16.2 Videos Guide

16.2a

Expressions below are given in $\mathbb{R}^{3}$. Analogous expressions exist for functions in $\mathbb{R}^{2}$ by simply leaving off the $z$- or $\mathbf{k}$-component

• Line integral of a scalar function $f$ with respect to arc length: over a curve $C$ parameterized by $\mathbf{r}(t)=\left\langle x(t),\ y(t),\ z(t)\right\rangle$

  • $\int^{\ }_{C}{f(x,\ y,\ z)\ ds}=\int^{b}_{a}{f\left(x(t),\ y(t),\ z(t)\right)\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}+\left(\frac{dz}{dt}\right)^{2}}\ dt}=\int^{b}_{a}{f\left(\mathbf{r}(t)\right)\left|\mathbf{r}'(t)\right|\ dt}$

• Note that $ds=\left|\mathbf{r}'(t)\right|\ dt$

• Line integral with respect to $x$

  • $\int^{\ }_{C}{f(x,\ y,\ z)\ dx}=\int^{b}_{a}{f\left(x(t),y(t),\ z(t)\right)\ x'(t)\ dt}$ (likewise for $y$ and $z$)

Exercises:

16.2b

• Evaluate the line integral, where $C$ is the given curve.

  • $\int^{\ }_{C}{xe^{y}\ ds}$, $C$ is the line segment from $(2,\ 0)$ to $(5,\ 4)$

16.2c

• $\int^{\ }_{C}{x^{2}\ dx+y^{2}\ dy}$, $C$ consists of the arc of the circle $x^{2}+y^{2}=4$ from $(2,\ 0)$ to $(0,\ 2)$ followed by the line segment from $(0,\ 2)$ to $(4,\ 3)$

16.2d

Expressions below are given in $\mathbb{R}^{3}$. Analogous expressions exist for functions in $\mathbb{R}^{2}$ by simply leaving off the $z$- or $\mathbf{k}$-component

• Line integral of a vector field $\mathbf{F}(x,\ y,\ z)=P\mathbf{i}+Q\mathbf{j}+R\mathbf{j}$ over a curve $C$ parameterized by
  $$\mathbf{r}(t)=\left\langle x(t),\ y(t),\ z(t)\right\rangle$$

  • $\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}=\int^ {}_{C}{\mathbf{F}\left(\mathbf{r}(t)\right)\cdot\mathbf{r}'(t)\ dt}=\int^{\ }_{C}{\mathbf{F}\cdot\mathbf{T\ }ds}=\int^{\ }_{C}{P\ dx+Q\ dy+R\ dz}$

  • $\left(\mbox{Thi}s\ \mbox{really}\ \mbox{means}\ \int^ {}_{C}{P\ dx+\int^ {}_{C}{Q\ dy}}+\int^ {}_{C}{R\ dz}\right)$

  • Note that $d\mathbf{r}=\mathbf{r}'(t)dt$

  • Also note that we generally parameterize $P$, $Q$, and $R$

• Final notes about orientation

  • $\int^{\ }_{-C}{f(x,\ y,\ z)\ dx}=-\int^{\ }_{C}{f(x,\ y,\ z)\ dx}$ (and likewise for $y$ and $z$)

and $\int^{\ }_{-C}{\mathbf{F}\cdot d\mathbf{r}}=-\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}$

but $\int^{\ }_{-C}{f(x,\ y,\ z)\ ds}=\int^{\ }_{C}{f(x,\ y,\ z)\ ds}$

Exercises:

16.2e

• Evaluate the line integral $\int^ {}_{C}{\mathbf{F}\cdot d\mathbf{r}}$, where $C$ is given by the vector function $\mathbf{r}(t)$.
  $\mathbf{F}(x,\ y,\ z)=\left(x+y^{2}\right)\mathbf{\ i}+xy\ \mathbf{j}+(y+z)\ \mathbf{k}$, $\mathbf{r}(t)=t^{2}\ \mathbf{i}+t^{3}\mathbf{\ j}-2t\ \mathbf{k}$, $0\leq t\leq2$

16.2f

• Find the work done by the force field $\mathbf{F}(x,\ y)=x^{2}\ \mathbf{i}+ye^{x}\ \mathbf{j}$ on a particle that moves along the parabola $x=y^{2}+1$ from $(1,\ 0)$ to $(2,\ 1)$.