Exam 18: Line Integrals

If $\vec { F } ( x , y ) = P ( x , y ) \vec { i } + Q ( x , y ) \vec { j }$ is a gradient vector field, then $\frac { \partial P } { \partial x } = \frac { \partial Q } { \partial y }$ .

Use Green's Theorem to evaluate $\int _ { c } \left( \left( \cos x ^ { 2 } + e ^ { x ^ { 2 } } + 3 y \right) \vec { i } + \left( e ^ { 2 \sin y } + 4 x + \cos y ^ { 4 } \right) \vec { k } \right) \cdot \overrightarrow { d r }$ where C is the circle of radius $\sqrt { 3 } ,$ centered at $\left( - 1 , e ^ { 10 } , \ln 13 \right)$ oriented in a counter-clockwise direction.

Given the graph of the vector field, $\vec{F}$ , shown below, list the following quantities in increasing order: $\int _ { C _ { 1 } } \bar { F } \cdot d \vec { r } , \quad \int _ { C _ { 2 } } \bar { F } \cdot d \vec { r } , \quad \int _ { C _ { 3 } } \bar { F } \cdot d \vec { r } .$

Evaluate $\int _ { - c } x \vec { i } + ( y + 10 x ) \vec { j } + ( z + 9 x ) \vec { k } \cdot \overrightarrow { d r }$ , where C is the curve $\vec { r } ( t ) = t \vec { i } + ( 1 - t ) \vec { j } + \left( t ^ { 2 } + 3 \right) \vec { k }$ for 0 $\le$ t $\le$ 1. Note that the line integral is around -C, not C.

Explain in words and symbols how to calculate the line integral $\int_{C} \vec{F} \cdot d \vec{r}$ given a parameterization, $\vec { r } = \vec { p } ( t )$ of the curve C.

Let $\vec{F}$ and $\vec { G }$ be two 2-dimensional fields, where $\vec{F}=3 x \vec{i}+5 y \vec{j}$ and $\vec { G } = 3 y \vec { i } + 5 x \vec { j }$ Let C₁ be the circle with center (2, 2)and radius 1 oriented counterclockwise. Let C₂ be the path consisting of the straight line segments from (0, 4)to (0, 1)and from (0, 1)to (3, 1). Find the line integral $\int_{c_{2}} \vec{F} \cdot d \vec{r}$ Use "pi" to represent $\pi$ if necessary.

If the length of curve C₁ is longer than the length of curve C₂, then $\int _ { C _ { 1 } } \vec { F } \cdot \overrightarrow { d r } \geq \int _ { C _ { 2 } } \vec { F } \cdot \overrightarrow { d r }$

Let f(x, y, z)be a function of three variables.Suppose that C is an oriented curve lying on the level surface f(x, y, z)= 2. Find the line integral $\int _ { C } \operatorname { grad } f \cdot \overline { d r }$

If $\vec{F}$ is a path-independent field, then $\int _ { C } \vec { F } \cdot d \vec { r } = 0$ where C has the parameterization $\vec { r } ( t ) = \cos t \vec { i } + \sin t \vec { j } , 0 < t < 3 \pi$

Let $\vec{F}$ be the constant vector field $2 \vec { i } - 2 \vec { j } + \vec { k }$ Calculate the line integral of $\vec{F}$ along a line segment L of length 9 at an angle $\pi$ /3 to $\vec{F}$

Consider the vector field $\vec{F}=5 x \vec{j}$ . Without using parametrization, calculate directly the line integral of $\vec{F}$ along the line from (3, 3)to (7, 3).

Which of the two vector fields shown below is not conservative?

Suppose a curve C is parameterized by $\vec { r } ( t )$ with $a \leq t \leq b$ and suppose $\vec { F }$ is a vector field $\vec{F}(t)=\vec{r}(t) \times \vec{r}^{\prime}(t)$ for $a \leq t \leq b$ .Explain why $\int _ { C } \vec { F } \cdot d \vec { r } = 0$

Is the following vector field is a gradient vector field? $\vec{F}=y \vec{i}+x \vec{j}$

If $\int _ { C } \bar { F } \cdot \overline { d r } = 0$ , then $\vec{F}$ is perpendicular to the curve C at every point.

Let $\vec{F}$ be a vector field with constant magnitude $\| \vec { F } \| = 8$ Suppose that $\vec { r } ( t ) ,$ 0 $\le$ t $\le$ 5, is a parameterization of a flow line C of $\vec{F}$ . Find $\int _ { C } \vec { F } \cdot d \vec { r }$

State the Fundamental Theorem of Calculus for Line Integrals.

Let $\vec{F}=\left(2+6 x e^{3\left(x^{2}+y^{2}\right)}\right) \vec{i}+\left(6 y e^{3\left(x^{2}+y^{2}\right)}\right) \vec{j}$ Use the curl test to check whether $\vec{F}$ is path-independent.