<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/8260e212-b29d-43aa-b9ae-43d0c9bbe73a/vector.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/8260e212-b29d-43aa-b9ae-43d0c9bbe73a/vector.png" width="40px" /> Vector:
$$ \vec v= v_1 \,\hat i + v_2 \,\hat j + v_3 \,\hat k \equiv \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} \equiv (v_1, v_2, v_3) $$
Vectors need to be written in terms of an orthonormal basis
$$ \begin{aligned} \hat i &=\hat x=\hat e_x =\hat e_1=\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \\ \hat j &= \hat y = \hat e_y = \hat e_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \\ \hat k &= \hat z = \hat e_z = \hat e_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \end{aligned} $$
Unit vectors are made using:
$$ \hat v=\frac{\vec v}{|\vec v|} $$
Magnitude of a vector is:
$$ v=|\vec v|=\sqrt{v_1^2+v_2^2+v_3^2} $$
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<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/614e5f9c-6f86-495b-87a0-34f4076d5aaa/scalar_field.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/614e5f9c-6f86-495b-87a0-34f4076d5aaa/scalar_field.png" width="40px" /> Scalar field:
$\phi(\vec r)$ is a quantity with magnitude at position $\vec r=(x,y,z)$
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<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/39137f22-ab77-41c4-a7ae-05f6c867515f/vector_field.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/39137f22-ab77-41c4-a7ae-05f6c867515f/vector_field.png" width="40px" /> Vector field:
$\vec v (\vec r)$ has magnitude and direction at $\vec r$
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Dot product
$$ \vec a\cdot \vec b= a_1b_1+ a_2b_2+a_3 b_3 = \sum^3_{i=1} a_ib_i=ab\cos(\theta) $$
Cross (vector) product
$$ \vec a\times \vec b=\left | \begin{matrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{matrix}\right |=ab\,\hat n \sin(\theta)=-\vec b \times \vec a $$
Combinations
$$ \begin{aligned} \vec a \cdot (\vec b \times \vec c)& = \left | \begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix}\right | =\vec c\cdot(\vec a \times \vec b) = \vec b \cdot (\vec c \times \vec a) \\ \vec a\times (\vec b\times \vec c)&=(\vec a \cdot \vec c)\vec b- (\vec a \cdot \vec b) \vec c \\ (\vec a\times \vec b)\cdot (\vec c\times \vec d)&=(\vec a\cdot \vec c)(\vec b\cdot \vec d)-(\vec a \cdot \vec d)(\vec b\cdot \vec c) \end{aligned} $$
$$ \begin{aligned} \vec \nabla \Phi&=\frac{\partial \Phi}{\partial x}\, \hat i+\frac{\partial \Phi}{\partial y}\, \hat j+\frac{\partial \Phi}{\partial z}\, \hat k=\sum^3_{i=1} \frac{\partial \Phi}{\partial x_i} \hat e_i \\ \nabla^2\Phi&=\frac{\partial^2 \Phi}{\partial x^2}+\frac{\partial^2 \Phi}{\partial y^2}+\frac{\partial^2 \Phi}{\partial z^2}
\end{aligned} $$
$$ \begin{aligned} \vec\nabla\cdot \vec v&= \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z} \\ \vec \nabla \times \vec v&= \left | \begin{matrix} \hat i & \hat j & \hat k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ v_x & v_y & v_z \end{matrix} \right | \\ \nabla^2\vec v&=\nabla^2 v_x \hat i+\nabla^2 v_y \hat j+\nabla^2 v_z \hat k \end{aligned} $$
