZYRlpGZm4vRmluUSwwLjE2NjY2NjdlbUYnL0Zcb0ZicEZJRkNGQ0ZDRkMvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRkxGXm9GSUZMRl5vLUZNNi1RIidGJ0ZDRlBGYW9GVEZWRlhGWkZmbi9GaW5RLDAuMTExMTExMWVtRidGYm8tRiw2JVEnb3V0cHV0RidGL0YyRmpwLUZNNi1RIj1GJ0ZDRlBGYW9GVEZWRlhGWkZmbi9GaW5RLDAuMjc3Nzc3OGVtRicvRlxvRmZxLUYsNiVRKmFuaW1hdGlvbkYnRi9GMkZMRl5vRmpwLUYsNiVRJnJhbmdlRidGL0YyRmpwRmJxLUZbcDYkUSIwRidGQ0Zeby1GTTYtUSMuLkYnRkNGUEZhb0ZURlZGWEZaRmZuL0ZpblEsMC4yMjIyMjIyZW1GJ0Zib0Zeb0Zqb0ZecC1GLDYlUScmIzk2MDtGJ0ZBRkNGTEZeb0ZqcC1GLDYlUSdmcmFtZXNGJ0YvRjJGanBGYnEtRltwNiRRIjVGJ0ZDRkwtRiw2JVElYXhlc0YnRi9GMkZicS1GLDYlUSZib3hlZEYnRi9GMkZDRkNGQw==plot:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpVE5CRnJhbWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ri1GNjYmLUYjNi0tRiw2JVEkY29zRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUY2NiQtRiM2JC1GLDYlUSJ0RidGL0YyRkNGQy1JI21vR0YkNi1RIixGJ0ZDLyUmZmVuY2VHRkIvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRk02LVEifkYnRkNGUC9GU0ZCRlRGVkZYRlpGZm5GaG4vRlxvRmpuLUYsNiVRJHNpbkYnRkFGQ0ZFRkxGXm9GPi1GNjYkLUYjNiYtSSNtbkdGJDYkUSIyRidGQy1GTTYtUSIqRidGQ0ZQRmFvRlRGVkZYRlpGZm4vRmluUSwwLjE2NjY2NjdlbUYnL0Zcb0ZicEZJRkNGQ0ZDRkMvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRkxGXm9GSUZMRl5vLUZNNi1RIidGJ0ZDRlBGYW9GVEZWRlhGWkZmbi9GaW5RLDAuMTExMTExMWVtRidGYm8tRiw2JVEnb3V0cHV0RidGL0YyRmpwLUZNNi1RIj1GJ0ZDRlBGYW9GVEZWRlhGWkZmbi9GaW5RLDAuMjc3Nzc3OGVtRicvRlxvRmZxLUYsNiVRJXBsb3RGJ0YvRjJGTEZeb0ZqcC1GLDYlUSZyYW5nZUYnRi9GMkZqcEZicS1GW3A2JFEiMEYnRkNGXm8tRk02LVEjLi5GJ0ZDRlBGYW9GVEZWRlhGWkZmbi9GaW5RLDAuMjIyMjIyMmVtRidGYm9GXm9Gam9GXnAtRiw2JVEnJiM5NjA7RidGQUZDRkxGXm9GanAtRiw2JVEnZnJhbWVzRidGL0YyRmpwRmJxLUZbcDYkUSI1RidGQ0ZMLUYsNiVRJWF4ZXNGJ0YvRjJGYnEtRiw2JVEmYm94ZWRGJ0YvRjJGQ0ZDRkM=maple calculates the tangent, normal and binormal values of tangent, normal, binormal: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 rest require immediate simplification because they are messy before terms are combined.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 you see that you are getting good results, you can supress the longer output with colons.However, the square roots in these formulas can lead to complex number problems. Compare the following outputs:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpVE5CRnJhbWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GNjYmLUYjNistSSVtc3VwR0YkNiUtRiw2JVEidEYnRi9GMi1GIzYkLUkjbW5HRiQ2JFEiMkYnL0YzUSdub3JtYWxGJ0ZKLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0ZKLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRlUvJSpzeW1tZXRyaWNHRlUvJShsYXJnZW9wR0ZVLyUubW92YWJsZWxpbWl0c0dGVS8lJ2FjY2VudEdGVS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUZQNi1RIn5GJ0ZKRlMvRldGVUZYRlpGZm5GaG5Gam5GXG8vRmBvRl5vLUkmbWZyYWNHRiQ2KEZGLUYjNiQtRkc2JFEiM0YnRkpGSi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZHAvJSliZXZlbGxlZEdGVS1GPzYlRkFGam9GTEZPRmJvRkFGSkZKLyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0ZPRmJvRkFGSkZKLUYsNiNRIUYnLyUrZXhlY3V0YWJsZUdGVUZKand now where we avoid the complex number complications: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 circle (helix example)If you do a search in help for "osculating circle", you will find a link to the Student[VectorCalculus] Example worksheet which shows how to use the RadiusOfCurvature command visualize the osculating circle for the standard helix: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The osculating circle lies in the plane of the unit tangent and normal (the binormal is perpendicular to this plane) with its center a distance equal to the reciprocal of the curvature (called the radius of curvature) along the normal vector.By editing the last component of the default example curve in the Tutor to become just t, you get the helix as well. Here we preload it instead: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calculus with functions of 2 or 3 variablesgraphing functions of 2 variables, contour plotsThe arrow definition of a MAPLE function of two variables: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Now we need the plot3d command: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Although you can click on the graph and use the toolbar to put in boxed axes, the surface just hangs in space with no reference point and no tickmarks for location or scale information, so it is usually a good idea to put the boxed axes into the plot command. Try the contour icon, rotate to see the contour plot from above, use the 1-1 icon to make axis units equal if relevant. The patchcontour style shows both the colored surface and the level curves.The contourplot command from the plots package gives the contour plot: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graphing level surfaces of functions of 3 variablesDefine a MAPLE function: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To plot the level surfaces of such functions in space, you need the plots package for the implicitplot3d