Exam 16: Vector Calculus

Find the value of the constant c such that the vector field is the curl of some vector field F.

Let S be the cube with vertices Approximate by using a Riemann sum as in Definition 1, taking the patches to be the squares that are the faces of the cube and the points to be the centers of the squares.Select the correct Answer

Select the correct Answer for each question. -Below is given the plot of a vector field F in the xy-plane. By studying the plot, determine whether is positive, negative, or zero.

Find the area of the surface S where S is the part of the surface that lies inside the cylinder

Find the exact value of where C is the line segment from

Determine whether or not vector field is conservative.If it is conservative, find a function such that

Select the correct Answer for each question. -Find the mass of the surface S having the given mass density.S is part of the plane in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.

Determine whether F is conservative.If so, find a function such that

Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone if its density function is

Match the vector field with its plot.

Find the mass of the surface S having the given mass density.S is the hemisphere the density at a point P on S is equal to the distance between P and the xy-plane.

Find a parametric representation for the part of the elliptic paraboloid that lies in front of the plane x = 0.Select the correct Answer

Select the correct Answer for each question. -Use a computer algebra system to compute the flux of F s the surface of the cube cut from the first octant by the planes

Select the correct Answer for each question. -Evaluate where C is the right half of the circle

Find the work done by the force field F on a particle that moves along the curve C.

Use Stokes' Theorem to evaluate is the part of the ellipsoid lying above the xy-plane and oriented with normal pointing upward.

Find the work done by the force field on a particle that moves along the parabola

Use Stoke's theorem to evaluate and C is the boundary of the part of the plane in the first octant.

Find the area of the part of the surface that lies between the planes x = 0, x = 4, , and

Use Green's Theorem to find the work done by the force in moving a particle in the positive direction once around the triangle with vertices and