Exam 15: Topics in Vector Calculus

Use Stokes' Theorem to evaluate $\int$ _C 11z dx + 11x dy + 11y dz over the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1) traversed in a counterclockwise manner.

F(x, y, z) = (4xy - 9) i + 3xyz j + z² k. Find div F.

Evaluate the surface integral where $\sigma$ is that portion of the paraboloid z = x² + y² enclosed by 1 $\le$ z $\le$ 9.

Find where F(x, y) = 2(2xy² + 1)i + 4x²y j and C is the curve r(t) = e ^t sin t i + e ^t cos t j; .

Use Green's Theorem to evaluate , where C is the circle .

Use Green's Theorem to evaluate $\int$ _C 2(2xy - y²)dx + 2(x² - y²)dy where C is the boundary of the region enclosed by y = x and y = x². Assume that the curve c is traversed in a counterclockwise manner.

Use a line integral to find the area of the region in the first quadrant enclosed by y = 12x and y = 12x³.

Evaluate the surface integral where $\sigma$ is the surface enclosed by z = x³, 0 $\le$ x $\le$ 2, and 0 $\le$ y $\le$ $\pi$ .

Sketch the vector field, F(x, y) = 4i - 8j, by drawing some typical non-intersecting vectors. The vectors need not be drawn to the same scale as the coordinate axes, but they should be in the correct proportions relative to each other.

Find the work done by the conservative force as it acts on a particle moving from P(0, 0) to .

Evaluate where F(x, y, z) = 2 i - z j + y k and $\sigma$ is that portion of the paraboloid x = y² + z² between x = 0 and x = 4. The surface is oriented by forward unit normals.

Evaluate where C is the boundary of the region in the first quadrant, enclosed by the circle and the coordinate axes.

Use Stokes' Theorem to evaluate $\int$ _C 3(4x - 2y)dx - 3yz²dy - 3y²z dz where C is the circular region enclosed by x² + y² = 4, z = 2.

Use the divergence theorem to evaluate where F(x, y, z) = 2y²x i + 2yz² j + 2x²y² k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the sphere x² + y² + z² = 4.

Use the divergence theorem to evaluate where F(x, y, z) = (x³ + 3xy²)i + z³ k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere of radius a centered at the origin.

For F(x, y) = 3x² i - 3y j the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(3, 2) is

Use the divergence theorem to evaluate where F(x, y, z) = 4x² i + 4y² j + 4z² k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the cube enclosed by the planes 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1, and 0 $\le$ z $\le$ 1.

Find the outward flux of F(x, y, z) = 5(x - 1)i + 5(y - 3)j + 5z k across the rectangular box with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (4, 0, 0), (4, 1, 0), (4, 0, 1), and (4, 1, 1).

Use a line integral to find the area of the region enclosed by 3x² + 12y² = 12.

Determine whether is conservative. If it is, find a potential function for it.( K is an arbitrary constant.)