command to implicitly plot each of the level surfaces that you specify (1 or more at a time):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Here is a single level surface, which is a sphere of radius 2: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Unfortunately for this case the level surfaces are contained within each other so one only sees the outer one unless one choses a window showing only half the surfaces to see inside, and since in this example they are spheres we need the 1-1 option to see them as spheres; you have to rotate this with your mouse to see inside: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transparency trickSometimes we really need to see inside a surface. Here is a trick to do so: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MAPLE functions versus expressions in partial differentiationMAPLE expressionsThis is just an expression; we cannot use function evaluation notation with it. Its derivative is also a MAPLE expression and we must substitute values into these expressions in order to evaluate them: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MAPLE functionsIf we make a MAPLE function then we can evaluate its values and its partial derivatives using usual function evaluation notation. The numbers in square brackets after the derivative operator D indicate which variable is doing the derivative (1 means the first one: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and 2 means the second one: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=):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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiZGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0Y4LyUqbGluZWJyZWFrR1EobmV3bGluZUYnRisvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGQUYrRkE=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: subscript notation in generalSubscripts on variables are indicated by square brackets in MAPLE 1d input so this is consistent with the calculus hand notation except the variable name is replaced by the number which indicates which of the variables it is (since MAPLE does not care what letters we use for the names of our variables, we need a notation that does not rely on them). We can directly use the 2D subscript notation for partial differentiation: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MAPLE functions versus expressions in partial differentiation:
higher derivativesMAPLE expressionsThis is just an expression; we cannot use function evaluation notation with it. Its derivative is also a MAPLE expression and we must substitute values into these expressions in order to evaluate them: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 get the contracted partial derivative notation you must copy the contracted output and paste it back into the input regionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KkYrLUkmbWZyYWNHRiQ2KC1JI21vR0YkNi1RKyZQYXJ0aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GIzYpRitGNC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4yZW1GJy8lJmRlcHRoR0ZWLyUqbGluZWJyZWFrR1ElYXV0b0YnRistRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJ0YrRjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZvLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjgvRjxGW3AvRj9GW3AvRkFGW3AvRkNGW3AvRkVGW3AvRkdGW3AvRklGW3BGSkZNLUZSNiZGVC9GWFEmMC40ZW1GJ0ZaRmZuRistRiM2KkYrLUYyNihGNC1GIzYpRitGNEZRRistRiw2JVEieEYnRlxvRl9vRitGOEZhb0Zkb0Znb0Zpb0ZccEZmcEYrLUYjNipGK0ZccUZccEZmcEYrLUYjNiYtRiw2JVEiR0YnRlxvRl9vRlxwLUkobWZlbmNlZEdGJDYkLUYjNiZGYHEtRjU2LVEiLEYnRjhGX3AvRj9GXm9GYXBGYnBGY3BGZHBGZXBGSi9GTlEsMC4zMzMzMzMzZW1GJ0ZpbkY4RjhGOEYrRjhGK0Y4RitGOEYrRjg=MAPLE functionsIf we make a MAPLE function then we can evaluate its values and its partial derivatives using usual function evaluation notation. The numbers in square brackets after the derivative operator D indicate which variable is doing the derivative (1 means the first one: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and 2 means the second one: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=):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 the subscripted multiple partial derivative notation will work: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ASIDE: both notations re-order derivative listingMaple D[,,,] automatically orders the subscripts in ascending order in contrast with the explicit partial derivative notation which retains the same order in which the derivatives are arranged (note the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= subscripts are only for show and do not work!)