NAME: YOUR NAME GOES HEREMath 2130: Maple Homework #4Due: Monday (August 6th)Do not email me your answers. I want a paper copy.Please use "ctrl+delete" or "command+delete" to delete unused commands & command prompts. Clean your homework up before printing!
HINT: Maple has built-in commands for computing things like divergence, curl, and flux and can even calculate a potential function for a conservative vector field. To see the full list of commands Maple has within the 'VectorCalculus' package, simply replace the below colon (with(VectorCalculus)":") with a semi-colon (with(VectorCalculus)";") and Maple will output the commands this package loads. Some of these may come in handy.restart;
with(plots):
with(VectorCalculus):#1 Define the vector field 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 (The 'SetCoordinates' command tells Maple how to define the components of the vectors in the field. Here we are using rectangular coordinates.)SetCoordinates('cartesian'[x,y,z]):
F := VectorField(???);(a) Plot the vector field LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KFEiRkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y0LyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y5LUkjbW9HRiQ2LlEifkYnRjUvRjhRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUkY1RkA=in the box LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JVEiMEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEmJmxlcTtGJ0YvRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnRjUtRiw2JVEjMTBGJ0YvRjJGL0Yy and 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. Include the options 'arrows = SLIM' (case sensitive), 'color = red' and 'axes = boxed'.fieldplot3d(???);(b) Compute the divergence and curl of F.(c) Is F a conservative vector field? Why or why not? If so, find a potential funciton for F.(d) Consider the curve parameterized by 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JVEiMEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEmJmxlcTtGJ0YvRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnRjUtRiw2JVEjMTBGJ0YvRjJGL0Yy. Plot this curve together with the vector field F. Use the same plot options for the vector field as in part (a). For "spacecurve" use the plot options: 'thickness=3', 'color=blue', and 'numpoints=5000'.r := t -> <???>:
'r(t)' = r(t)';plotVF := fieldplot3d(???):
plotC := spacecurve(???):
display({plotC,plotVF});(e) Calculate the work done by F moving a particle along this helix directly using the parameterization. Find the exact answer and a decimal approximation. If the Fundamental Theorem of Line Integrals applies, use it and verify your answers match. (Hint: The command to plug LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= into F is 'evalVF( F, r(t) )'.) Recall that Work LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRS1GLDYuUSJ+RidGL0YyRjVGN0Y5RjtGPUY/RkEvRkRRJjAuMGVtRicvRkdGTC1JJW1zdWJHRiQ2JS1GLDYuUSsmSW50ZWdyYWw7RidGL0YyRjVGNy9GOlEldHJ1ZUYnRjsvRj5GVUY/RkFGS0ZNLUYjNiYtSSNtaUdGJDYmUSJDRicvJSdpdGFsaWNHRlVGLy9GM1EnaXRhbGljRidGZ25GL0Zpbi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUZaNihRIkZGJy8lJWJvbGRHRlUvRmhuRjFGLy9GM1ElYm9sZEYnLyUrZm9udHdlaWdodEdGZW8tRiw2LlEpJm1pZGRvdDtGJ0YvRjJGNUY3RjlGO0Y9Rj9GQS9GRFEsMC4xNjY2NjY3ZW1GJy9GR0ZccC1GWjYmUSJkRidGZ25GL0Zpbi1GWjYoUSJyRidGYW9GY29GL0Zkb0Zmby1GLDYwRkpGYW9GL0Zkb0Zmb0Y1RjdGOUY7Rj1GP0ZBRktGTS1GLDYwRi5GYW9GL0Zkb0Zmb0Y1RjdGOUY7Rj1GP0ZBRkNGRkZkcC1JKG1zdWJzdXBHRiQ2KEZRLUYjNiYtRlo2JlEiYUYnRmduRi9GaW5GZ25GL0Zpbi1GIzYmLUZaNiZRImJGJ0ZnbkYvRmluRmduRi9GaW4vJTFzdXBlcnNjcmlwdHNoaWZ0R0Zdb0Zbby9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRidGXm8tSShtZmVuY2VkR0YkNiUtRiM2JkZhcC1GW3I2JS1GIzYlLUZaNiZRInRGJ0ZnbkYvRmluRi9GMkYvRjJGL0YyRi9GMkZob0ZhcC1GLDYuUSInRidGL0YyRjVGN0Y5RjtGPUY/RkEvRkRRLDAuMTExMTExMWVtRidGTUZfckZILUZaNiZRI2R0RidGZ25GL0Zpbi1GLDYuUSIuRidGL0YyRjVGN0Y5RjtGPUY/RkFGS0ZNRkhGL0Yy