Multivariable calculus and vector analysis

An example of a conservative vector field

If a vector field $\mathbf{F}$ is a gradient field, meaning $\mathbf{F}=\nabla f$ for some scalar-valued function $f$, then we can compute the line integral of $\mathbf{F}$ along a curve $C$ from some point $\mathbf{a}$ to some other point $\mathbf{b}$

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}=f\left(\mathbf{b}\right)-f\left(\mathbf{a}\right).& & \hfill \end{array}$$

Note that this integral does not depend on the entire curve $C$; it depends on only the endpoints $\mathbf{a}$ and $\mathbf{b}$. If we replaced $C$ by another curve with the same endpoints, the integral would be unchanged. Hence $\mathbf{F}$ is path-independent (or conservative).

As an example, consider $f\left(x,y\right)=x{y}^2$, and let $\mathbf{F}\left(x,y\right)=\nabla f\left(x,y\right)=\left(y^2,2xy\right)$. We know that $\mathbf{F}$ must be conservative.

What is ${\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}$ where $C$ is a path $\mathbf{c}\left(t\right)=\left(t^2,t^3\right)$ for $1\le t\le 2$? The starting point is $\mathbf{a}=\mathbf{c}\left(1\right)=\left(1,1\right)$, and the ending pont is $\mathbf{b}=\mathbf{c}\left(2\right)=\left(4,8\right)$. Hence the integral must be

$$\begin{array}{llll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}& =f\left(4,8\right)-f\left(1,1\right)& \hfill & \\ \hfill & =4\left(8^2\right)-1\left(1^2\right)=256-1=255& \hfill & \end{array}$$

We could also compute ${\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}$ the direct way using the parametrization $\mathbf{c}\left(t\right)$:

$$\begin{array}{llll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}& ={\int \; <!--nolimits\; -->}_a^b\mathbf{F}\left(\mathbf{c}\left(t\right)\right)\cdot {\mathbf{c}}^{\text{'}}\left(t\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_1^2\mathbf{F}\left(t^2,t^3\right)\cdot \left(2t,3t^2\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_1^2\left({\left(t^3\right)}^2,2t^2{t}^3\right)\cdot \left(2t,3t^2\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_1^{2}8t^{7}dt={\left.t^8\right|}_1^2=256-1=255,& \hfill & \end{array}$$

which agrees with the first answer.

We should get the same answer for any path from $\left(1,1\right)$ to $\left(4,8\right)$. Since $\left(4,8\right)-\left(1,1\right)=\left(3,7\right)$, we let the curve $B$ be the straight line path parametrized by $\mathbf{p}\left(t\right)=\left(1,1\right)+t\left(3,7\right)=\left(1+3t,1+7t\right)$ for $0\le t\le 1$. (Note that this is not a reparametrization of $C$. The curve $C$ was not a straight line, so $B$ is a completely different curve.)

The integral along $B$ is

$$\begin{array}{llll}\hfill {\int \; <!--nolimits\; -->}_B\mathbf{F}\cdot d\mathbf{s}& ={\int \; <!--nolimits\; -->}_0^1\mathbf{F}\left(\mathbf{p}\left(t\right)\right)\cdot {\mathbf{p}}^{\text{'}}\left(t\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^1\mathbf{F}\left(1+3t,1+7t\right)\cdot \left(3,7\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^1\left({\left(1+7t\right)}^2,2\left(1+3t\right)\left(1+7t\right)\right)\cdot \left(3,7\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^1\left(1+14t+49t^2,2+20t+42t^2\right)\cdot \left(3,7\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^1\left(17+182t+441t^2\right)dt& \hfill & \\ \hfill & ={\left.\left(17t+91t^2+147t^3\right)\right|}_0^1=17+91+147=255.& \hfill & \end{array}$$

Indeed, we got the same answer again.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

5a. Green's theorem

5b. The theorem for conservative vector fields

• Path-independent or conservative vector fields
• An example of a conservative vector field
• Path-independence implies no circulation
• Understanding the conditions for path-independence
• Finding the potential function
• Path-independence example in three dimensions

5c. Stokes' theorem

5d. The divergence theorem

6. Review material