Exam 18: Line Integrals

Let $\vec{F}=\left(y^{2}+y \sin (x y)\right) \vec{i}+(4 x y-y+x \sin (x y)) \vec{j}$ Is $\vec{F}$ path-independent?

Find the work done by the force field $\overrightarrow { \vec { F } } = \left( 12 y + e ^ { x } - 6 \sin x \right) \vec { i } + 3 x ^ { 2 } y \vec { j }$ along the parabola y = 2x² from (0, 0)to (1, 2).

Let $\vec{F}=(3 x+2 y) \vec{i}+(3 x-3 y) \vec{j}$ Evaluate $\int_{C} \vec{F} \cdot d \vec{r}$ , where C is the line from (0, 0)to (2, 2).

On an exam, students were asked to evaluate $\int _ { C } \left( x ^ { 2 } \vec { i } + x \vec { j } \right) \cdot d \vec { r }$ , where C has the parameterization $x = \cos t , y = \sin t , 0 \leq t \leq \pi$ .One student wrote: "Using Green's Theorem, $\int _ { C } \left( x ^ { 2 } \vec { i } + x \vec { j } \right) \cdot d \vec { r } = \int _ { D } \left( \frac { \partial x } { \partial x } - \frac { \partial \left( x ^ { 2 } \right) } { \partial y } \right) d A = \int _ { D } 1 d A =$ Area of the semi-circle = $\frac { \pi } { 2 }$ ." Do you agree with the student?

If C₁ and C₂ are two curves with the same starting and ending points, then $\int _ { c _ { 1 } } \vec { F } \cdot \overline { d r } = \int _ { c _ { 2 } } \vec { F } \cdot \overline { d r }$ , for any vector field $\vec{F}$

Let $\vec { F } = \sqrt { x ^ { 2 } + y ^ { 2 } } \vec { i } + \sqrt { x ^ { 2 } + y ^ { 2 } } \vec { j }$ Is the line integral of $\vec{F}$ around the unit circle traversed counterclockwise: positive, negative, or zero?

Let $\vec { F } = ( 4 x + 5 y ) \vec { i } + ( 4 x - 4 y ) \vec { j }$ Let $I _ { 1 } = \int _ { C _ { 1 } } \bar { F } \cdot \overline { d r }$ , where C₁ is the line from (0, 0)to (2, 2). Let $I_{2}=\int_{c_{2}} \vec{F} \cdot \overrightarrow{d r}$ , where C₂ is parameterized by $\vec { r } ( t ) = t \vec { i } + \frac { 1 } { 2 } t ^ { 2 } \vec { j } , \quad 0 \leq t \leq 2$ Notice that both C₁ and C₂ go from (0, 0)to (2, 2), but is $I _ { 1 } = I _ { 2 } ?$ Explain.

Let $\vec { F } = x \vec { j }$ Let C₁ be the line from (0, 0)to (2, 0), C₂ the line from (2, 0)to (2,-1), C₃ the line from (2,-1)to (0,-1), and C₄ the line from (0,-1)to (0, 0). (A)Using the definition of line integral only, without parameterizing the curves, show that the line integral of $\vec{F}$ along C = C₁ + C₂ + C₃ + C₄ is -2.That is, show $\int_{C} \vec{F} \cdot \overline{d r}=-2$ (B)The rectangle, R, enclosed by the lines C₁, C₂, C₃ and C₄ is of area 2.So, by Green's Theorem $\int _ { C } \vec { F } \cdot \overline { d r } = \int _ { R } \left( \frac { \partial x } { \partial x } - 0 \right) d A = \text { Area of } R = 2$ Is something wrong?

Suppose that $\int_{c} \vec{F} \cdot d \vec{r}=6$ , where C is the circle of radius 1, centered at the origin, starting at (1, 0)and traveling counter-clockwise back to (1, 0). $\int _ { G _ { 1 } } 2 \vec { F } \cdot d \vec { r } = 12$

Let $\vec { F } ( x , y ) = \left( 9 x ^ { 2 } - 3 y ^ { 2 } \right) \vec { i } - 6 x y \vec { j }$ Evaluate the line integral $\int _ { C } \vec { F } \cdot d \vec { r }$ where C the path from (0, 0)to (1, 1)that goes along the x-axis to (1, 0), and then vertically up to (1, 1).

