Multivariate Calculus Review 3

    This topic is a collection of problems and concepts that might help someone understand their working knowledge of multivariate calculus.

(1) Find the Jacobian given and

(2) Let be the region in the -plane that is bounded by the coordinate axes and the line Use the change of variables to compute the integral  

(3) Evaluate where is the region bounded by the parallelogram with vertices and

(4) A rotation of the -plane through the fixed angle is given by



Compute the Jacobian Let denote the region bounded by the ellipse Use a rotation of to obtain an integral that is equivalent to



Evaluate the transformed integral.

(5) If show that if and only if

(6) Show that the curl of the gradient of a function is always

(7) Show that the divergence of the curl of a vector field is

(8) Let Either find a vector field such that or show that no such exists.

(9) Evaluate the line integral where and is the boundary of the triangle with vertices and , transversed once clockwise, as viewed from above.

(10) A 5,000-lb satellite orbits the earth in a circular orbit 5,000 mi from the center of the earth. How much work is done as the satellite moves through one complete revolution?

(11) Suppose a particle with charge and mass moves with velocity under the influence of an electric field and a magnetic field Then the total force on the particle is called the Lorentz force. Use Newton's second law of motion, to show that



and then evaluate the line integral where is the trajectory of a particle traveling with constant speed.  

(12) Let and let and be the following two paths joining to

   and   

Show that Explain what this means?

(13) Show that the vector field is conservative and find a scalar potential for Then evaluate the line integral where is any smooth path connecting to

(14)  Show that the vector field



is conservative and find a scalar potential for

(15) Show that the vector field



is conservative and then evaluate the line integral where is any piecewise smooth path connecting to

(16) The gravitational force field between two particles of masses and separated by a distance is modelled by



where and is the gravitational constant.

(a) Show that is conservative by finding a scalar potential for The scalar potential function is often called the Newtonian potential.

(b) Compute the amount of work done against the force field in moving an object from the point to

(17) Let



(a) Compute the line integral where is the upper semicircle transversed counterclockwise. What is the value of if is the lower semicircle also transversed counterclockwise?

(b) Show that if ,  then



but is not conservative on the unit disk

(18) Evaluate the closed line integral where is the boundary of the region between the -axis and the semicircle traversed counterclockwise (including the -axis).

(19) Evaluate



where is any Jordan curve whose interior does not contain the point

(20) If is a Jordan curve, show that



where is the region enclosed by

(21) Suppose is continuously differentiable in a doubly-connected region and that



throughout How many distinct values of are there for the integral



where is a piecewise smooth Jordan curve in

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