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viewing tangent planes and normal lines to a surfacewhen the surface is a graphFor the upper hemisphere 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 [or 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], using implicit differentiation one finds 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNihGKy1JJm1mcmFjR0YkNigtRiM2JC1JI21vR0YkNi1RKyZQYXJ0aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUEY8LUYjNiZGKy1GIzYlRjgtRiw2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y9USdpdGFsaWNGJ0Y8RitGPC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXm8vJSliZXZlbGxlZEdGQS1GOTYtUTEmSW52aXNpYmxlVGltZXM7RidGPEY/RkJGREZGRkhGSkZMRk5GUS1GIzYnLUYsNiVRInpGJ0ZaRmduLUY5Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjxGP0ZCRkRGRkZIRkpGTEZORlEtRiM2Ji1GOTYtUSIoRidGPC9GQEZmbkZCL0ZFRmZuRkZGSEZKRkwvRk9RLDAuMTY2NjY2N2VtRicvRlJGZnAtRiM2Ji1GLDYlUSJ4RidGWkZnbi1GOTYtUSIsRidGPEY/L0ZDRmZuRkRGRkZIRkpGTEZOL0ZSUSwwLjMzMzMzMzNlbUYnRldGPC1GOTYtUSIpRidGPEZjcEZCRmRwRkZGSEZKRkxGZXAvRlJRLDAuMTExMTExMWVtRidGPEYrRjxGK0Y8LUY5Ni1RIj1GJ0Y8Rj9GQkZERkZGSEZKRkwvRk9RLDAuMjc3Nzc3OGVtRicvRlJGXHItRiM2Ji1GOTYtUSomdW1pbnVzMDtGJ0Y8Rj9GQkZERkZGSEZKRkxGTkZRLUY0NigtRiM2JEZXRjwtRiM2JEZob0Y8RmluRlxvRl9vRmFvRitGPEYrRjxGK0Y8 so at the point (1,1,1), the linear approximation gives the tangent plane 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrRisvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnfrom which we can read off a normalLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiYtRiM2Jy1JI21uR0YkNiRRIjFGJ0ZALUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVRLDAuMzMzMzMzM2VtRictRiM2JkZlbkZpbkZlbkZARitGQEZALyUlb3BlbkdRMyZMZWZ0QW5nbGVCcmFja2V0O0YnLyUmY2xvc2VHUTQmUmlnaHRBbmdsZUJyYWNrZXQ7RidGK0ZARitGQEYrRkA=If we only plot the upper hemisphere and this tangent plane over the first quadrant:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JJm1zcXJ0R0YkNiMtRiM2KS1JI21uR0YkNiRRIjNGJ0ZALUY9Ni1RKCZtaW51cztGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGXW8tRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJ0Y2RjktRmZuNiRRIjJGJ0ZALyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkBGaW4tRiM2JkYrLUZibzYlLUYsNiVRInlGJ0Y2RjlGZ29Gam9GK0ZARitGQEYrRkBGK0ZARitGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2LEYrLUZXNiYtRiM2KEYrLUkmbXNxcnRHRiQ2Iy1GIzYpLUkjbW5HRiQ2JFEiM0YnRkAtRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZmby1GIzYmRistSSVtc3VwR0YkNiUtRiw2JVEieEYnRjZGOS1GX282JFEiMkYnRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGQEZiby1GIzYmRistRltwNiUtRiw2JVEieUYnRjZGOUZgcEZjcEYrRkBGK0ZALUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJy1GIzYoRl5vRmJvRl1wRmJvRmpwRkBGK0ZARkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGXXEtRiM2J0ZdcC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmFyLUYjNictRl9vNiRGZXBGQC1GPTYtUSMuLkYnRkBGQkZFRkdGSUZLRk1GT0Zlb0ZULUZqbjYjRl5vRitGQEYrRkBGXXEtRiM2J0ZqcEZdckZjckYrRkBGXXEtRiM2Ji1GLDYlUSVheGVzRidGNkY5Rl1yLUYsNiVRJmJveGVkRidGNkY5RkBGK0ZARkBGQEYrRkAtRj02LVEiO0YnRkBGQkZgcUZHRklGS0ZNRk9GUUZickZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjcDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==We have to rotate this and put in boxed axes to see where these surfaces are and that they are indeed tangent. Next we add in the normal line: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjcDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==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 the plot with the mouse rotates it and shows that indeed the normal line passes through the point of tangency and is perpendicular to the tangent plane there.when the surface is a level surfaceFor the sphere as a level surface LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNipGKy1GIzYmRistSSVtc3VwR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GP1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGRS1JI21vR0YkNi1RIitGJ0ZFLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZQLyUpc3RyZXRjaHlHRlAvJSpzeW1tZXRyaWNHRlAvJShsYXJnZW9wR0ZQLyUubW92YWJsZWxpbWl0c0dGUC8lJ2FjY2VudEdGUC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmluLUYjNiZGKy1GNjYlLUYsNiVRInlGJ0Y7Rj5GQUZHRitGRUZKLUYjNiZGKy1GNjYlLUYsNiVRInpGJ0Y7Rj5GQUZHRitGRUYrRkUtRks2LVEiPUYnRkVGTkZRRlNGVUZXRllGZW4vRmhuUSwwLjI3Nzc3NzhlbUYnL0Zbb0ZecC1GQjYkUSIzRidGRUZFRitGRQ==, one finds the gradient vector is 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 so at the point (1,1,1) one finds the gradient <LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRitGMkYrLUYzNi1RIj5GJ0YvRjYvRjpGOEY8Rj5GQEZCRkQvRkdRLDAuMjc3Nzc3OGVtRicvRkpGUUYv = 2 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNigtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMEY3RjBGNEY0LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0Y0, so we can use the 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to write the equation of the tangent plane, which passes through the point (1,1,1):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This derails due to one of the packages loaded and must be corrected in a future release of Maple. By hand we simplify this correctly to but still the output is