16.3 Videos Guide

16.3a

Theorem (statement and proof):

• Fundamental Theorem for Line Integrals: If $C$ is a smooth curve and $f$ is a differentiable function whose gradient $\mathbf{\nabla}f$ is continuous on $C$, then

$\int^{\ }_{C}{\mathbf{\nabla}f\cdot d\mathbf{r}}=f\left(\mathbf{r}(b)\right)-f\left(\mathbf{r}(a)\right)$, where $\mathbf{r}(t),\ \ \ a\leq t\leq b$ describes $C$

$=f\left(x_{2},\ y_{2},\ z_{2}\right)-f\left(x_{1},\ y_{1},\ z_{1}\right)$ (an analogous expression exists for the $\mathbb{R}^{2}$ case)

16.3b

• Description of path independence

  • $\int^{\ }_{c_{1}}{\mathbf{F}\cdot d\mathbf{r}}=\int^{\ }_{c_{2}}{\mathbf{F}\cdot d\mathbf{r}}$ for any two paths $C_{1}$ and $C_{2}$ that connect the same two points

16.3c

Theorems (statement and proof):

• $\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}=0$ for all closed paths $\Longleftrightarrow\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}$ is path independent

$\Longrightarrow\ \ \mathbf{F}$ is a conservative vector field (this means there exists a potential function $f$ of $\mathbf{F}$)

$\Longrightarrow\ \ \frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}$

Exercises:

16.3d

• Find a function $f$ such that $\mathbf{F}=\mathbf{\nabla}f$ and (b) use part (a) to evaluate $\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}$ along the given curve $C$.

  • $\mathbf{F}(x,\ y)=\left(3+2xy^{2}\right)\ \mathbf{i}+2x^{2}y\ \mathbf{j}$,
    $C$ is the arc of the hyperbola $y=\frac{1}{x}$ from $(1,\ 1)$ to $\left(4,\frac{1}{4}\right)$

16.3e

• $\mathbf{F}(x,\ y,\ z)=\left(y^{2}z+2xz^{2}\right)\ \mathbf{i}+2xyz\ \mathbf{j}+\left(xy^{2}+2x^{2}z\right)\ \mathbf{k}$,
  $C$: $x=\sqrt{t}$, $y=t+1$, $z=t^{2}$, $0\leq t\leq1$

16.3f

• Law of Conservation of Energy (statement and proof)