Line Integral of Conservative Vector Fields

Extension of Line Integrals.

Recall $\vec{F}$ is a conservative vector field iff $\vec{F}=\nabla f$ for some function f.

If $\vec{F}$ is a conservative vector field, the path is indpendent — it doesn’t matter which path we move from a to b because the same amount of wokr is done

Fundamental Theorem of Calculus Part 1 §

$$\begin{array}{rl}{\int }_a^{b}f(x)dx& =F(x){|}_b^a=F(b)-F(a)\end{array}$$

For some $f(x,y,z)$

$$\vec{F}=\nabla f=<f_x,f_y,f_z>$$

Fundamental Theorem of Line Integral §

$$\begin{array}{rl}W& ={\int }_a^b\vec{F}dr={\int }_a^b\nabla fdr=f(x,y,z){\mid }_a^b\\ & =f(b)-f(a)\end{array}$$

Note: Line integral of conservative vector field can be done without knowing the path $\vec{r}(t)$

Examples §

Example 1 §

Find the work done by the vector field

$$\vec{F}(x,y)=<x{e}^{2y},x^2{e}^{2y}>\text{from }(0,0)\to (-1,1)$$

Solution §

First Option §

Test if conservative:

$$\begin{array}{r}{f}_{xy}=2x{e}^{2y}\\ f_{yx}=2x{e}^{2y}\end{array}$$

Find potential function:

$$\begin{array}{rl}f(x,y)& =\int x{e}^{2y}dx=\frac{x^2}{2}{e}^{2y}+g(y)\\ & =\int x^2{e}^{2y}dy=\frac{x^2}{2}{e}^{2y}+g(x)\\ & ⟹f(x,y)=\frac{x^2}{2}{e}^{2y}+C\end{array}$$

Apply fundamental theorem of Line Integrals:

$$\begin{array}{rl}W& =\int \nabla fdr\\ & =f(x,y){|}_{(0,0)}^{(-1,1)}\\ & =\frac{x^2}{2}{e}^{2y}+C{|}_{(0,0)}^{(-1,1)}\\ & =\frac{1}{2}{e}^2\end{array}$$

Second Option §

Find the path from $\vec{r}(t)$ from $(0,0)\to (-1,1)$:

Since we know it’s conservative, we can assume that vector is linear

$$\begin{array}{rl}\vec{r}(t)& =<-t,t>\\ {\vec{r}}^{\prime }(t)& =<-1,1>\\ W& ={\int }_0^1\vec{F}(t)\cdot {\vec{r}}^{\prime }(t)dt\\ & ={\int }_0^{1}t{e}^{2t}+t^2{e}^{2t}dt\\ & =3.695\end{array}$$

Equivalent Condition §

iF $\vec{F}=<P,Q,R>$ have first partial derivative on an open connected region $R$, and $C$ is piecewise smoth curve then the following are true:

1. $\vec{F}$ is conservative iff $\nabla f=\vec{F}$ for some $f$
2. ${\int }_C\vec{F}$ is independent path
3. ${\int }_C\vec{F}dr=\vec{0}$ for every closed curve C in R