wrong!LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiZGKy1GIzYoLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIrRicvRjxRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlUtRiw2JVEieUYnRjhGO0Y+LUYsNiVRInpGJ0Y4RjtGQkYrRkJGK0ZCRitGQg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiUtSSNtb0dGJDYtUSI9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkstSSNtbkdGJDYkUSIzRidGN0Y3RitGN0YrRjc=If we only plot the upper hemisphere and this tangent plane over the first quadrant: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjcDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjcDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==Now we display this with its tangent plane:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRI3AxRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSNwMkYnRi9GMkZBRkFGQQ==We have to rotate this and put in boxed axes to see where these surfaces are and that they are indeed tangent. Next we add in the normal line: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjcDNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==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 the plot shows that indeed the normal line passes through the point of tangency and is perpendicular to the tangent plane there.calculating the linear approximation (ZOOM) and
adding in the cross-sections of the surface and the tangent 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Equation of tangent plane at a point: starting value plus "rise = slope times run" increments from both coordinates, defining the linear approximation function: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVElcGZMZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiw2JVEiJUYnRjZGOUZALUY9Ni1RIjpGJ0ZARkJGRUZHRklGS0ZNRk9GUUZURkBGK0ZAZoom in to see both the graph and its tangent plane almost indistinguishable (rotate around to make sure, looking at the edges):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integration (functions of two variables, over rectangles, VecCalc muint command)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiktRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2KEYrLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEieEYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0Y2RjlGQEZALUY9Ni1RJyZyYXJyO0YnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtRiM2KEYrLUYjNidGKy1JJW1zdXBHRiQ2JUZnbi1JI21uR0YkNiRRIjJGJ0ZALyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGQEZCRkVGR0ZJRktGTUZPRl5vL0ZVRl9vRmJvRkAtRj02LVEiK0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZecS1GIzYmRistRl1wNiVGYm9GX3BGY3BGK0ZARitGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGX28vJSZkZXB0aEdGXHIvJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGK0ZARitGQEYrLUYjNiZGKy1GIzYmRistRiM2JkYzLUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GT0Zeb0ZpcEZYRkBGK0ZARitGQEYrRkA=Partial integration: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Iterated integral over a rectangular region:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNitGKy1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYkUSIwRidGOi1GUjYkUSIxRidGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIyRicvJS9zdWJzY3JpcHRzaGlmdEdGVEYrLUYjNitGKy1GNDYnRjZGVS1GUjYkRlpGOkZYRmVuRistRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjtRJ2l0YWxpY0YnLUY3Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjpGPUZARkJGREZGRkhGSi9GTVEmMC4wZW1GJy9GUEZbcC1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0Zib0Zlby1GNzYtUSIsRidGOkY9L0ZBRmRvRkJGREZGRkhGSkZqby9GUFEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGYm9GZW9GOkY6RjpGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0ZjcS8lKmxpbmVicmVha0dRJWF1dG9GJy1GNzYtUTAmRGlmZmVyZW50aWFsRDtGJ0Y6Rj1GQEZCRkRGRkZIRkpGam9GXHBGW3FGOkYrRl5xRlxyRmJwRjpGK0Y6RitGOg==step by step integration details It is easy to check each step of a multiple integration as 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=visualizing the 3-d solid corresponding to a double integralConsider a double integral, which can be interpreted as the signed volume between the graph of the integrand and the x-y plane LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNitGKy1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEiMUYnRjotRlQ2JFEiMkYnRjovJTFzdXBlcnNjcmlwdHNoaWZ0R0ZZLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1GIzYrRistRjQ2J0Y2LUZUNiRGaG5GOi1GLDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGP0Y6RlpGZm5GKy1GIzYnLUYsNiVRInlGJy9GY29GQi9GO1EnaXRhbGljRictRjc2LVExJkludmlzaWJsZVRpbWVzO0YnRjpGPS9GQUY/RkNGRUZHRklGS0ZNL0ZRRk8tRiM2Ji1GLDYlUSRzaW5GJ0Zib0Y6LUY3Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjpGPUZfcEZDRkVGR0ZJRktGTUZgcC1JKG1mZW5jZWRHRiQ2JC1GIzYmRistRiM2Ji1GLDYlUSJ4RidGaW9Gam9GXHBGZm9GOkYrRjpGOkY6RitGOkYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmhxLyUqbGluZWJyZWFrR1ElYXV0b0YnLUY3Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjpGPUZfcEZDRkVGR0ZJRktGTUZgcEZmb0Y6RitGY3FGYXJGYHFGOkYrRjpGK0Y6By boxing the plot of the