5.5 Path-Independent Vector Fields and the Fundamental Theorem of Calculus for Line Integrals (continued)

Line Integrals Along Closed Curves

What other vector fields are path-independent?

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 $\int_{C_2}\vF\cdot d\vr$?

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

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

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

Now let us suppose that $\vG$ is a continuous vector field on a region $D$ for which $\oint_C\vG\cdot d\vr = 0$ for all closed curves $C$. Pick two points $P$ and $Q$ in $D$. Let $C_1$ and $C_2$ be oriented curves from $P$ to $Q$. What type of curve is $C = C_1 - C_2$?

What is $\oint_C\vG\cdot d\vr$? Why?

What does that tell you about the relationship between $\int_{C_1}\vG\cdot d\vr$ and $\int_{C_2}\vG\cdot d\vr$?

Explain why this shows that $\vG$ is path-independent.

Since $D$ is open, there is a disc (perhaps very small) surrounding $(x,y)$ that is contained in $D$, so fix a point $(a,b)$ in that disc. Since $D$ is path-connected, there is a path $C_1$ from $(x_0,y_0)$ to $(a,b)$. Let $C_y$ be the line segment from $(a,b)$ to $(a,y)$ and let $C_x$ be the line segment from $(a,y)$ to $(x,y)$. (See Figure.) Rewrite $f(x,y)$ as a sum of $f(x_0,y_0)$ and line integrals along $C_1$, $C_y$, and $C_x$.

Notice that we can parametrize $C_y$ by $\langle a,t\rangle$ for $b\leq t\leq y$. Find a similar parametrization for $C_x$.

Use the parametrization from above to write $\int_{C_y}\vF\cdot d\vr$ and $\int_{C_x}\vF\cdot d\vr$ as single variable integrals in the manner of Section 5.4. Use the fact that $\vF(x,y) = \langle F_1(x,y),F_2(x,y)\rangle$ to express your integrals in terms of $F_1$ and $F_2$ without any dot products.

Rewrite your expression for $f(x,y)$ using a line integral along $C_1$ and the single variable integrals above.

Notice that your expression for $f(x,y)$ from the previous part only depends on $x$ as the upper limit of an single variable integral. Use the Second Fundamental Theorem of Calculus to calculate $f_x(x,y)$.

To calculate $f_y(x,y)$, we continue to consider a path $C_1$ from $(x_0,y_0)$ to $(a,b)$, but now let $L_x$ be the line segment from $(a,b)$ to $(x,b)$ and let $L_y$ be the line segment from $(x,b)$ to $(y,b)$. Modify the process you used to find $f_x(x,y)$ to find $f_y(x,y)$.

What can you conclude about the relationship between $\grad f$ and $\vF$? What does this tell you about $\vF$ beyond that it is path-independent and continuous?

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 $1\leq t\leq 2$.