Let C be the curve x = 2t + cos t, y = 4t, z = 2 sin t for 0 $\le$ t $\le$ 3 $\pi$ /2. Use a potential function to evaluate $\int_{C} \vec{F} \cdot d \vec{r}$ exactly, where $\overrightarrow { \vec { F } } = 4 x ^ { 3 } \cos ( 4 y z ) \vec { i } - 4 x ^ { 4 } z \sin ( 4 y z ) \vec { j } - 4 x ^ { 4 } y \sin ( 4 y z ) \vec { k } \text {. }$

Let $\vec { F } = \sqrt { x ^ { 2 } + y ^ { 2 } } \vec { i } + \sqrt { x ^ { 2 } + y ^ { 2 } } \vec { j }$ For a fixed $\theta$ , let C _$\theta$ be the line segment from (0, 0)to the point (cos $\theta$ , sin $\theta$ )on the unit circle. Find a parameterization of C _$\theta$ and compute $\int_{C_{0}} \vec{F} \cdot d \vec{r}$ .(Your answer will depend on $\theta$ .)

Let $\vec { F } = \frac { y } { 4 x ^ { 2 } + 4 x y + y ^ { 2 } } \vec { i } - \frac { x } { 4 x ^ { 2 } + 4 x y + y ^ { 2 } } \vec { j }$ Check that $\frac { \partial F _ { 1 } } { \partial y } = \frac { \partial F _ { 2 } } { \partial x }$ .Is $\vec{F}$ is path-independent?

Answer true or false, giving a reason for your answer.Let $\phi ( x , y ) = \ln ( | x y | )$ so that $\nabla \phi = \frac { 1 } { x } \vec { i } + \frac { 1 } { y } \vec { j }$ . Then $\int _ { C } \nabla \phi \cdot d \vec { r } = \phi ( 1,1 ) - \phi ( - 1,1 )$ where C is given by the parametrization $x = t , y = 1$ for $- 1 \leq t \leq 1$ .

Explain what is meant by saying a vector field is conservative.

Let C be a segment 9 units long of the contour f(x, y)= 5.What is the work done by the gradient field of f along C?

Consider the vector field $\vec { F } = ( A x + B y ) \vec { i } + ( B x + C y ) \vec { j }$ for certain constants A, B and C. Show that $\vec{F}$ is path-independent by finding its potential function.

Let $\overrightarrow { \vec { F } } = ( 3 x + 15 y ) \vec { i } + ( 3 x - 3 y ) \vec { j }$ Evaluate $\int_{C} \vec{F} \cdot d \vec{r}$ , where C is parameterized by $\vec { r } ( t ) = t \vec { i } + \frac { 1 } { 2 } t ^ { 2 } \vec { j } , \quad 0 \leq t \leq 2$

Given that $\vec{G}=3 x^{2} \vec{i}+10 y^{4} \vec{j}$ find a function g so that $\vec{G}=\operatorname{grad} g .$ Use the function g to compute $\int _ { C } \vec { G } \cdot d \vec { r }$ where C is a curve beginning at the point (2, 4)and ending at the point (0, 1).

Let C₁ be the rectangular loop consisting of four line segments: from (0, 0)to (1, 0), then to (1, 2), then to (0, 2), then back to (0, 0).Suppose C₂ is the triangular loop joining (0, 0)to (1, 0), then to (1, 2)then back to (0, 0), and C₃ is another triangular loop joining (0, 0)to (1, 2), then to (0, 2)and then back to (0, 0). Is it true that $\int _ { c _ { 2 } } \vec { F } \cdot \overline { d r } = \int _ { C _ { 3 } } \vec { F } \cdot \overline { d r } + \int _ { C _ { 1 } } \vec { F } \cdot \overline { d r }$ for any vector field $\vec{F}$ defined on the xy-plane?