function over the rectangle of integration, and plotting it together with the zero function, one gets an idea of its 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 that the contribution to the integral from the part above the x-y plane must be less than about 2*3*4 = 24 since these are the dimensions of the box which contains this part of the graph. In this case, plotting the absolute value we see 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so that the part above the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= plane and the part below exactly cancel and each is some small fraction of the total box volume, namely half the integral of the absolute value. However, the absolute value ("abs" in Maple) derails using antiderivatives to evaluate the integral and Maple fails, as does the 2d input with the absolute value signs. One would have to use variable limits of integration and integrate separately over the region where the function is positive and negative to see the individual contributions.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, it can numerically evaluate this:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSIlRidGL0YyL0YzUSdub3JtYWxGJ0Y9Rj0=multiple integration with palettes and step-by-step checkingMultiple integration is just successive partial integration, a calc 2 operation. If one finds an antiderivative at each integration step and then evaluates it between the upper and lower limits and then proceeds to the next partial integration step, one can check the hand work required to perform the successive integrations. The palette insertion of integrals is the most efficient. Here is an example:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZwbHVzO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInlGJ0YvRjJGOQ==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For a triple integral, one just continues this one more step. The Maple command MultiInt does this step by step process as an option.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 (functions of two variables, between function graphs)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Here we put LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= first and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= second so that the usual orientation of the axes appears in the plot: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The integral represents the signed volume of the region between the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= plane and the graph (positive above, negative below). In this case the function is nonnegative over the region of integration (rotate the graph to check) so it is the physical volume of the region between the graph and the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= plane.re-iterating a double integralEven Maple cannot yet understand how simple this result is: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But in other order of integration after re-iteration by hand: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRIiVGJ0Y2RjlGQEZARkBGK0ZARitGQA==integration (functions of three variables, over rectangular boxes)A triple integral with constant limits over a rectangular region: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step by step integration detailsIf you have the VecCalc package described above in the double integration section, you can use the MUltiple 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Since 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Funny that this did not come up in the Cartesian integral. Probably this is because in the outer integral there, this parameter could be negative and the final result would be negative.spherical 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visualizing the 3-d solid corresponding to a triple integralThe region of integration in space for a triple integral can be visualized by plotting the graphs of the innermost limits of integration as functions of the remaining outer variables of integration. This will not show the sides of the region, only the top and bottom. More sophistication is needed to get the sides shown as well, but since they should be clear from the context, this is not a big problem.
