MA 2005 A-95 Sample Exam 3 Name here
Show your work in the space provided. Unsupported answers may not receive full credit. Use the backs of the pages if needed. (In the real exam, there will be plenty of space for your work.)
compute the divergence, , and the curl, .
Answer: Write . Then
Expand the derivatives using the chain rule. Then compute the two terms on the right hand side of the identity and compare.
where and C is the straight line y= -3x+2 for .
Answer: The curve can be parametrized by , for . Then
where C is the helix for and the vector field is given by
Next, compute the dot product
So we have
Answer: Since is a potential for , we know that
Now we write our parametric curve as
So we have
Note that the integrand is the same as
so we have
The final step is to realize that , , , and , which gives the desired result.
Answer: First, we need to compute dS. The easiest way is to use the formula
So the area is given by
where the vector field is given by
and the surface S is the unit cube , , and , and is the outward unit normal to the surface.
Answer: The divergence theorem says that
where R is the region bounded by the surface S. We have
which gives us
where is the single face of the unit cube defined by z=1, , and .
Answer: For this face of the cube, we have z=1, , and . So we have
Answer: You are on your own with this one.