Stoke’s Theorem and Divergence Theorem Academic Essay

Stoke’s Theorem and Divergence Theorem

1. Verify that Stokes’ theorem is true for the given vector field F and surface S:
F(x, y, z) = h-y, x, -2i, S is the cone z
2 = x
2 + y
2
, 0 = z = 4, oriented downward.
2. (a) Use Stokes’ theorem to evaluate R
C F · dr. Assume C is oriented
counterclockwise as viewed from above. F(x, y, z) = h1, x + yz, xy –
v
zi and C
is the boundary of the part of the plane 3x + 2y + z = 1 in the first octant.
(b) Use Stokes’ theorem to evaluate RR
S
curl F · n dS.
F(x, y, z) = h2y cos z, ex
sin z, xey
i, and S is the hemisphere
x
2 + y
2 + z
2 = 9, z = 0, oriented upward.
3. (a) Verify that the divergence theorem is true for the given vector field F on the
region E:
F(x, y, z) = hx
2
, xy, zi, E is the solid bounded by the paraboloid
z = 4 – x
2 – y
2
, and the xy plane.
(b) Use the divergence theorem to calculate the surface integral RR
S F · n dS (i.e
calculate the flux of F across S) where F(x, y, z) = ||r||2
r, r = hx, y, zi and S is
the sphere with radius R and center the origin.
Suppose the density function of a solid object that occupies the region E is ?(x, y, z),
in units of mass per unit volume, at any given point (x, y, z), then its mass is
m =
Z Z Z
E
?(x, y, z) dV
4. Find the mass of the solid E where E is the cube given by 0 = x = a, 0 = y = a,
0 = z = a, and the density function is ?(x, y, z) = x
2 + y
z + z
2
.
5. Find the Fourier series of f on the given interval:
f(x) = (
x -p = x < 0
0 0 = x < p