For cylindrical and spherical coordinates we have to parametrize the position vector 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 of the surfaces represented by the inner limits of integration using those coordinates.Cartesian coordsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNitGKy1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tRiM2JS1GNzYtUSomdW1pbnVzMDtGJ0Y6Rj1GQEZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBRLDAuMTExMTExMWVtRictRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y7USdpdGFsaWNGJ0Y6RlovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1GIzYrRistRjQ2J0Y2LUYjNiZGUy1JJm1zcXJ0R0YkNiMtRiM2KEYrLUYjNiZGKy1JJW1zdXBHRiQ2JUZaLUkjbW5HRiQ2JEZeb0Y6L0Zdb0Zhb0YrRjotRjc2LVEoJm1pbnVzO0YnRjpGPUZARkJGREZGRkhGSi9GTVEsMC4yMjIyMjIyZW1GJy9GUEZqcC1GIzYmRistRmBwNiUtRiw2JVEieEYnRmduRmpuRmJwRmVwRitGOkYrRjpGK0Y6RmhvRlxvRl9vRistRiM2KUYrLUY0NidGNi1GIzYmRlMtRmlvNiMtRiM2KkYrRl1wRmZwRlxxRmZwLUYjNiZGKy1GYHA2JS1GLDYlUSJ5RidGZ25Gam5GYnBGZXBGK0Y6RitGOkYrRjpGaXFGXG9GX28tRmNwNiRRIjFGJ0Y6LUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRlxzLyUqbGluZWJyZWFrR1ElYXV0b0YnLUY3Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjpGPUZARkJGREZGRkhGSkZWL0ZQRlctRiw2JVEiekYnRmduRmpuRjpGK0ZnckZlc0ZhckY6RitGZ3JGZXNGYHFGOkYrRjpGK0Y6LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIxRidGQEZARitGQC1GPTYtUSI7RidGQEZCL0ZGRjhGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnRlRGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2Ny1JJm1zcXJ0R0YkNiMtRiM2KkYrLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSJhRidGNkY5LUkjbW5HRiQ2JFEiMkYnRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGQC1GPTYtUSgmbWludXM7RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRl1wLUYjNiZGKy1GXW82JS1GLDYlUSJ4RidGNkY5RmJvRmZvRitGQEZpby1GIzYmRistRl1vNiUtRiw2JVEieUYnRjZGOUZib0Zmb0YrRkBGK0ZALUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJ0ZqcC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmdxLUY9Ni1RKiZ1bWludXMwO0YnRkBGQkZFRkdGSUZLRk1GT0ZccEZecC1GZm42Iy1GIzYoRitGam5GaW9GX3BGK0ZALUY9Ni1RIy4uRidGQEZCRkVGR0ZJRktGTUZPRlxwRlRGXHJGXXEtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGUy8lJmRlcHRoR0Zoci8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0ZjcEZjcUZpcUZfb0ZgckZfb0ZdcS1GLDYlUShzY2FsaW5nRidGNkY5RmNxLUYsNiVRLGNvbnN0cmFpbmVkRidGNkY5RkBGQEZARitGQC1GPTYtUSI7RidGQEZCRmBxRkdGSUZLRk1GT0ZRRmhxRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Ji1GPTYtUSInRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjExMTExMTFlbUYnL0ZVUSYwLjBlbUYnRjNGWEZARitGQEYrRkBGK0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Somehow the following grid refinement for the whole sphere removed these glitches.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIxRidGQEZARitGQC1GPTYtUSI7RidGQEZCL0ZGRjhGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnRlRGQA==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 sphere is a bit tricky since the vertical tangent planes at the equator cause gaps in the surface in these coordinates (rotate the first plot to see this) as well as at the east pole from this perspective. Another example is instructive, this time a tetrahedron. We plot the top and bottom of this solid: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The side walls of the region of integration are vertical surfaces from the top edges to the bottom edges.cylindrical 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIxRidGQEZARitGQC1GPTYtUSI7RidGQEZCL0ZGRjhGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnRlRGQA==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This does a much better job, but does not show the axis labels without adding the labels option.The cylindrical coordinate option using a parametrized surface approach is much simpler:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIxRidGQEZARitGQC1GPTYtUSI7RidGQEZCL0ZGRjhGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnRlRGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2QS1GVzYmLUYjNihGKy1GVzYmLUYjNiktRiw2JVEickYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictRiw2JVEnJiM5NTI7RicvRjdGREZARmBvLUYjNiYtRj02LVEqJnVtaW51czA7RidGQEZCRkVGR0ZJRktGTUZPRlEvRlVRLDAuMTExMTExMWVtRictSSZtc3FydEdGJDYjLUYjNihGKy1GIzYmRistSSVtc3VwR0YkNiUtRiw2JVEiYUYnRjZGOS1JI21uR0YkNiRRIjJGJ0ZALyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkAtRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZpcS1GIzYmRistRmlwNiVGXW9GXnFGYnFGK0ZARitGQEYrRkBGK0ZARkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGYG8tRlc2Ji1GIzYpRl1vRmBvRmZvRmBvRmFwRitGQEZARl9yRmJyRitGQEZAL0ZgclEifGZyRicvRmNyUSJ8aHJGJ0Zgb0Zdby1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmFzLUZfcTYkRmRxRkAtRj02LVEjLi5GJ0ZARkJGRUZHRklGS0ZNRk9GaHFGVEZbcUZgb0Zmb0Zdc0Zjc0Zlcy1GIzYmRl5xLUY9Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYsNiVRJyYjOTYwO0YnRmlvRkBGQEZgby1GLDYlUSdjb29yZHNGJ0Y2RjlGXXMtRiw2JVEsY3lsaW5kcmljYWxGJ0Y2RjlGYG8tRiw2JVElYXhlc0YnRjZGOUZdcy1GLDYlUSZib3hlZEYnRjZGOUZgby1GLDYlUShzY2FsaW5nRidGNkY5Rl1zLUYsNiVRLGNvbnN0cmFpbmVkRidGNkY5RmBvLUYsNiVRJ2xhYmVsc0YnRjZGOUZdcy1GVzYmLUYjNipGKy1JI21zR0YkNiNRInhGJ0Zgby1GanU2I1EieUYnRmBvLUZqdTYjUSJ6RidGK0ZARkBGX3JGYnJGK0ZARkBGQEYrRkAtRj02LVEiO0YnRkBGQkZjb0ZHRklGS0ZNRk9GUUZic0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Ji1GPTYtUSInRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjExMTExMTFlbUYnL0ZVUSYwLjBlbUYnRjNGWEZARitGQEYrRkAtRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlRGQA==changing the default dependent variable in cylindrical coordinatesWhen you plot a function with the cylindrical coordinate option, it assumes you are plotting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn versus a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y2LyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIixGJ0Y3RjkvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJ6RicvRjVGREY3L0Y6USdpdGFsaWNGJ0Y3RjlGN0Y5RjdGOQ==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To plot more naturally LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn versus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUScmIzk1MjtGJy9GNUY5RjdGQUY3RkFGN0ZBRjdGQQ==, the plot3d,coords help page gives the suggestion (strangely they plot the first coordinate versus the remaining ones!):