$$ \begin{aligned} \vec\nabla\times\vec\nabla\Phi&=0 \\ \vec \nabla \cdot ( \vec \nabla \times \vec v)&= 0 \\ \vec \nabla \times(\vec \nabla \times \vec v)&=\vec \nabla(\vec \nabla \cdot \vec v)-\nabla^2\vec v
\end{aligned} $$
Plane polar coordinates
$$ \hat r=\cos(\theta)\,\hat i+\sin(\theta) \, \hat j \quad \quad \hat \theta=-\sin(\theta)\, \hat i+\cos(\theta) \, \hat j \\ \text dx \text dy\to r\, \text dr\text d\theta\;\; \\ \vec r=r\hat r=\begin{pmatrix} r\cos\theta \\ r \sin\theta \end{pmatrix} $$
Cylindrical polar coordinates
$$ (\hat i, \hat j, \hat k)\to (\hat r, \hat \theta , \hat z) \quad \text dx \text dy \text dz\to r\,\text dr\text d\theta\text dz=\text dV \quad r=r\hat r+ z\hat z=\begin{pmatrix} r\cos\theta \\ r\sin\theta \\ z \end{pmatrix} \\ \begin{aligned} \vec \nabla \Phi & =\frac{\partial \Phi}{\partial r} \hat{r}+\frac{1}{r} \frac{\partial \Phi}{\partial \theta} \hat{\theta}+\frac{\partial \Phi}{\partial z} \hat{z} \\ \vec \nabla \cdot \vec{v} & =\frac{1}{r} \frac{\partial}{\partial r}\left(r v_r\right)+\frac{1}{r} \frac{\partial v_\theta}{\partial \theta}+\frac{\partial v_z}{\partial z} \\ \vec \nabla \times \vec{v} & =\left(\frac{1}{r} \frac{\partial v_z}{\partial \theta}-\frac{\partial v_\theta}{\partial z}\right) \hat{r}+\left(\frac{\partial v_r}{\partial z}-\frac{\partial v_z}{\partial r}\right) \hat{\theta}+\frac{1}{r}\left(\frac{\partial r v_\theta}{\partial r}-\frac{\partial v_r}{\partial \theta}\right) \hat{z} \\ \nabla^2 \vec\Phi & =\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \vec \Phi}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 \vec \Phi}{\partial \theta^2}+\frac{\partial^2 \vec \Phi}{\partial z^2} \end{aligned} $$
Spherical polar coordinates
$$ \vec v=v_r\hat r+ v_\theta \hat \theta +v_\phi \hat \phi \qquad \text dV=r^2\sin\theta \text dr\text d\theta \text d\phi \\ \begin{aligned}\nabla \Phi= & \frac{\partial \Phi}{\partial r} \hat{r}+\frac{1}{r} \frac{\partial \Phi}{\partial \theta} \hat{\theta}+\frac{1}{r \sin \theta} \frac{\partial \Phi}{\partial \phi} \hat{\phi} \\\vec \nabla \cdot \vec{v}= & \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 v_r\right)+\frac{1}{r \sin \theta} \frac{\partial}{\partial \theta}\left(\sin (\theta)\, v_\theta\right)+\frac{1}{r \sin \theta} \frac{\partial v_\phi}{\partial \phi} \\\vec \nabla \times \vec{v}= & \frac{1}{r \sin \theta}\left(\frac{\partial}{\partial \theta}\left(\sin (\theta)\, v_\phi\right)-\frac{\partial v_\theta}{\partial \phi}\right) \hat{r}+\frac{1}{r}\left(\frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi}-\frac{\partial}{\partial r}\left(r v_\phi\right)\right) \hat{\theta} \\& +\frac{1}{r}\left(\frac{\partial}{\partial r}\left(r v_\theta\right)-\frac{\partial v_r}{\partial \theta}\right) \hat{\phi} \\\nabla^2 \vec \Phi= & \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial \vec \Phi}{\partial r}\right)+\frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \vec \Phi}{\partial \theta}\right)+\frac{1}{r^2 \sin ^2 \theta} \frac{\partial^2 \vec \Phi}{\partial \phi^2} \end{aligned} $$
Visual representations of coordinate systems:
$$ \int^b_a \frac{\text df}{\text dx}\,\text dx=f(b)-f(a)
$$
