Vectors

1.

In 2 dimensions, $\vec{r} = x\hat{i} + y\hat{j}$ where $\hat{i}$ and $\hat{j}$ are unit vectors in the i (or x) and j (or y) directions.

2.

In 3 dimensions, $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ where $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors in the i (or x),j (or y), and k (or z) directions.

3.

The dot product of two vectors

$$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$$

if $\vec{A} = A_x\hat{i} + A_y\hat{j}+A_z\hat{k}$, all in 3 dimensions.

4.

The cross product of two three dimensional vectors

$$\vec{A} \otimes \vec{B} = \vert A\vert \vert B\vert \cos \theta$$

where |A| is the magnitude of $\vec{A}$, and $\cos \theta$ is the angle between the two vectors. The resultant is itself a vector perpendicular to both $\vec{A}$ and $\vec{B}$.

5.

An alternative representation of the cross product is

$$\vec{A} \otimes \vec{B} = det \left \vert \begin{array}{ccc} ... ...\\ A_x & A_y & A_z\\ B_x & B_y & B_z \end{array}\right \vert$$

6.

In 2 dimensions

$$\nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y}$$

7.

In 3 dimensions

$$\nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z}$$

8.

In 3 dimensions

$$\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$$

which can be re-written in spherical polar coordinates as

$$\nabla^2 = \frac{1}{r^2} \left ( \frac{\partial r^2\frac{ \pa... ...ac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \right )$$