Define a new cylindrical system so 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 instead of 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: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 is a parabola of revolution: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spherical 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIxRidGQEZARitGQC1GPTYtUSI7RidGQEZCL0ZGRjhGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnRlRGQA==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A more interesting example would have a nontrivial radial function. We move the sphere up the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis so that it is tangent at the origin, add an inner sphere, and show only a half view: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Keep in mind that the region of integration is always from the inner radius surface to the outer radius surface.The spherical coordinate option is much simpler: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integration in polar coordinatesWe have to remember to include the area factor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEkaW50Ric=. Here is a region of integration: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and if we integrate the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JVEiMkYnRjgvRjxRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjhGQg== over it we get: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Aside
This is the volume under the graph of the function 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 in cylindrical coordinates, or visualizing the floor and ceiling of this solid: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integration in spherical coordinatesWe have to remember to include the volume factor 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:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ki1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RKyZJbnRlZ3JhbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTS1JI21uR0YkNiRRIjBGJ0Y4LUYjNiYtRlE2JFEiMkYnRjgtRjU2LVEnJnNkb3Q7RidGOEY7L0Y/RkIvRkFGPUZDL0ZGRj1GR0ZJRksvRk9RLDAuMzMzMzMzM2VtRictRiw2JVElJnBpO0YnLyUnaXRhbGljR0Y9RjhGOC8lMXN1cGVyc2NyaXB0c2hpZnRHRlMvJS9zdWJzY3JpcHRzaGlmdEdGU0YrLUYjNjAtRjI2J0Y0RlAtRiM2JkYrLUkmbWZyYWNHRiQ2KEZbby1GUTYkUSI0RidGOC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZXAvJSliZXZlbGxlZEdGPUYrRjhGYG9GYm8tRjI2J0Y0RlAtRiM2KEYrLUY1Ni1RJGNvc0YnRjhGO0ZmbkZnbkZDRmhuRkdGSUZLRmluLUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjhGO0Y+RmduRkNGaG5GR0ZJRktGTi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJiZwaGk7RidGXm9GOC8lK2V4ZWN1dGFibGVHRj1GOEY4RitGOEZgb0Ziby1GUTYkRmJwRjhGWS1JJW1zdXBHRiQ2JS1GLDYlUSYmcmhvO0YnRl5vRjhGVkZgby1GNTYtUSJ+RidGOEY7Rj5GZ25GQ0ZobkZHRklGS0ZOLUYsNiVRJHNpbkYnRl5vRjhGZHEtRjU2LVEwJkRpZmZlcmVudGlhbEQ7RidGOEY7Rj5GZ25GQ0ZobkZHRklGS0ZORmNyLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjVlbUYnLyUmZGVwdGhHRmRzLyUqbGluZWJyZWFrR1ElYXV0b0YnRlxzRmlxRjhGK0Zfc0Zccy1GLDYlUSgmdGhldGE7RidGXm9GOEY4RitGOA==This represents the volume of the following snow cone shaped solid. From ?coords
Maple spherical coordinates (u,v,w)=(\317\201,\316\270,\317\225) in Stewart Calculus:
x = u*cos(v)*sin(w)
y = u*sin(v)*sin(w)
z = u*cos(w)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 functions of two or three 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For 2-D vector field plots, it is probably best to reduce the default number of gridpoints from 20 to say 8 to avoid overcluttering the figure: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For 3-D vector field plots, it is probably helpful to reduce the default number of gridpoints from 8 to maybe 5 to avoid overcluttering the figure: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 this one you have to interact with it by rotating it around to get a feel for the more complicated 3-d information in the sampled vectors.gradient fieldplots and contour plotsThe gradient of a function is the vector whose components are the corresponding partial derivatives of that function. We can visualize