FoldUnfold Table of Contents Properties of The Divergence and Curl of a Vector Field Properties of The Divergence and Curl of a Vector Field 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 … [Read more...]

## The Divergence and Curl of a Vector Field In Two Dimensions

FoldUnfold Table of Contents The Divergence and Curl of a Vector Field In Two Dimensions Example 1 Example 2 Example 3 Example 4 The Divergence and Curl of a Vector Field In Two Dimensions From The Divergence of a Vector Field and The Curl of a Vector Field pages we gave formulas for the divergence and for the curl of a vector field $\mathbf{F}(x, y, z) = P(x, y, z) … [Read more...]

## The Curl of a Vector Field Examples 1

FoldUnfold Table of Contents The Curl of a Vector Field Examples 1 Example 1 Example 2 Example 3 The Curl of a Vector Field Examples 1 Recall from The Curl of a Vector Field page that if $\mathbf{F} (x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a vector field on $\mathbb{R}^3$ and the appropriate partial derivatives of $P$, $Q$, and $R$ … [Read more...]

## The Curl of a Vector Field

FoldUnfold Table of Contents The Curl of a Vector Field Example 1 Example 2 The Curl of a Vector Field Recall from The Divergence of a Vector Field page that if $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a vector field on $\mathbb{R}^3$ and the partial derivatives $\frac{\partial P}{\partial x}$, $\frac{\partial Q}{\partial y}$ … [Read more...]

## The Divergence of a Vector Field Examples 2

FoldUnfold Table of Contents The Divergence of a Vector Field Examples 2 Example 1 Example 2 The Divergence of a Vector Field Examples 2 Recall from The Divergence of a Vector Field page that if $\mathbf{F} (x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a vector field on $\mathbb{R}^3$ then the divergence of $\mathbf{F}$ is given by the … [Read more...]

## The Divergence of a Vector Field Examples 1

FoldUnfold Table of Contents The Divergence of a Vector Field Examples 1 Example 1 Example 2 Example 3 The Divergence of a Vector Field Examples 1 Recall from The Divergence of a Vector Field page that if $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a vector field on $\mathbb{R}^3$ and if $\frac{\partial P}{\partial x}$, … [Read more...]

## The Divergence of a Vector Field

FoldUnfold Table of Contents The Divergence of a Vector Field Example 1 Example 2 The Divergence of a Vector Field Recall that if $w = f(x, y, z)$ is a three variable real-valued function, then the gradient of $f$ denoted $\nabla f$ is given by: (1) \begin{align} \quad \nabla f(x, y, z) = \left ( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial … [Read more...]

## Surface Integrals Review

FoldUnfold Table of Contents Surface Integrals Review Surface Integrals Review We will now review some of the recent content regarding surface integrals. On the Parametric Surfaces we began to look at parametric surfaces. We saw that for $u$ and $v$ are parameters that we can define some neat surfaces as $\vec{r}(u, v) = (x(u, v), y(u, v), z(u, v))$ for $a ≤ u ≤ b$ and … [Read more...]

## Surface Integrals of Vector Fields Examples 2

FoldUnfold Table of Contents Surface Integrals of Vector Fields Examples 2 Example 1 Example 2 Surface Integrals of Vector Fields Examples 2 Recall from the Surface Integrals of Vector Fields page that if $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a continuous vector field and that if $\delta$ is a smooth orientable surface … [Read more...]

## Surface Integrals of Vector Fields Examples 1

FoldUnfold Table of Contents Surface Integrals of Vector Fields Examples 1 Example 1 Example 2 Surface Integrals of Vector Fields Examples 1 Recall from the Surface Integrals of Vector Fields page that if $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a continuous vector field and that if $\delta$ is a smooth orientable surface … [Read more...]