Exam 13: Vector Calculus

Use Green's Theorem to find the work done by the force in moving a particle from the origin along the x-axis to (1,0)then along the line segment to (0,1)and then back to the origin along the y-axis.

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

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

Let where . Which of the following equations does the line segment from to satisfy?

A thin wire is bent into the shape of a semicircle If the linear density is 4 ,find the exact mass of the wire.

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. in the first octant, with orientation toward the origin.

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

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

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.

Match the equation with one of the graphs below.

A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C.Its moments of inertia about the axes are Find the moments of inertia about the axes,if C is a rectangle with vertices (0,0),(4,0), (4,5)and .

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

Find the curl of the vector field F.

The temperature at the point in a substance with conductivity is Find the rate of heat flow inward across the cylindrical

Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above.

Use Stokes' Theorem to evaluate where and is the triangle with vertices is oriented counterclockwise as viewed from above.

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

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?

Use Stokes' Theorem to evaluate is the part of the paraboloid that lies inside the cylinder oriented upword.