Exam 16: Vector Calculus

Find the curl of the vector field F.

Find a parametric representation for the part of the sphere that lies above the cone

Use Stoke's theorem to evaluate C is the curve of intersection of the plane z = x + 9 and the cylinder

Evaluate the line integral where and C is the arc of the circle traversed counterclockwise from ( , 0) to (0, ). Round your answer to two decimal places.

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

The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines. The flow lines of the vector field satisfy the differential equations and Solve these differential equations to find the equations of the family of flow lines.

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

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.

Let R be a plane region of area A bounded by a piecewise-smooth simple closed curve C. Using Green's Theorem, it can be shown that the centroid of R is , where Use these results to find the centroid of the given region. The triangle with vertices , , and .

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

Find the area of the surface S where S is the part of the sphere that lies to the right of the xz-plane and inside the cylinder

Find the exact value of where C is the line segment from (0, 0, 0) to (1, , ).

Suppose that F is an inverse square force field, that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin.

Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S.

Find the correct identity, if f is a scalar field, F and G are vector fields.

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

Evaluate . ; S is the part of the plane in the first octant.

Show that F is conservative and find a function f such that , and use this result to evaluate , where C is any path from to . ; and

Find the exact mass of a thin wire in the shape of the helix if the density is 5.

Find a parametric representation for the part of the plane that lies inside the cylinder