Multivariable Calculus (Math 214) Assignment

due November 29, 1999

1. Discussion

The substitution rule for multiple integrals states that if

is a bi-single-valued differentiable transformation, S a reasonable subset of R^{N}, T(S) the transform of S under T, and f a continuous function, then

where j_{T} is the Jacobian (determinant) of T.

2. Exercises

1. Use the substitution rule with the transformation

   x = u cos v , y = u sin v

   to evaluate the integral

   INT[INT[_{D} e^{- x^{2} - y^{2}} dx dy]]

   taken over the disk D of given radius a > 0 with center at the origin.

2. Use the substitution rule with a suitable transformation to compute the volume of the solid ellipsoid

   {x^{2}}/{a^{2}} + {y^{2}}/{b^{2}} + {z^{2}}/{c^{2}} <= 1

   for given positive constants a, b, and c.

3. Recalling that there is a formula for the curve integral of a gradient vector field, show that if f is a scalar-valued function which is differentiable in the ball x^{2} + y^{2} + z^{2} < a^{2}, and if g is the function defined in the ball whose value at the point (x, y, z) is the integral over the line segment from the origin to (x, y, z) of \nabla f, then g differs from f by a constant.

4. Noting that the textbook defines a scalar-valued function div F, called the divergence of F, for each vector field F, compute div F when F is each of the four vector fields in exercise 3 of the assignment due November 15.

5. Apply Green's Theorem to evaluate the path integral over the circle of radius a with center at the origin, when traversed once counter-clockwise, of the vector fields

   1. F(x, y) = (x - y, x + y).

   2. F(x, y) = (-y/rho, x/rho), where rho = SQRT{x^{2} + y^{2}}.