vector calculus
Vector Calculus Product Rules
$$\nabla (fg) = f(\nabla g) + g(\nabla f)$$
Curl
$$\scriptsize \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\\\ F_x & F_y & F_z \end{vmatrix}$$
Divergence
$$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$
Gradient
$$\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}$$
Green's Theorem
$$\small \oint_C Pdx + Qdy = \iint_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dA$$
Stokes' Theorem
$$\int_C \vec{F} \cdot d\vec{r} = \iint_S \nabla \times \vec{F} \cdot d\vec{S}$$
Cross Product
$$\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$
Divergence Theorem
$$\iiint_V (\nabla \cdot \vec{F}) dV = \oiint_S (\vec{F} \cdot \hat{n}) dS$$
Dot Product
$$\vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i$$