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THIRD HOUR EXAM, MATH 223
Instructor: Kris Green, May 1, 1997

1.

Multiple Choice (5 points each):

(a)

The divergence of the vector field $\vec{F} = r^2 \vec{r}$ at the point (1,1,1) is:

$$5\hat{i} + 5\hat{j} + 5\hat{k}$$ a. 5r² b. 1 c. 15

(b)

The potential function for the vector field $\vec{G} = \frac{2}{x}\hat{i} + \frac{1}{y}\hat{j} + \frac{1}{x} \hat{k}$ (with x > 0, y> 0, z > 0) is

$$\ln (x^2yz) + 2\ln (xyz) +$$

c. Neither of the above d. This vector field does not have a potential

(c)

The vector $d\vec{A} ( =\hat{n} dS)$ for the upper hemisphere of x² + y² + z² = 4 oriented toward +z is

$$\left(-\frac{x}{\sqrt{4-x^2 - y^2}}\hat{i} - \frac{y}{\sqrt{4 - x^2 - y^2}}\hat{j} + \hat{k}\right)dx dy$$ a.

$$\left(\frac{x}{\sqrt{4-x^2 - y^2}}\hat{i} + \frac{y}{\sqrt{4 - x^2 - y^2}}\hat{j} + \hat{k}\right)dx dy$$ b.

$$\left(-\frac{x}{\sqrt{4-x^2 - y^2}}\hat{i} - \frac{y}{\sqrt{4 - x^2 - y^2}}\hat{j} - \hat{k}\right)dx dy$$ c.

$$\left(\frac{x}{\sqrt{4-x^2 - y^2}}\hat{i} + \frac{y}{\sqrt{4 - x^2 - y^2}}\hat{j} - \hat{k}\right)dx dy$$ d.

(d)

Suppose you know that a vector field $\vec{F}$ has the following properties: (1) div$(\vec{F})$ = 5, and (2) curl$(\vec{F}) = \hat{i} + 2\hat{j}$. What is the flux of $\vec{F}$ through the cube covering the region $-1 \le x \le 1$, $-1 \le y \le 1$, and $-1 \le z \le 1$?

a. 0 b. 40 c. -40 d. 5

2.

Calculate the flux of the vector field $\vec{F} = -z\hat{i} + \frac{z^3}{y} \hat{j} + x\hat{k}$ through the upper hemisphere of x² + y² + z² = 4 that lies above the rectangle $-1 \le x \le 1$, $0.5 \le y \le 1$.

3.

Consider the vector field

$$\vec{F} = \left(\frac{1}{y} - \frac{y}{x^2} - 2xy\right)\hat{i} + \left( \frac{1}{x} - \frac{x}{y^2} + y\right) \hat{j}$$

(a)

Show that $\vec{F}$ can be written in the form $\vec{F} = \nabla f + \vec{G}$ where f(x,y) = (x² + y²)/xy and $\vec{G}$ is a non-conservative vector field.

(b)

Use the previous part to compute the line integral $\int_C \vec{F} \cdot d\vec{r}$ where C is the portion of the parabola y = x² from x = 1 to x = 3.

4.

Consider the circulation of the vector field

$$\vec{G} = (y + 2z)\hat{i} + (2x + z)\hat{j} + (2x + y)\hat{k}$$

around the boundary of the sphere x² + y² + z² = 1 in the first octant (ie. x,y,z are all positive.) The figure below illustrates this. Note that the curve C₁ is part of the circle x² + z² = 1, C₂ is part of the circle x² + y² = 1 and C₃ is part of the circle y² + z² = 1.

(a)

Set up the line integral by parameterizing the path.

(b)

Stoke it. (ie. Apply Stokes' theorem.)

(c)

Calculate one of these two integrals that you have set up.

5.

Compute the flux of the vector field $\vec{F} = 4z\hat{i} + e^{z^2}\hat{j} + \sin (x^2 z) \hat{k}$ through the triangle in the yz plane with vertices at (0,0,0), (0,2,4) and (0,0,4) and oriented in the +x direction.