Mat 241 Semester Final Fall, 2015 Key Name KEY

Directions: Show all work for each question and make sure your answers are clearly identified. You may use the back side of pages if needed.

#1. (5 points) While the Mars rover “Curiosity” was landing on Mars, a force of

acted on the rover while it was descending on the path

.

Find the work done by the force over the curve of descent.

Solve by direct calculation:

#2. (5 points) Suppose we have a sphere of radius 1 with the first octant removed as shown below.

Also assume the force shown below acts on it.

Find the Flux through the bounded solid (assume outward orientation).

Use the Divergence Theorem.

#3. (5 points). The beak-shaped surface shown below is subject to the force:

The surface is bounded below by the curve in the xy-plane given by:

Compute: by using the line integral: .Solve by Stokes’ Theorem.

#4. (5 points) Skippy has a cousin, “Spanky”. Find the area of Spanky given that Spanky is defined by the closed parametric curve:

Find the Area of Spanky!

Solve by Green’s Theorem in Reverse.

#5. (2 points each part)

A. Show that the given force field is a gradient field (conservative).

B. Find a potential function, f, for the force field given in part A.

C. Suppose we have a spacecurve given by:

Carefully find the following:

=

=

D. Using your potential function from part B, and your results from part C, compute the work of our force field from part A on the curve using the Fundamental Theorem of Line Integrals.

By the way, here is the curve. It would make a fun roller-coaster!

BONUS: Optional

We use Stokes’ Theorem.

Choose the surface z = 100. The upward normal to this surface is

Have a nice break and stay classy!