Vector calculus

$f=\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]$

Nabla operator

$\nabla =\left[\begin{array}{c}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{array}\right]$

Gradient

$\nabla f=\left[\begin{array}{c}\frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y}\\ \frac{\partial f}{\partial z}\end{array}\right]$

Divergence

$\nabla \cdot f=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}$

Curl

$\nabla \times f=\left[\begin{array}{c}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{array}\right]\times \left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]=\left[\begin{array}{c}\frac{\partial f_z}{\partial y}-\frac{\partial f_y}{\partial z}\\ \frac{\partial f_x}{\partial z}-\frac{\partial f_z}{\partial x}\\ \frac{\partial f_y}{\partial x}-\frac{\partial f_x}{\partial y}\end{array}\right]$

Divergence theorem (Gauss theorem)

${\iiint }_{\Omega }(\nabla \cdot F)\cdot dV={\iint }_{\partial \Omega }F\cdot Nds$

Stokes theorem

Line integrals

There are three types of line integrals.

1. Line integral over a scalar field ${\int }_a^{b}f(x(t))\cdot ||x^{\prime }(t)||dt$
2. Line integral over a vector field (Work) ${\int }_{\Gamma }F\cdot ds={\int }_a^{b}F(x(t))\cdot \frac{x^{\prime }(t)}{||x^{\prime }(t)||}\cdot ||x^{\prime }(t)||={\int }_a^{b}F(x(t))\cdot x^{\prime }(t)dt$ Where $\frac{x^{\prime }(t)}{||x^{\prime }(t)||}$ is the unit normal vector, of $\vec{x}$.
3. Line integral over a complex function ${\int }_{\Gamma }fdz={\int }_a^{b}f(z(t))\cdot z^{\prime }(t)dt$