# Vector Calculus

## Differentiation of vectors

Def: vector $\bold{a}$ that is a function of a scalar $\bold{a}(u)$,

### Differentiation of compose vector expression

Let $\bold{a}, \bold{b}$ be vector function of $\bold{u}$, $\phi$ be scalar function of $\bold{u}$

Then,

### Differentiation of a vector

We use a small change $\Delta \bold{a}$ in a vector function $a(u)$ resulting from a small change $\Delta u$ in its argument. When $\Delta u \rightarrow 0$,

For example, for velocity $\bold{v}$,

### Intergration of a vector

1. If $\bold{a}(u)= \frac{d \bold{A}u}{du}$, then $\int \bold{a}(u) du = \bold{A}(u) +\bold{ b}$
2. $\int_{u_1}^{u_2} \bold{a}(u)du = \bold{A}(u_2) -\bold{A}(u_1)$

### Vector fuctions of several argument

1. Chain rule: If $\bold{a}=\bold{a}(u_1, u_2, \dots), u_i=u_i(v_1, v_2, \dots)$, then

2. Special case: $\bold{a}=\bold{a}(v, u_1, u_2, \dots), u_i=u_i(v)$

3. Differential:

### Scalar and vector field

Focus on

1. Some region $\Omega \subset \mathbb{R}^3$
2. 3-dimension only
3. $\phi, \bold{a}$ continous and diffentiable
4. line, surface, volume

## Vector Operators

Def: $\nabla = \bold{i}\frac{\partial}{\partial x} + \bold{j}\frac{\partial}{\partial y} + \bold{k}\frac{\partial}{\partial z}$

Def: $\text{grad. } \phi =\nabla \phi$

Remark:

1. Infinitesimal change in $\phi$ from $\bold{r}$ to $\bold{}r+d \bold{r}$ is

where $\bold{r}$ is the position in the field.

2. If $\bold{r}(u)$ represents a curve in space, then total deraviative of $\phi (\bold{r})$ respected to $u$ along the curve is

3. The change rate of $\phi$ w.r.t (was respected to) distance $s$ along $\bold{a}$ is given by

is called directional dedrivative

4. $\nabla \phi$ lies on the direction of the fast icrease in $\phi$.

Chain rule:

### Divergence

Def:

Remark: $\nabla \cdot \bold{a} =0 \Leftrightarrow \text{solenodial}$

Def: Laplacian operator

for a scalar field$\phi$, $\nabla^2 \phi = \nabla\cdot(\nabla\phi)$

### Curl

Def:

Remark: $\nabla \times \bold{a}=0 \Leftrightarrow \text{irrotional}$

Remark: Vector operator formaula: $\phi, \psi$ scalar, $a, b$ vector

### Combination of grad. div. curl

All posibilities:

Remark:

## Cylindrical and spherical polar coordiates

### Cylindrical polar coordinates

Def:

The position vector:

The basis vectors:

where $\bold{\hat{e}_u}=\frac{\bold{\hat{e}_u}}{|\bold{\hat{e}_u}|}$, $\bold{e_u}=\frac{\partial \bold{r}}{\partial u}$

therefore,

The vector operators:

the volume $dV$ of the parallelpiped:

### Spherical polar coordinates

Def:

the basis vector:

General displacement vector

the volume of infinitesimal parallelpiped

The vector operators:

Remark: for simplify the calculation,

### General curvilinear coordinates

Def: the position of a point P having Cartesian coordinates x,y,z may be expressed in terms of the three curvilinear coordinates u1,u2,u3,where

and similarly

If $\bold{r}(u_1, u_2, i_3)$ be the position vector $P$ then $\bold e_i =\frac{\partial \bold r}{\partial u_i}$ be the vector tangent to $u_i$ curve at $P$. The unit vector

where $h_i$ is the length (scale factors)

infinitesimal vector displacement:

The linear operators are given by

Last Updated on 2 years by Yichen Liu