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THIRD HOUR EXAM, MATH 223
Instructor: Kris Green, December 5, 19967

1.

Multiple Choice (5 points each):

(a)

The divergence of the vector field $\vec{H}(x,y,z) = x^2y\hat{i} + y^2z\hat{j} + z^2x\hat{k}$ at the point (1,1,-1) is:

$$2\hat{i} - 2\hat{j} -2\hat{k} -\hat{j} - \hat{k}$$ a. -2 d. 2xy + 2yz + 2xz

(b)

The curl of $\vec{F} = xz\hat{i} - y^2\hat{j} + 2x^2y\hat{k}$ is:

$$z\hat{i} - 2y \hat{j} 2x\hat{i} + 3y\hat{j} - 2y\hat{k}$$

$$2x^2 \hat{i} - (4xy - x)\hat{j} 2x^2\hat{i} + (4xy - x)\hat{j}$$

(c)

Choose the expression that best describes the vector field shown below: (include vector field)

$$y\hat{i} + \frac{1}{x}\hat{j} \frac{x\hat{i} + y\hat{j}}{\sqrt{x^2 + y^2}}$$

$$-y\hat{i} + x\hat{j} -y\hat{i} - \frac{1}{x}\hat{j}$$

(d)

Consider the following line integral:

$$\int_C \left(\frac{1}{x-2} \hat{i} + \sqrt{x^2 + y^2} \hat{j}\right) \cdot d\vec{r}.$$

Over which of the following parameterized paths can Green's Theorem be applied to this integral?

$$C: \vec{r}(t) = \cos(t)\hat{i} + \sin(t)\hat{j}, \quad 0 \le t \le 2\pi$$ a.

$$C: \vec{r}(t) = [2\cos(t) - 3]\hat{i} + [2\sin(t)-3]\hat{j}, \quad 0 \le t \le 2\pi$$ b.

$$C: \vec{r}(t) = [\cos(t) + 2]\hat{i} + \sin(t)\hat{j}, \quad 0 \le t \le 2\pi$$ c.

$$C: \vec{r}(t) = [2\cos(t)+5]\hat{i} + [2\sin(t)+5]\hat{j}, \quad 0 \le t \le 2\pi$$ d.

2.

Short answer (10 points each):

(a)

Give 3 characterizations of a conservative vector field. If you name more than 3 correctly, then extra credit will be given. (Hint: One of these can be the definition of a conservative vector field.)

(b)

State Stokes' Theorem and describe the conditions under which it can be applied.

3.

(10 points) Compute the line integral $\int_C \vec{F} \cdot d\vec{r}$for the vector field

$$\vec{F}(x,y) = \cos \left( \frac{y}{x}\right)\hat{i} + \sin(y) \hat{j}$$

along the path y=x² from (1,1) to (2,4).

4.

(10 points) Compute the line integral $\int_C \vec{F} \cdot d\vec{r}$for the vector field

$$\vec{F}(x,y,z) = \frac{1}{x}\hat{i} + \frac{1}{y} \hat{j} + \frac{1}{z} \hat{k}$$

along the path $C: \vec{r}(t) = (5 + \cos(\pi t))\hat{i} + (1 - t(t-2)) \hat{j} + e^t \hat{k}$ for $0 \le t \le 2$.

5.

(10 points) Compute the line integral $\int_C \vec{F} \cdot d\vec{r}$for the vector field

$$\vec{F}(x,y) = x^2 e^y \hat{i} + xy^2 \hat{j}$$

along the square with vertices (2,2), (-2,2), (-2,-2), and (2,-2) in the counterclockwise direction.

6.

(10 points) Find the flux of $\vec{F}(x,y,z) = x^3y^2\hat{i} - x^4y \hat{j} + z\hat{k}$ over the portion of the surface z = 16 - x² - y² that lies above the disk $x^2 + y^2 \le 16$.

7.

Let $\vec{G}(x,y,z) = (2x + y) \hat{i} + (x+y+z)\hat{j} + (3z - 5)\hat{k}$. Compute the flux of $\vec{G}$ through the sphere of radius 4 centered at (1,-1,0).

8.

Find the circulation of $\vec{F}(x,y,z) = xz \hat{i} - y^2 \hat{j} + 2x^2y \hat{k}$ around the triangle in the plane z = y with vertices (3,0,0), (0,2,2) and (0,0,0) oriented in this order.