 Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q. 15

Expert-verified Found in: Page 1140 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Given an integral of the form ${\int }_{c}\mathbit{F}\mathbf{·}d\mathbit{r}$, what considerations would lead you to evaluate the integral with Stokes’ Theorem?

If right-hand side of Stokes' Theorem is used, the result obtained is the same with a single area integral.

See the step by step solution

## Step 1. Given Information

An integral of the form ${\int }_{C}\mathbit{F}·d\mathbit{r}$.

## Step 2. Stokes' Theorem

"Let $S$ be an oriented, smooth or piecewise-smooth surface bounded by a curve$C$. Suppose that $\mathbit{n}$ is an oriented unit normal vector of $S$and $C$ has a parametrization that traverses $C$ in the counterclockwise direction with respect to$\mathbit{n}$. If a vector field$\mathbit{F}\left(x,y,z\right)={\mathbit{F}}_{\mathbf{1}}\left(x,y,z\right)\mathbit{i}+{\mathbit{F}}_{\mathbf{2}}\left(x,y,z\right)\mathbit{j}+{\mathbit{F}}_{\mathbf{3}}\left(x,y,z\right)\mathbit{k}$ is defined on $S,$ then, ${\int }_{c}\mathbit{F}\left(x,y,z\right)·d\mathbit{r}=\int {\int }_{S}curl\mathbit{F}\left(x,y,z\right)·\mathbit{n}dS"$.

## Step 3. Understading the orientation

If $S$ is an oriented, smooth or piecewise-smooth surface bounded by a curve $C$, Stokes' Theorem relates a line integral of a vector field around the boundary curve $C$ to a surface integral of the curl of the vector field. When the line integral has piecewise-continuous boundary, for example, if the boundary curve $C$ is a rectangle or triangle , it requires the evaluation of several smooth pieces.If you use the right-hand side of Stokes' Theorem, you obtain the same result with a single area integral. ### Want to see more solutions like these? 