Vector calculus (review)

Gradients

Suppose $u(x,y,z)$ is a scalar valued function. Gradient of $u$, denoted by $\nl(u)$, is defined as

$$\nl(u) = \left[\begin{array} {rrrr} \frac{\p u}{\p x} \\ \frac{\p u}{\p y} \\ \frac{\p u}{\p z} \end{array} \right]$$

Divergence

Suppose $F(x,y,z) = F_1 \mathbf{i} + F_2 \mathbf{j} + F_2 \mathbf{k}$ is a vector valued function. Divergence of $F$, denoted by $\text{div}(F)$ of $\nl \cdot (F)$ is defined as

$$\textrm{div} (F) = \frac{\p F_1}{\p x} + \frac{\p F_2}{\p y} + \frac{\p F_3}{\p z}$$

Laplacian

Laplacian of a scalar valued function $u(x,y,z)$, denoted by $\Dt(u)$, is defined as

$$\Dt (u) = \textrm{div} (\nl u) = \frac{\p^2 u}{\p x^2} + \frac{\p^2 u}{\p y^2} + \frac{\p^2 u}{\p z^2}$$

Divergence Theorem

Recall divergence theorem from vector calculus. If $\vec{F}$ is a vector field, $V$ is a given volume, $\p V$ is the the boundary of $V$, then the flux of $\vec{F}$ across $\p V$ is equal to the volume integration of $\divc(\vec{F})$.

$$\int_{\p V} \vec{F} \cdot \hat{n} \, dS = \int_V \divc (\vec{F}) d V$$