Exam 15: Topics in Vector Calculus

Use Stokes' Theorem to evaluate $\int$ _C (x + y)dx + (2x - 3)dy + (y + z)dz over the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) traversed in a counterclockwise manner.

Find if = x³ + 3y² + z

Let F(x, y, z) = 10x i + 10y j + 10z k and $\sigma$ be the portion of the surface z = 5 - x² - y² that lies above the xy-plane. Find the magnitude of the flux of the vector field across $\sigma$ .

Evaluate where F(x, y, z) = 4x i + 4y j + 4z k and $\sigma$ is that portion of the plane 2x + 3y + 4z = 12 which lies in the first octant and is oriented by upward unit normals.

Evaluate the surface integral where $\sigma$ is the surface enclosed by y = x², 0 $\le$ x $\le$ 2, and -1 $\le$ z $\le$ 2.

Use Stokes' Theorem to evaluate $\int$ _C 30y²dx + 30x²dy - 30(x + z)dz where C is a triangle in the xy-plane with vertices (0, 0, 0), (1, 0, 0), and (1, 1, 0) with a counterclockwise orientation looking down the positive z axis.

Let . Find cirlF.

Use Green's Theorem to evaluate $\int$ _C 11(3x² + y)dx + 44xy dy where C is the triangular region with vertices (0, 0), (2, 0) and (0, 4). Assume that the curve is traversed in a counterclockwise manner.

Let F(x, y, z) = 3y i. The flux outward between the planes z = 0 and z = 2 is

F (x, y, z) = 8x³ i + 16y² j + 24z² k. Find the outward flux of the vector field F across the unit cube in the first octant and including the origin as a vertex.

Find the surface area of the cone that lies between the planes z = 6 and z = 7.

Evaluate $\int$ _C 6y²dx - 6x²dy where C is the line segment from (0, 1) to (1, 0).

Find the outward flux of F(x, y, z) = 5x i + (y + 3)j + 8z² k across the unit cube in the first octant that has a vertex at the origin.

Find the outward flux of the vector field across the sphere .

Use Stokes' Theorem to evaluate where F(x, y, z) = 8y k and $\sigma$ is that portion of the ellipsoid 4x² + 4y² + z² = 4 for which z $\ge$ 0.

Use the divergence theorem to evaluate where , n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere by x² + y² + z² = 36.

Use the divergence theorem to evaluate where , n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere by x² + y² + z² = 9.

Let . The work done by the force field on a particle moving along an arbitrary smooth curve from P(1, 1) to Q(0, 0) is

Evaluate the surface integral where $\sigma$ is that portion of 3x + 3y + 3z = 3 which lies in the first octant.

Evaluate . Note: This integral is independent of path.