them: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjcDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjcDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRIiVGJ0Y2RjlGQC1GPTYtUSI6RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZARitGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiktRiw2JVEjcDFGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRI3AyRidGNkY5RmhuLUYjNiYtRiw2JVEoc2NhbGluZ0YnRjZGOS1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmpvLUYsNiVRLGNvbnN0cmFpbmVkRidGNkY5RkBGK0ZARkBGQEYrRkBGK0ZAgrad, div, 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2d example (pretends to be in the x-y plane in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JVEiM0YnRjgvRjxRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjhGQg==)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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiw2JVEpR3JhZGllbnRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJmRidGMkY1L0Y2USdub3JtYWxGJ0ZARitGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUStEaXZlcmdlbmNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJGRidGNEY3Rj5GPkY+RitGPg==same 2d example in 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiw2JVEpR3JhZGllbnRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJmRidGMkY1L0Y2USdub3JtYWxGJ0ZARitGQA==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real 3d 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiw2JVEpR3JhZGllbnRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJmRidGMkY1L0Y2USdub3JtYWxGJ0ZARitGQA==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line integralsLine integrals evaluate the work done by a force in moving a point mass along a curve, for example.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRKFN0dWRlbnRGJ0YvRjItRjY2Ji1GIzYkLUYsNiVRL1ZlY3RvckNhbGN1bHVzRidGL0YyL0YzUSdub3JtYWxGJ0ZELyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUYsNiNRIUYnRkRGREZMRkQ=2d 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 exampleLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUS9TZXRDb29yZGluYXRlc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmRistSShtYWN0aW9uR0YkNiQtRiM2JkYrLUknbXN0eWxlR0YkNiotRiw2KFEqY2FydGVzaWFuRidGNC8lK2ZvcmVncm91bmRHUStbOTUsMCwyMTddRicvJSxwbGFjZWhvbGRlckdGNi8lMGZvbnRfc3R5bGVfbmFtZUdROE1hcGxlfklucHV0flBsYWNlaG9sZGVyRidGNy1GOzYpLUYjNistRiw2KFEieEYnRjRGSkZNRk9GNy1JI21vR0YkNjBRIixGJ0ZKRk1GTy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRltvLyUpc3RyZXRjaHlHRltvLyUqc3ltbWV0cmljR0Zbby8lKGxhcmdlb3BHRltvLyUubW92YWJsZWxpbWl0c0dGW28vJSdhY2NlbnRHRltvLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGam8tRiw2KFEieUYnRjRGSkZNRk9GNy1GWjYwRmZuRkpGTUZPRmduRmluL0Zdb0Y2Rl5vRmBvRmJvRmRvRmZvL0Zpb1EmMC4wZW1GJy9GXHBRLDAuMzMzMzMzM2VtRictRiw2KFEiekYnRjRGSkZNRk9GN0ZKRk1GT0ZnbkZKRk1GT0Znbi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0YrRkovJStleGVjdXRhYmxlR0Zbb0ZNRk9GZ25GK0Znbi8lK2FjdGlvbnR5cGVHUTttYXBsZXNvZnQ6dGFzay1wbGFjZWhvbGRlckYnRitGZ25GZ24tRlo2LVEiOkYnRmduRmluRlxvRl5vRmBvRmJvRmRvRmZvRmhvRltwRmduRitGZ24=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoTGluZUludEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRLFZlY3RvckZpZWxkRidGL0YyLUY2NiQtRiM2JC1GNjYmLUYjNigtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGUC8lKnN5bW1ldHJpY0dGUC8lKGxhcmdlb3BHRlAvJS5tb3ZhYmxlbGltaXRzR0ZQLyUnYWNjZW50R0ZQLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkZILUYsNiVRInpGJ0YvRjJGTEZMLyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0ZMRkxGSC1GLDYlUSVQYXRoRidGL0YyLUY2NiQtRiM2Ki1GNjYmLUYjNigtRiw2JVEidEYnRi9GMkZILUklbXN1cEdGJDYlRmRwLUYjNiUtSSNtbkdGJDYkUSIyRidGTC1GLDYjUSFGJ0ZMLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZILUZocDYlRmRwLUZdcTYkUSIzRidGTEZjcUZMRkxGY29GZm9GSEZkcC1GSTYtUSI9RidGTEZOL0ZSRlBGU0ZVRldGWUZlbi9GaG5RLDAuMjc3Nzc3OGVtRicvRltvRmByLUZdcTYkRmVxRkwtRkk2LVEjLi5GJ0ZMRk5GXnJGU0ZVRldGWUZlbi9GaG5RLDAuMjIyMjIyMmVtRicvRltvRmluLUZdcTYkUSIxRidGTEZMRkxGTEZMRkw=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEsZmllbGRwbG90M2RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ni1GNjYmLUYjNigtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkZBLUYsNiVRInpGJ0YvRjJGRUZFLyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0ZBRj4tRkI2LVEiPUYnRkVGRy9GS0ZJRkxGTkZQRlJGVC9GV1EsMC4yNzc3Nzc4ZW1GJy9GWkZnby1JI21uR0YkNiRRIjBGJ0ZFLUZCNi1RIy4uRidGRUZHRmVvRkxGTkZQRlJGVC9GV1EsMC4yMjIyMjIyZW1GJy9GWkZYLUZqbzYkUSIxRidGRUZBRmZuRmJvRmlvRl1wRmNwRkFGaW5GYm9GaW9GXXBGY3BGRUZFLUZCNi1RIjpGJ0ZFRkdGZW9GTEZORlBGUkZURmZvRmhvLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRlgvJSZkZXB0aEdGXnEvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRiw2JVErc3BhY2VjdXJ2ZUYnRi9GMi1GNjYkLUYjNjItRjY2Ji1GIzYoLUYsNiVRInRGJ0YvRjJGQS1JJW1zdXBHRiQ2JUZhci1GIzYlLUZqbzYkUSIyRidGRS1GLDYjUSFGJ0ZFLyUxc3VwZXJzY3JpcHRzaGlmdEdGXHBGQS1GZXI2JUZhci1Gam82JFEiM0YnRkVGX3NGRUZFRlxvRl9vRkFGYXJGYm9GaW9GXXBGY3BGQS1GLDYlUSp0aGlja25lc3NGJ0YvRjJGYm9GaXJGQS1GLDYlUSZjb2xvckYnRi9GMkZiby1GLDYlUSRyZWRGJ0YvRjJGRUZFRmZwRmlwLUYsNiVRKGRpc3BsYXlGJ0YvRjItRjY2JC1GIzYrLUZCNi1RIiVGJ0ZFRkdGSkZMRk5GUEZSRlRGVkZZRmZ0RkFGZnRGQS1GLDYlUSVheGVzRidGL0YyRmJvLUYsNiVRJmJveGVkRidGL0YyRkVGRUZF[in progress?]Does anyone get this far anyway?