Math 234 — Final Exam Review

About line integrals

1. Given two points $A(0, 3)$ and $B(4,0)$, let $\cC$ be the line segment from $A$ to $B$. Suppose we also have a constant vector field $\vF(x, y) = 2\ve_1 - \ve_2$
   1. Draw the points, the line segment, and the vector field.
   2. Compute \[ L_1 = \int_{\cC} 3 ds,\qquad L_2 = \int_{\cC} \vF \cdot \vT\; ds,\qquad L_3 = \int_{\cC} \vF \cdot \vN\; ds \] without finding a parametrization of $\cC$, and without finding an antiderivative.
   Answers
2. The distance from the point $A$ to $B$ is 7, the length of the constant vectorfield $\vF$ is $\|\vF\|=3$, and the angle between the line segment $AB$ and the vector $\vF$ is $\theta$. Compute the flux of the vectorfield $\vF$ across the line segment $AB$, and the work done by the force $\vF$ if it acts on a particle that moves from $A$ to $B$. Answers
3. If \[ \vF(x, y) = \begin{pmatrix} P(x, y) \\ Q(x, y) \end{pmatrix}, \] then which of these integrals \begin{align*} A &= \int_{\cC} \vF \cdot \vT\; ds\\ B &= \int_{\cC} \vF \cdot \vN\; ds\\ C &= \int_{\cC} \vF \cdot d\vx \\ D &= \int_{\cC} P(x,y) dx + Q(x,y) dy\\ E &= \int_{\cC} -Q(x, y) dx + P(x, y)dy \end{align*} are equal to each other?? Answers
4. 1. State Green's theorem for line integrals.
   2. State Green's theorem for flux integrals.
5. Suppose that $\cC$ is a closed curve that encloses some domain $\cR$. Let $\vF(x, y) = P(x, y)\ve_1 +Q(x, y)\ve_2$, and suppose $\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 0$ holds everywhere in the domain $\cR$. Then \[ \int_\cC \vF\cdot\vN\; ds =0,\quad\text{True or False?} \] Or \[ \int_\cC \vF\cdot\vT\; ds =0,\quad\text{True or False?} \] Answers
6. Let $f(x, y) = xy\cos(x)$, and let $\cC$ be the line segment from the point $(0,0)$ to the point $(\pi, 1)$. Compute \[ \int_\cC \vec\nabla f\cdot\vT\; ds \] Answer

7. The following figure shows some level sets of a function $f(x,y)$ (in grey), some points $K$, $L$, $M$, $N$, and $O$, and four paths $\cC_1$, … , $\cC_4$.

   
   1. True/False? $\int_{\cC_2} f ds = 0.2$ ?
   2. True/False? $\int_{\cC_2} \vec\nabla f\cdot d\vec x = 0.2$ ?
   3. True/False? $\int_{\cC_2} f ds \gt 0$ ?
   4. True/False? $\int_{\cC_2} \vec\nabla f\cdot d\vec x \gt 0$ ?
   5. True/False? $\int_{\cC_2} f ds = 0$ ?
   6. True/False? $\int_{\cC_1} \vec\nabla f\cdot d\vec x \gt 0$ ?
   7. True/False? $\int_{-\cC_1} f ds \lt 0$ ?
   8. True/False? $\int_{- \cC_1} \vec\nabla f\cdot d\vec x \gt 0$ ?
   9. True/False? $\int_{\cC_1+\cC_2} \vec\nabla f\cdot d\vec x = \int_{\cC_2} \vec\nabla f\cdot d\vec x$ ?
   10. True/False? $\int_{\cC_1+\cC_2} \vec\nabla f\cdot d\vec x = \int_{\cC_4} \vec\nabla f\cdot d\vec x$ ?
   11. True/False? $\int_{\cC_3} \vec\nabla f\cdot d\vec x \gt 0$ ?
   Answers
8. Consider the line integral $\int_\cC \sin(x) dx + e^{xy} dy $, where $\cC$ is some curve in the plane. We write $\vT$ for the unit tangent of $\cC$, and $\vN$ for the unit normal (assume that $\vN$ is obtained by rotating $\vT$ clockwise by $90^\circ$.) The line integral $I$ can be written in the following three ways \[ \int_\cC \sin(x) dx + e^{xy} dy = \int_\cC \vF\cdot d\vx = = \int_\cC \vF\cdot\vT\, ds = \int_\cC \vG\cdot\vN\, ds =\int_\cC \vH\cdot d\vx, \] provided you choose the right vector fields $\vF$, $\vG$, and $\vH$. Find these vector fields.
9. Let $\vF$ be the vector field \[ \vF(x,y) = \begin{pmatrix} xy^2 \\ x^2y \end{pmatrix}, \] and let $\cC$ be a simple closed curve in the plane. Consider the integrals \[ I = \int_\cC \vF\cdot\vT ds, \qquad J = \int_\cC \vF\cdot\vN\, ds. \] Here $\vT$ is the unit tangent that gives the curve the clockwise orientation, and $\vN$ is the outward unit normal to the curve.
   1. Which of the following conditions always holds?

      (a) $I=0$; (b) $I\lt 0$; (c) $I\gt 0$; (d) any of these, depending on the curve $\cC$.

   2. Which of the following conditions always holds?

      (a) $J=0$; (b) $J\lt 0$; (c) $J\gt 0$; (d) any of these, depending on the curve $\cC$.

10. If $\cC$ is a curve in the plane, then what do the following integrals represent? \[ I_1 = \int_\cC \vT\cdot\vT\, ds\qquad I_2 = \int_\cC \vN\cdot\vN\, ds\qquad I_3 = \int_\cC \vT\cdot\vN\, ds. \] Here $\vT$ and $\vN$ are unit tangent and unit normal vectors for the curve.