Math 225: Calculus III   Assignment 10

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Instructions

This assignment contains problems involving vector fields, line integrals, flow integrals, and Green's Theorem, see Chapter 15 in Finney & Thomas or Chapter 5 in the Lecture Notes. Before you begin, you may want to review the notebook AdvancedCalculusDemo in the Notebook folder or the back of the Lecture Notes. 

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Initialization

<<Calculus`Math225`

i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};

Problem 1

Compute Div[F], Curl[F], and Div[Curl[F]] for the following vector fields.
a) F = (x^2 y + z) i + (y^2 z + x) j + (z^2 x + y) k
b) F = Cos[x+y] i + Sin[x-y] j
c) F = y E^z i + z E^x j + x E^y k
d) "Prove" using Mathematica that for an arbitrary vector field,
    F = m[x,y,z]i + n[x,y,z]j + p[x,y,z]k
     we always get Div[Curl[F]] == 0.

Solution

Problem 2

Let F = (y-z) i + (x-z) j - (x+y) k. Compute the flow integral "F.dr" over each of the following curves by using the appropriate formula with Integrate[]. Compare your answer to FlowIntegrate[]:
a) r[t_] = (2t^3-t) i + (t^2+t)/2 j + t k, 0<= t <=1
b) r[t_] = (1-Cos[t])/2 i + Sin[t/2]j + t/Pi k, 0<= t <= Pi
c) Explain why the answers to a) and b) have to be the same, even though the curves are quite different.

Solution

Problem 3

a) Integrate
(y^3 - y + z)dx + (3x y^2 - x + 2y z)dy + (y^2 + x + 2z)dz
over the curve C defined by t^2 i + t j + (t^3-1)k, 0 <= t <= 1,
(using Integrate[], or FlowIntegrate[]).
b) Find a function f[x,y,z] such that
Grad[f[x,y,z]] == 
    (y^3 - y + z)i + (3x y^2 - x + 2y z)j + (y^2 + x + 2z)k
c) Use the Fundamental Theorem of Line Integrals to evaluate the the integral in a) using the result of b).

Solution

Problem 4

Let C be the boundary of the "triangular" region R in the first quadrant enclosed by the x-axis, the line x==1, and the curve x==y^3. Use Green's Theorem to transform the integral of
2x y^2dx + x^2 y dy over C into a double integral over R and then evaluate this double integral.

Solution

Problem 5

Produce a plot of a surface that you think has interesting features. The surface does not have to be the graph of a function and can be done in any coordinate system you wish. Explain what features of the surface are interesting and why.

Solution