MTH 255 Sample Midterm 2

This was the second midterm given by another instructor.

1.

In this problem the curve C is the quarter of the circle of radius 2, centered at the origin, which lies in the first quadrant, oriented from (2,0) to (0,2).

(a)

Compute the integral along C of f ds, where f(x,y) = xy/(x²+y²).

(b)

Compute the integral along C of F · dr for F = y i-x j.

2.

In this problem C is the half of the circle x²+y² = 9 in the right half plane (x > 0) together with the segment on the y-axis from (0,3) to (0,-3) oriented counterclockwise, and F = -x²y i+xy² j.

(a)

Compute the integral along C of F · dr.

(b)

Compute the integral along C of F · Nds, where N is the outer unit normal.

3.

Compute the integral along C of F · dr where F = grad((x+2y+3z)e^xz) and C is the piecewise smooth curve consisting of the straight line segments from (0,1,1) to (1,2,1) to (2,1,0).

4.

Let F = (3y²-3x²+y) i+(6xy+x+z) j+y k.

(a)

Compute curl F.

(b)

Compute div F.

(c)

Is F conservative (everywhere)? Justify your answer.

5.

(This question deals with material we have not covered.)