Differentiation of Real Vector Fields

Divergence

Definition: Divergence

Let $\mathbf{F}:\mathcal{D}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be a real vector field which is differentiable at $\mathbf{p}\in \mathcal{D}$ and let $F_1,\cdots F_n$ be the component functions of $\mathbf{F}$.

The divergence of $\mathbf{F}$ at $\mathbf{p}$ is a number defined using the partial derivatives of $F_1,\cdots F_n$ with respect to Cartesian coordinates as follows:

$$\sum _{i=1}^n\frac{\partial F_i}{\partial x_i}(\mathbf{p})=\frac{\partial F_1}{\partial x_1}(\mathbf{p})+\frac{\partial F_2}{\partial x_2}(\mathbf{p})+\cdots +\frac{\partial F_n}{\partial x_n}(\mathbf{p})$$

NOTATION

$$\nabla \cdot \mathbf{F}(\mathbf{p})\mathrm{div}\mathbf{F}(\mathbf{p})$$

Tip: Divergence as Dot Product

The formula for divergence can be easily remembered as a “dot product” between ${\left[\begin{array}{ccc}\frac{\partial }{\partial x_1}& \cdots & \frac{\partial }{\partial x_n}\end{array}\right]}^{\mathsf{T}}$ and ${\left[\begin{array}{ccc}{F}_1& \cdots & F_n\end{array}\right]}^{\mathsf{T}}$.

Despite not being apparent from this definition, the divergence of $\mathbf{F}$ at $\mathbf{p}$ gives us an idea of how much $\mathbf{F}$ tends to point towards or away from $\mathbf{p}$, i.e. how much $\mathbf{F}$ tends to “diverge” away from $\mathbf{p}$. A positive divergence indicates that $\mathbf{F}$ tends to point away from $\mathbf{p}$, while a negative divergence is an indication of $\mathbf{F}$ tendency to point towards $\mathbf{p}$. If $\nabla \cdot \mathbf{F}(\mathbf{p})$ is zero, then $\mathbf{F}$ tends to point as much towards $\mathbf{p}$ as away from it.

Theorem: Linearity of the Divergence

Let $\mathbf{u}:{\mathcal{D}}_{\mathbf{u}}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $\mathbf{v}:{\mathcal{D}}_{\mathbf{v}}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be real vector fields.

If $\mathbf{u}$ and $\mathbf{v}$ are both differentiable at $\mathbf{p}\in {\mathcal{D}}_{\mathbf{u}}\cap {\mathcal{D}}_{\mathbf{v}}$, then divergence has the following property for all $\lambda ,\mu \in \mathbb{R}$:

$$\mathrm{div}(\lambda \mathbf{u}+\mu \mathbf{v})(\mathbf{p})=\lambda \mathrm{div}\mathbf{u}(\mathbf{p})+\mu \mathrm{div}\mathbf{v}(\mathbf{p})$$

PROOF

TODO

Curl

Definition: Curl

Let $\mathbf{F}:\mathcal{D}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be a real vector field which is differentiable at $\mathbf{p}\in \mathcal{D}$ and let $F_1,F_2,F_3$ be the component functions of $\mathbf{F}$.

The curl of $\mathbf{F}$ at $\mathbf{p}$ is a real vector defined via the partial derivatives of $F_1,F_2,F_3$ as follows:

$$\left[\begin{array}{c}\frac{\partial F_3}{\partial y}(\mathbf{p})-\frac{\partial F_2}{\partial z}(\mathbf{p})\\ \frac{\partial F_1}{\partial z}(\mathbf{p})-\frac{\partial F_3}{\partial x}(\mathbf{p})\\ \frac{\partial F_2}{\partial x}(\mathbf{p})-\frac{\partial F_1}{\partial y}(\mathbf{p})\end{array}\right]$$

NOTATION

$$\nabla \times \mathbf{F}(\mathbf{p})\mathrm{curl}\mathbf{F}(\mathbf{p})\mathrm{rot}\mathbf{F}(\mathbf{p})$$

Tip: Curl as Cross Product

The formula for curl can be remembered easily by imagining it as the “cross product” between ${\left[\begin{array}{ccc}\frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\end{array}\right]}^{\mathsf{T}}$ and ${\left[\begin{array}{ccc}{F}_1& F_2& F_3\end{array}\right]}^{\mathsf{T}}$. This shortcut is reflected in the notation $\nabla \times \mathbf{F}$.

The curl of $\mathbf{F}$ at $\mathbf{p}$ is indicative of $\mathbf{F}$‘s tendency to swirl around $\mathbf{p}$. The higher the magnitude of $\nabla \times \mathbf{F}(\mathbf{p})$, the more $\mathbf{F}$ tends to rotate around $\mathbf{p}$.

Theorem: Divergence of Curl

Let $\mathbf{F}:\mathcal{D}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be a real vector field.

If $\mathbf{F}$ is differentiable at $\mathbf{p}\in \mathcal{D}$, then the divergence of its curl there is zero.

$$\mathrm{div}\mathrm{curl}\mathbf{F}(\mathbf{p})=0$$

PROOF

TODO