Multivariate Calculus with Matrices, Math 2720 Maple Assignment 2 The purpose of this notebook is to revisit Maple and use it to think about the geometry of linear transformations and parametrization. You must choose a single parter with whom to complete this assignment. This notebook has a number of sections. The last section contains a quiz. To receive credit, you must write explainations for your answers. You must enter explanatory text linking the Maple commands. The purpose of this quiz is to have you rethink material we have covered which is relevant to our study of integration. You should be able to work the quiz by studying the examples in the preceeding sections, your first Maple assignment, and your calculus books. This is an open-communication quiz, meaning that you may ask anyone (including your instructor, but especially your partner) for help at any time.

First we need to load the linear algebra package. We also load the plots package. restart; with(LinearAlgebra): with(plots): Here is how we can name an arbitrary vector, enter a matrix, and multiply them together. X := Vector(2,symbol=x); A := Matrix([[3,1],[-4,2]]); A.X;

Recall that we can parametrize a curve or a surface. Maple can help us visualize the image of these parametrizations. We can plot a curve in two dimensions as follows. plot([sin(t), cos(t), t = 0 .. 3*Pi]); If we load VectorCalculus, we can also plot a curve in three dimensions. with(VectorCalculus): SpaceCurve(<cos(t),sin(t),t>,t=0..3*Pi); If we use plots and linalg, we can also parametrize surfaces. with(plots): with(linalg): x:=(u,v)->3*sin(u)*cos(v): x(u,v); y:=(u,v)->3*sin(u)*sin(v): y(u,v); z:=(u,v)->3*cos(u): z(u,v); plot3d([x(u,v),y(u,v),z(u,v)],u=0..2*Pi,v=0..Pi);

For each of the following, write Maple commands linked by text explanations of what you are doing. Write your answers in a new file and include the name of your single team-mate. (a) Consider the region bounded by the lines x=0, y=0, x=-4, and y=2. Plot this region and find its area. Then plot the image of the region under the transformation 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 calculate its area. (b) Consider the region bounded by the curves 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4=. Plot this region and find its area. Then plot the image of the region under the transformation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUY2NiYtRiM2Ji1GLDYjUSFGJy1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNictSSRtdGRHRiQ2KC1GIzYlLUkjbW5HRiQ2JFEiMkYnRj1GL0YyLyUpcm93YWxpZ25HRmduLyUsY29sdW1uYWxpZ25HRmduLyUrZ3JvdXBhbGlnbkdGZ24vJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGX3AtRl9vNigtRiM2JS1GZG82JFEiM0YnRj1GL0YyRmdvRmlvRltwRl1wRmBwRmdvRmlvRltwLUZcbzYnLUZfbzYoLUZkbzYkUSIwRidGPUZnb0Zpb0ZbcEZdcEZgcC1GX282KC1GZG82JEZfcEY9RmdvRmlvRltwRl1wRmBwRmdvRmlvRltwLyUmYWxpZ25HUSVheGlzRicvRmhvUSliYXNlbGluZUYnL0Zqb1EmcmlnaHRGJy9GXHBRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0YxLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGYXIvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGXHMvJSZmcmFtZUdGXHMvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZFLyUtZXF1YWxjb2x1bW5zR0ZFLyUtZGlzcGxheXN0eWxlR0ZFLyUlc2lkZUdGanEvJTBtaW5sYWJlbHNwYWNpbmdHRmlyRmVuRj1GPS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GQDYtUTEmSW52aXNpYmxlVGltZXM7RidGPUZDRkZGSEZKRkxGTkZQL0ZTUSYwLjBlbUYnL0ZWRmh0RjotRkA2LVEifkYnRj1GQ0ZGRkhGSkZMRk5GUEZndEZpdEY9and calculate its area. (c) What is the image of the parametrization with x(t)=(t^2+1)/(t^2-1) and y(t)=2t/(t^2+1)? (d) Parametrize the surface of an ellipsoid which is not a sphere, and use the parametrization to ask Maple to draw the ellipoid. (e) Ask Maple to plot one eighth of the surface of the sphere.