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FOURTH HOUR EXAM
MATH 223
Instructor: Kris Green
May 1, 1998
9:00-9:50

1.

Consider the motion of a particle described by the parametric curve

$$x(t) = t + \cos t, \qquad y(t) = \sin t, \qquad 0 \le t \le 4\pi,$$

(a)

Is the motion periodic? (ie. does it ever repeat its movements?)

(b)

Does the motion ever come to a complete stop? If so, when?

(c)

Is the motion ever purely horizontal (x-direction)? If so, when?

(d)

Is the motion ever purely vertical (y-direction)? If so, when?

(e)

Using what you know about the motion of a particle along the path $(x(t), y(t)) = (\cos t, \sin t)$ for $0 \le t \le 2\pi$, describe in complete sentences the motion of the particle above.

2.

For the vector field

$$\vec{F} = \left(4y + \frac{32}{yz}\right)\hat{i} + \left(2xz... ...2}\right) \hat{j} - \left(1 + \frac{32x}{yz^2}\right)\hat{k},$$

(a)

Show that $\vec{F} = \nabla f + \vec{G}$ where

$$f(x,y,z) = \frac{32x}{yz},$$

and $\vec{G}$ is a non-conservative vector field.

(b)

Use part (a) (the fact that $\vec{F} = \nabla f + \vec{G}$, and you know what f and $\vec{G}$ are) to compute the work done by the force $\vec{F}$ along the path C from (1,1,1) to (2,8,4) along the curve (t,t³,t²).

3.

Given the vector field

$$\vec{G} = xz^2 \hat{i} + (2x + 3y) \hat{j} + x^2z \hat{k}$$

and the closed path, C, formed by the semicircles $y = \sqrt{1 - x^2}$ (in the xy-plane) and $z = \sqrt{1 - x^2}$ (in the xz-plane) and oriented counter-clockwise when viewed from the +z axis, calculate the circulation of $\vec{G}$ around C, using Stokes' Theorem. Note that there is an obvious surface, S, which has the curve C as its boundary. Graphing these curves on the same set of xyz-axes this may help.

4.

For each of the following situations, explain (using complete sentences, possibly supported by calculations and/or diagrams) why Green's Theorem can not be directly applied. If a modification to Green's Theorem (such as the addition of a negative) will correct the problem, explain this.

(a)

$\int_C (-y\hat{i} + xy \hat{j}) \cdot d\vec{r}$, C is the path $(t, \sin t)$ for $0 \le t \le 2\pi$.

(b)

$\int_C (2xy\hat{i} - y^2 \hat{j}) \cdot d\vec{r}$, where $\vec{r}(t) = \cos (t)\hat{i} + \sin (2t) \hat{j}$ for $0 \le t \le 2\pi$.

(c)

$\int_C \left( \frac{1}{x-2} \hat{i} + \frac{3x}{y - 1}\hat{j} \right) \cdot d\vec{r}$ where C is the circle of radius 4 centered at (1,1) traversed counter-clockwise.

5.

For the illustrated paths and vector fields below, decide whether the line integral $\int_C \vec{F} \cdot d\vec{r}$ is positive, negative, or zero.

(a)

(b)

(c)

(d)