Math 225: Calculus III   Assignment 9

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Instructions

This assignment contains problems involving multiple integrals, see Chapter 14 in Finney & Thomas or Chapter 4 in the Lecture Notes. Before you begin, you may want to review the notebooks MultipleIntegrals, SphericalCoords and AdvancedCalculusDemo in the Notebook folder or the back of the Lecture Notes. There are also many examples of how to do these problems in the Quizzes folder. In Problem 1, Jacobian determinants can be computed using the function J[]:
?J
J[f,g,{u,v}] gives the Jacobian determinant of the coordinate
    transformation x = f(u,v), y = g(u,v). J[f,g,h,{u,v,w}]
   gives  the Jacobian determinant of the coordinate
   transformation  x = f(u,v,w), y = g(u,v,w), z = h(u,v,w).
To create plots in cylindrical coordinates in Problem 4, make a substitution like
{x,y,z} /. {x -> r Cos[theta], y -> r Sin[theta]}
and plug it into ParametricPlot3D[..,{r,a,b},{t,c,d}].
In Problem 5, use a similar procedure for spherical coordinates:
{x,y,z} /.
    {x->rho Cos[theta]Sin[phi],
     y->rho Sin[theta]Sin[phi],
     z->rho Cos[phi]}
and plug it into ParametricPlot3D[..,{theta,a,b},{phi,c,d}].

Be sure to type in comments explaining what you are doing.

Remember to uncheck Show In/Out Names in the Kernel menu and close this group before printing.

Initialization

<<Calculus`Math225`

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

Problem 1

Compute the Jacobian determinant of the following coordinate transformations. Note that you must first solve for x and y as functions of u and v (use
Solve[{u==..., v==...},{u,v}]), then apply D[] and Det[] in an appropriate way. Finally compute the Jacobian using J[].
a) u == x + y, v == x^2 - y^2
b) u == 3x - y + z, v == x + 2y + z, w == -x + y + z

Solution

Problem 2

Plot the following vector fields using VectorField[F,{x,-5,5},{y,-5,5}].
a) F = { 0.2 x, 0.1 y}
b) F = { 0.1 y, 0.2 x}
c) F = {-0.1 y, 0.2 x}

Solution

Problem 3

Let C be the curve defined by
    r[t_] = t(t^2-1) i + 2t j - t^2 k
for -1<= t <= 2.
Integrate the function f[x_,y_,z_] = z^2 - x y over C by using the appropriate formula with NIntegrate[]. Compare the result with LineIntegrate[]//N.

Solution

Problem 4

Use cylindrical coordinates and ParamtericPlot3D[] to plot the surfaces
a) z = theta, 0 <= theta <= 2Pi, -2 <= r <= 2
b) z = E^(-r^2/4)Cos[Pi r], 0 <= theta <= 2Pi,
    0 <= r <= 3

Solution

Problem 5

Use spherical coordinates and ParamtericPlot3D[] to plot the surface
rho = 1/(phi^2(phi-Pi)^2), Pi/4 <= phi <=3Pi/4,
0 <= theta <= 2Pi.

Solution