Vector Field | Sahithyan's Notes

A vector field in $\mathbb{R}^n$ is a function that assigns a vector to each point in its domain $D \subseteq \mathbb{R}^n$ ,

\boldsymbol{F}(x,y,z) = f(x,y,z)\hat{i} + g(x,y,z)\hat{j} + h(x,y,z)\hat{k}

Here $f$ , $g$ , $h$ are real-valued component functions of $\boldsymbol{F}$ on $D$ .

$\boldsymbol{F}$ is continuous iff each component function is continuous. $\boldsymbol{F}$ is differentiable iff each component function is differentiable.

Inverse-Square Field

Let $\boldsymbol{r} \ne \boldsymbol{0}$ be the position vector of a point and $c$ a constant. A vector field of the below form is an inverse-square field.

\boldsymbol{F}(\boldsymbol{r}) = \frac{c}{\lVert \boldsymbol{r} \rVert^3}\boldsymbol{r}

For example, the electric field given by Coulomb’s law is an inverse-square field

Conservative Field

A vector field $\boldsymbol{F}$ in 2-space or 3-space is conservative on a region $D$ iff it is the gradient field of some scalar function $\phi$ defined on $D$ ,

\boldsymbol{F} = \nabla \phi

$\phi$ is a potential function, or scalar potential, of $\boldsymbol{F}$ .

Divergence

Let $\boldsymbol{F}(x,y,z) = f\hat{i} + g\hat{j} + h\hat{k}$ . The divergence of $\boldsymbol{F}$ is

\text{div}\,\boldsymbol{F} = \nabla \cdot \boldsymbol{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z}

$\text{div}\,\boldsymbol{F}$ is a scalar field.

Curl

Let $\boldsymbol{F}(x,y,z) = f\hat{i} + g\hat{j} + h\hat{k}$ . The curl of $\boldsymbol{F}$ is

\text{curl}\,\boldsymbol{F} = \nabla \times \boldsymbol{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\[4pt] \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\[4pt] f & g & h \end{vmatrix}

$\text{curl}\,\boldsymbol{F}$ is a vector field.