Exam 16: Vector Calculus

Use a computer algebra system to compute the flux of F across S. S is the surface of the cube cut from the first octant by the planes

A particle is moving in a velocity field At time t = 1 the particle is located at the point (1, 5, 5). a). What is the velocity of the particle at t = 1? b). What is the approximate location of the particle at t = 1.01?

Let

Evaluate the line integral over the given curve C. ; ,

Use Gauss's Law to find the charge contained in the solid hemisphere , if the electric field is

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ; u = ln 5, v = 0

Find the gradient vector field of the scalar function f (That is, find the conservative vector field F for the potential function f of F.)

Use Stokes' Theorem to evaluate S consists of the four sides of the pyramid with vertices (0, 0, 0), (3, 0, 0), (0, 0, 3), (3, 0,3) and (0, 3, 0) that lie to the right of the xz-plane, oriented in the direction of the positive y-axis.

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

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

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

Evaluate the line integral over the given curve C. ; ,

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

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

Evaluate the line integral over the given curve C. , where C is the line segment joining (-4, -5) to (5, 4)

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.

Find the exact value of where and C is given by

Use Stoke's theorem to evaluate C is the boundary of the part of the paraboloid in the first octant. C is oriented counterclockwise as viewed from above.

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

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