FoldUnfold Table of Contents The Divergence Theorem The Divergence Theorem We will now look at a counterpart to both Green's Theorem and Stoke's Theorem known as The Divergence Theorem. Theorem 1 (The Divergence Theorem): Let $E$ be a regular domain in $\mathbb{R}^3$ and let $S$ be the closed surface boundary of $E$ that is oriented outward and with unit normal field … [Read more...]

## Stokes’ Theorem Examples 2

FoldUnfold Table of Contents Stokes' Theorem Examples 2 Example 1 Example 2 Stokes' Theorem Examples 2 Recall from the Stokes' Theorem page that if $\delta$ is an oriented surface that is piecewise-smooth, and that $\delta$ is bounded by a simple, closed, positively oriented, and piecewise-smooth boundary curve $C$, and if $\mathbf{F} (x, y, z) = P(x, y, z) \vec{i} + Q(x, … [Read more...]

## Stokes’ Theorem Examples 1

FoldUnfold Table of Contents Stokes' Theorem Examples 1 Example Example 2 Stokes' Theorem Examples 1 Recall from the Stokes' Theorem page that if $\delta$ is an oriented surface that is piecewise-smooth, and that $\delta$ is bounded by a simple, closed, positively oriented, and piecewise-smooth boundary curve $C$, and if $\mathbf{F} (x, y, z) = P(x, y, z) \vec{i} + Q(x, … [Read more...]

## Stokes’ Theorem

FoldUnfold Table of Contents Stokes' Theorem Stokes' Theorem Recall from the Green's Theorem page that if $D$ is a regular and closed region in the $xy$-plane whose boundary is a positively oriented, piecewise smooth, simple, and closed curve $C$ and if $\mathbf{F}(x, y) = P(x, y) \vec{i} + Q(x, y) \vec{j}$ is a smooth vector field in $\mathbb{R}^2$ … [Read more...]

## Green’s Theorem Examples 2

FoldUnfold Table of Contents Green's Theorem Examples 2 Example 1 Example 2 Green's Theorem Examples 2 Recall from the Green's Theorem page that if $D$ is a regular closed region in the $xy$-plane and the boundary of $D$ is a positively oriented, piecewise smooth, simple, and closed curve, $C$ and if $\mathbb{F} (x, y) = P(x, y) \vec{i} + Q(x, y) \vec{j}$ is a smooth … [Read more...]

## Green’s Theorem Examples 1

FoldUnfold Table of Contents Green's Theorem Examples 1 Example 1 Example 2 Green's Theorem Examples 1 Recall from the Green's Theorem page that if $D$ is a regular closed region in the $xy$-plane and the boundary of $D$ is a positively oriented, piecewise smooth, simple, and closed curve, $C$ and if $\mathbb{F} (x, y) = P(x, y) \vec{i} + Q(x, y) \vec{j}$ is a smooth … [Read more...]

## Green’s Theorem

FoldUnfold Table of Contents Green's Theorem Other Ways to Write Green's Theorem Green's Theorem Recall that if $y = f(x)$ is a single variable function that is continuous on the interval $[a, b]$ and differentiable on $(a, b)$, then both parts of The Fundamental Theorem of Calculus Part 1 and The Fundamental Theorem of Calculus Part 2 has that: (1) \begin{align} \quad … [Read more...]

## Curl Identities

FoldUnfold Table of Contents Curl Identities Curl Identities Let $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ be a vector field on $\mathbb{R}^3$ and suppose that the necessary partial derivatives exist. Recall from The Divergence of a Vector Field page that the divergence of $\mathbf{F}$ can be computed with the following … [Read more...]

## Divergence Identities

FoldUnfold Table of Contents Divergence Identities Divergence Identities Let $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ be a vector field on $\mathbb{R}^3$ and suppose that the necessary partial derivatives exist. Recall from The Divergence of a Vector Field page that the divergence of $\mathbf{F}$ can be computed with the … [Read more...]

## The Curl Of Conservative Vector Fields

FoldUnfold Table of Contents The Curl Of Conservative Vector Fields The Curl Of Conservative Vector Fields Recall from the Conservative Vector Fields page that a vector field $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{j}$ on $\mathbb{R}^3$ is said to be conservative if there exists a potential function $\phi$ such that $\mathbf{F} = … [Read more...]