formulas

$$\det(A)$$

$$\sin(2\theta) = 2 \cdot \sin(\theta) \cdot \cos(\theta)$$

$$\scriptsize \sin(a \pm b) = \sin(a) \cos(b) \pm \cos(a) \sin(b)$$

$$proj_{\vec{b}}(\vec{a})=\frac{\vec{a}\cdot\vec{b}}{\|\vec{b}\|^2}\vec{b}=(\vec{a}\cdot\hat{b})\hat{b}$$

$$\small \oint_C Pdx + Qdy = \iint_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dA$$

$$\frac{F}{A} = E\frac{\Delta L}{L}$$

$$F = -kx$$

$$\sinh(x) = \frac{e^x - e^{-x}}{2}$$

$$\int_C \vec{F} \cdot d\vec{r} = \iint_S \nabla \times \vec{F} \cdot d\vec{S}$$

$$\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$

$$\iiint_V (\nabla \cdot \vec{F}) dV = \oiint_S (\vec{F} \cdot \hat{n}) dS$$

$$\vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i$$

$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I$$

$$\oiint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$$

$$c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(\theta)$$

$$\frac{d}{dx}f(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

$$\sin^2(x) + \cos^2(x) = 1$$

$$\oint_C f(z)dz = 0$$

$$U = mgh$$

$$E_k = \frac{1}{2}mv^2$$

$$\nabla^2 f = \nabla \cdot \nabla f = 0$$

$$\int_a^b f(x) dx = F(b) - F(a)$$

$$a^2 + b^2 = c^2$$

$$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$

$$p = mv$$

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

$$F = ma$$

$$e^{i\pi} = -1$$

$$\small{\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}, \cos(x) = \frac{e^{ix} + e^{-ix}}{2}}$$

$$PV = nRT$$