Path-Independent Vector Fields and the Fundamental Theorem of Calculus for Line Integrals

Calculate each of the following line integrals.

1. $\int_C \nabla f\cdot d\vr$ if $f(x,y) = 3xy^2 - \sin(x) + e^y$ and $C$ is the top half of the unit circle oriented from $(-1,0)$ to $(1,0)$.

2. $\int_C \nabla g\cdot d\vr$ if $g(x,y,z) = xz^2 - 5y^3\cos(z) + 6$ and $C$ is the portion of the helix $\vr(t) = \langle 5\cos(t),5\sin(t),3t\rangle$ from $(5,0,0)$ to $(0,5,9\pi/2)$.

3. $\int_C \nabla h\cdot d\vr$ if $h(x,y,z) = 3y^2e^{y^3} - 5x\sin(x^3z) + z^2$ and $C$ is the curve consisting of the line segment from $(0,0,0)$ to $(1,1,1)$, followed by the line segment from $(1,1,1)$ to $(-1,3,-2)$, followed by the line segment from $(-1,3,-2)$ to $(0,0,10)$.

Calculate each of the following line integrals.

1. $\int_C \vF\cdot d\vr$ if $\vF(x,y) = \langle 2x,2y\rangle$ and $C$ is the line segment from $(1,2)$ to $(-1,0)$.

2. $\int_C \vG\cdot d\vr$ if $\vG(x,y) = \langle 4x^3-12y\cos(xy),9y^2-12x\cos(xy)\rangle$ and $C$ is the portion of the unit circle from $(0,-1)$ to $(0,1)$.

3. $\int_C \vH\cdot d\vr$ if $\vH(x,y,z) = \langle H_1,H_2,H_3\rangle$ with and $C$ is the curve consisting of the line segment from $(1,1,1)$ to $(3,0,3)$, followed by the line segment from $(3,0,3)$ to $(1,5,-1)$, followed by the line segment from $(1,5,-1)$ to $(0,0,0)$.

Suppose that $\vF$ is a continuous path-independent vector field (in $\R^2$ or $\R^3$) on some region $D$.

1. Let $P$ and $Q$ be points in $D$ and let $C_1$ and $C_2$ be oriented curves from $P$ to $Q$. What can you say about $\int_{C_1}\vF\cdot d\vr$ and $\int_{C_2}\vF\cdot d\vr$?

2. Let $C = C_1 - C_2$. Explain why $C$ is a closed curve.

3. Calculate $\oint_C\vF\cdot d\vr$.

4. Write a sentence that summarizes what we can conclude about line integrals of $\vF$ at this point in the activity.

5. Now let us suppose that $\vG$ is a continuous v…

Let $\vF=\langle F_1,F_2,F_3\rangle$, where $F_1= 3xy^2+z$ and $F_3 = yz^2+\cos(z)$.

1. Give a component function $F_2$ such that $\vF$ is a gradient vector field.

2. Give a component function $F_2$ such that $\vF$ is not a gradient vector field.

Compute $\int_C ye^z\, dx +xe^z\, dy+xye^z\, dz$ where $C$ is given by $\langle t^2,t^3,t-1\rangle$ for $1\leq t\leq 2$.

Let $\vF(x,y,z)$ be a vector field such that $\vF=\nabla f$ for a smooth function $f(x,y,z)$. Let $S$ be a level surface of $f$ and let $C_1$ be curve that stays on the surface $S$. Let $C_2$ be a curve that begins and ends on $S$ but does have points not on $S$.

1. What is the value of $\int_{C_1} \vF \cdot d\vr$?

2. What is the value of $\int_{C_2} \vF \cdot d\vr$?