Exam 16: Vector Calculus

A thin wire is bent into the shape of a semicircle If the linear density is find the exact mass of the wire.Select the correct Answer

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

Evaluate the surface integral. S is the part of the plane that lies in the first octant.Select the correct Answer

Select the correct Answer for each question. -Evaluate S is the part of the cone between the planes and

Below is given the plot of a vector field F in the xy-plane.(The z-component of F is 0.) By studying the plot, determine whether div F is positive, negative, or zero.Select the correct Answer

Determine whether F is conservative.If so, find a function such that Select the correct Answer

Find a function such that and use it to evaluate along the given curve C.

Calculate the work done by the force field when a particle moves under its influence around the edge of the part of the sphere that lies in the first octant, in a counterclockwise direction as viewed from above.

Select the correct Answer for each question. -Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. where C is the cardioid

Show that F is conservative and find a function f such that and use this result to evaluate where C is any path from Select the correct Answer

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.)

Evaluate the line integral.

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

Find the area of the part of paraboloid that lies inside the cylinder

Use Stokes' Theorem to evaluate S is the part of the paraboloid lying below the plane and oriented with normal pointing downward.

Evaluate the line integral over the given curve C.

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find , where a is the constant vector.

Select the correct Answer for each question. -Find the exact mass of a thin wire in the shape of the helix if the density is 5.

Below is given the plot of a vector field F in the xy-plane.(The z-component of F is 0.) By studying the plot, determine whether div F is positive, negative, or zero.

Evaluate that is, find the flux of F across S. S is the hemisphere n points upward.