16.3 - Conservative Vector Fields

Abstract

Recall from 16.1 - Vector Fields that a vector field $\vec{F}$ is conservative if there exists a function $f$ such that $\vec{F}=\nabla f$. The function $f$ is called the potential function.

Example

$\vec{F}(x,y)=y\vec{i}+x\vec{j}$ is conservative. The potential function is $f=xy$, because $\nabla f=⟨y,x⟩=\vec{F}$.

Fundamental Theorem for Line Integrals

Let $\vec{F}$ be a conservative vector field, i.e. $\vec{F}=\nabla f$, where $f$ is differentiable and $\nabla f$ is continuous. Let $C$ be a path parameterized by $\vec{r}(t)$, $t\in [a,b]$.

Then:

$${\int }_C\vec{F}\cdot d\vec{r}={\int }_C\nabla f\cdot d\vec{r}=f(\vec{r}(b))-f(\vec{r}(a))$$

Remarks

1. This theorem says that to evalutate the line integral of a conservative vector field, we only need to evalutate the potential function at the end points of the path. For example, if $\vec{r}=⟨x(t),y(t)⟩$, $t\in [a,b]$, then:

$$\begin{array}{rl}{\int }_C\nabla f\cdot dr& =f(\vec{r}(b))-f(\vec{r}(a))\\ & =f(x(b),y(b))-f(x(a),y(a))\\ & =f(P_1)-f(P_0)\end{array}$$

Sketch

2. Compare with the Fundamental Theorem of Calculus:

$$\begin{array}{rl}{\int }_a^b{f}^{\prime }(x)dx& =f(b)-f(a)\\ {\int }_C\nabla f\cdot d\vec{r}& =f(P_1)-f(P_0)\end{array}$$

In both cases, we integrate a “derivative” by evaluating an “antiderivative” at end points.

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Example

Example

Evaluate ${\int }_C\vec{F}\cdot d\vec{r}$, where $\vec{F}$ is the vector field $\vec{F}=y\vec{i}+x\vec{j}$ and $C$ is: