# Overview



Vector calculus is about differentiation and integration of vector fields, primarily in $$\mathbb{R}^3$$ with coordinates $$x,y,z$$ and unit vectors $$\hat{\imath},\hat{\jmath},\hat{k}$$. Here we will focus on differentiation.

## Definitions

### Parametric curve

a function with one-dimensional input and a multi-dimensional output.

Parametric curves may be expressed as a set of equations, such as $$f(t)= \left\{ \begin{array}{l} f_x(t)=t^3-3t \\ f_y(t)=3t^2 \end{array} \right. \nonumber$$ or as a vector $$f(t) = \left\langle \;t^3-3t,\; 3t^2\; \right\rangle \label{eq:parmcurve}$$

### Multi-variable functions

A multi-variable function is a function with more than one argument. This concept extends the idea of a function of one variable to several variables.

In other words, let $$f$$ be a function of variables $$x, y, \cdots$$, then function $$f(x,y.\cdots)$$ is a multi-variable function.

#### Scalar field

When a multi-variable function returns a scalar value for each point, it is called a scalar field.

A scalar field maps $$n$$-dimensional space to real numbers. Scalar fields are commonly visualized as values on a grid of points in the plane. For instance, a weather map showing the temperature $$T$$ at each point $$(x,y)$$ on a map.

For example: scalar field $$z=\sin x + \cos y$$, can be plotted with the result encoded as color, or on the $$z$$-axis.

#### Vector field

When a multi-variable function assigns a vector to each point $$(x,y)$$, it is called a vector field.

Vector fields are commonly visualized as arrows from a grid of points in the plane. This allows a $$n$$-dimensional input and output to be visualized in a $$n$$-dimensional drawing, where the arrows further give an intuition of e.g. fluid or air flow. An example of a vector field is a weather map where the magnitude and angle of the vectors represent the speed and direction of the wind at each point $$(x,y)$$.

In other words, let $$M,N,\cdots$$ be functions of variables $$x,y,\cdots$$. Then the function $$\vec{F}$$ defined below, is called a vector field. $$\vec{F}=\hat\imath M + \hat\jmath N +\;\cdots = \left\langle M, N, \cdots \right\rangle$$

The vectors are drawn starting at input $$(x,y)$$ where the magnitude and direction is determined by $$\vec{F}(x,y)$$. For example, the plots for $$\vec{F}=\left\langle\; x,\; y \right\rangle$$, and $$\vec{F}=\left\langle\; -y,\; x \right\rangle$$ are shown below.

### Rotation

If an object is rotating in two dimensions, you can describe the rotation completely with a single value: the angular velocity, $$\omega=\phi/t$$. Where a positive value indicates a counter-clockwise​ rotation.

For an object rotating in three dimensions, the direction can be described using a 3D vector, $$\vec{\omega}$$. The magnitude of the vector indicates the angular speed; the direction indicates the axis around which it tends to swirl.

The direction of the angular velocity is determined by the convention called the right-hand rule for rotation:

When the object is rotating counter-clockwise, the direction of angular velocity is along with the circular path directed upwards.

## Differentiation

Del ($$\nabla$$) is a shorthand form to simplify long mathematical expressions such as the Maxwell equations. Think of this symbol as loosely representing a vector of partial derivative operators $$\newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \nabla = \hat{\imath}\pdv{x} + \hat{\jmath}\pdv{y} + \hat{k}\pdv{z}$$ Or, in vector notation $$\newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \nabla = \left\langle \pdv{x}, \pdv{y}, \pdv{z} \right\rangle$$

Depending how $$\nabla$$ is applied, it may denote: a gradient scalar field; the divergence of a vector field; or the curl of a vector field. Each of these are described below.

Let $$f$$ be a scalar field with variables $$x,y,z$$. The vector derivative of the scalar field $$f(x,y,z)$$ is defined as the gradient. Denoted as the $$\nabla$$ “multiplied” by a scalar field $$f$$ \newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \begin{align} \nabla f &=\left\langle \pdv{x}, \pdv{y}, \pdv{z} \right\rangle f \\ &= \left\langle \pdv{x}f, \pdv{y}f, \pdv{z}f \right\rangle \end{align}

The gradient of $$f$$ at point $$(x,y)$$ is a vector that points in the direction that makes the function $$f$$ increase the fastest. The magnitude of the gradient at point $$(x,y)$$ equals the slope in that direction.

#### Example

Find the gradient for scalar field $$f(x,y,z)=x+y^2+z^3$$ \newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \begin{align*} \nabla f &=\left\langle \pdv{x}, \pdv{y}, \pdv{z} \right\rangle f \\ &=\left\langle \pdv{x}f, \pdv{y}f, \pdv{z}f \right\rangle \\ &=\left\langle 1, 2y, 3z^2 \right\rangle \end{align*}

### Divergence

For intuition, picture the vector field as a fluid where each vector describes the velocity at that point. Around some points, where all vectors point outward, the fluid just springs in to existence, as if there is a source. A positive divergence tells you how much of a source it is. Divergence is also positive if there is more flowing out than in that point.

Let $$\vec{v}$$ be a vector field where $$v_x,v_y,v_z$$ are each functions of variables $$x,y,z$$. $$\vec{v} = \left\langle v_x, v_y, v_z \right\rangle$$

The divergence of vector field $$\vec{v}$$ is written as a dot-product \newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \begin{align} \nabla \cdot \vec{v} &=\left\langle \pdv{x}, \pdv{y}, \pdv{z} \right\rangle \cdot \left\langle v_x, v_y, v_z \right\rangle \\ &=\pdv{x}v_x + \pdv{y}v_y + \pdv{z}v_z \end{align}

When the divergence at point $$(x,y)$$ is positive, the density increases. In other words, more is coming in than is leaving at that point. For example the electric field of two electric charges

#### Example

Find the divergence for vector field $$\vec{v}(x,y,z)=\left\langle xy,yz,xz\right\rangle$$ \newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \begin{align*} \nabla \cdot \vec{v} &=\left\langle \pdv{x}, \pdv{y}, \pdv{z} \right\rangle \cdot \left\langle xy,yz,xz\right\rangle \\ &= \pdv{x}xy + \pdv{y}yz + \pdv{z}xz \\ &= y + z + x = x + y + z \end{align*} The result is a scalar.

### Curl

For intuition, think about the vector field as a fluid flow. Imagine placing a tiny paddlewheel into the vector field at a point. Would it spin around? If it spins clockwise, it is said to have positive curl.

Let $$\vec{v}$$ be a vector field where $$v_x,v_y,v_z$$ are each functions of variables $$x,y,z$$. $$\vec{v} = \left\langle v_x, v_y, v_z \right\rangle$$

The curl (rotation) of vector field $$\vec{v}$$ is written as a cross-product. \newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \begin{align} \nabla \times \vec{v} &=\left\langle \pdv{x}, \pdv{y}, \pdv{z} \right\rangle \times \left\langle v_x, v_y, v_z \right\rangle \end{align}

The cross product can be computed using the pseudo-determinant. \require{color} \newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \begin{align} \nabla \times \vec{v} &=\begin{vmatrix} \color{red}{\hat{\imath}} & \color{green}{\hat{\jmath}} & \color{blue}{\hat{z}} \\ \color{red}{\pdv{x}} & \color{green}{\pdv{y}} & \color{blue}{\pdv{z}} \\ \color{red}{v_x} & \color{green}{v_y} & \color{blue}{v_z} \end{vmatrix} \\ &=\color{red}{\hat\imath} \begin{vmatrix} \color{green}{\pdv{y}} & \color{blue}{\pdv{z}} \\ \color{green}{v_y} & \color{blue}{v_z} \end{vmatrix} – \color{green}{\hat\jmath} \begin{vmatrix} \color{red}{\pdv{x}} & \color{blue}{\pdv{z}} \\ \color{red}{v_x} & \color{blue}{v_z} \end{vmatrix} + \color{blue}{\hat z} \begin{vmatrix} \color{red}{\pdv{x}} & \color{green}{\pdv{y}} \\ \color{red}{v_x} & \color{green}{v_y} \end{vmatrix} \\ &=\left\langle \begin{array}{c} \color{green}{\pdv{y}} \color{blue}{v_z} – \color{blue}{\pdv{z}} \color{green}{v_y} \\ \color{blue}{\pdv{z}} \color{red}{v_x} – \color{red}{\pdv{x}} \color{blue}{v_z} \\ \color{red}{\pdv{x}} \color{green}{v_y} – \color{green}{\pdv{y}} \color{red}{v_x} \end{array} \right\rangle \end{align}

#### Example

Find the curl for vector field $$\vec{v}(x,y,z)=\left\langle xy,yz,xz\right\rangle$$ \newcommand{pdv}[1]{\tfrac{\partial}{\partial #1}} \begin{align*} \nabla \times \vec{v} &= \left\langle \pdv{x}, \pdv{y}, \pdv{z} \right\rangle \times \left\langle xy,yz,xz\right\rangle \\ &= \begin{vmatrix} \hat\imath & \hat\jmath & \hat z \\ \pdv{x} & \pdv{y} & \pdv{z} \\ xy & yz & xz \end{vmatrix} \\ &= \left\langle \pdv{y}xz – \pdv{z}yz, -\left(\pdv{x}xz – \pdv{z}xy\right), \pdv{x}yz – \pdv{y}xy \right\rangle \\ &= \left\langle 0 – y, -(z – 0), 0 – x \right\rangle \\ &=\left\langle -y, -z, -x \right\rangle \end{align*} The result is a vector.

## Notes

If you prefer a visual explanation of divergence and curl, refer to YouTube

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