We can use the divergence theorem to write Gauss’ law as follows:

$$ \begin{aligned} \int_S \vec E\cdot \text d\vec S&=\frac{1}{\epsilon_0}\int_V\rho\,\text dV \\ \int_V\vec\nabla\cdot\vec E\,\text dV&=\frac{1}{\epsilon_0}\int_V\rho\,\text dV \\ \vec \nabla \cdot \vec E&=\frac{\rho}{\epsilon_0} \end{aligned} $$

The continuity equation

$$ \frac{\rm d\rho}{\rm dt} + \boldsymbol{\nabla} \cdot {\boldsymbol{J}} = 0 $$

• where $\rho$ is the amount of a quantity $q$ (e.g. mass of fluid) per unit of volume and $J$ is the flux per unit time

Line integrals

$$ \int_C f\, \rm dl $$

• For a simple 2D curve we can write

$$ \begin{aligned} (\text dl)^2&=(\text dx)^2+(\text dy)^2 \\ \text dl&=\sqrt{1+\left(\frac{\text dy}{\text dt} \right)^2}\,\text dx \\ l&=\int_c\text dl=\int^{x_1}_{x_0}\sqrt{1+\left(\frac{\text dy}{\text dt} \right)^2}\,\text dx \end{aligned} $$

• If we add a time dependence, i.e. $x=g(t)$ and $y=h(t)$ then we get:

$$ l = \int_{t_0}^{t_1} \sqrt{\left(\frac{\rm dx}{\rm dt}\right)^2 + \left( \frac{\rm dy}{\rm dt}\right)^2}\,\rm dt $$

• Or in polar coordinates, i.e. $x=r\cos\theta$ and $y=r\sin\theta$ then we get:

$$ \rm dl = \sqrt{r^2 + \left(\frac{\text dr}{\text d\theta} \right)^2}\,\text d\theta $$

• In a more real example you could use this to find the mass of a line

$$ \int_C\rho_l(x,y)\,\text dl $$

Forces and work

$$ {\rm Work~done} = \int_C \vec{F} \cdot \text d \vec {l} $$

• If the path is a closed path (loop) the line integral is called the circulation and we denotate this with a contour integral $\oint\vec F\cdot \text d \vec l$

Steps to evaluate a vector line integral

• 🗒️ Define $\text d\vec l$ in the appropriate coordinate system

$$ \begin{aligned} \text {d}\vec l &= \text dx\,\hat{i} + \text dy\,\hat{j} \\ \text d \vec l &= \text dr\,\hat{r} + r\text d\theta\,\hat{\theta} \\ \end{aligned} $$