## Differentiation of vectors

**Def:** vector

### Differentiation of compose vector expression

Let

Then,

### Differentiation of a vector

We use a small change

For example, for velocity

,

### Intergration of a vector

- If
, then

### Vector fuctions of several argument

Chain rule: If

, then Special case:

Differential:

### Scalar and vector field

Focus on

- Some region
- 3-dimension only
continous and diffentiable - line, surface, volume

## Vector Operators

### Gradient

**Def:**

**Def:**

**Remark:**

Infinitesimal change in

from to is where

is the position in the field. If

represents **a curve in space**, then total deraviative ofrespected to along the curve is The change rate of

w.r.t (was respected to) distance along is given by is called directional dedrivative

lies on the direction of the fast **icrease**in.

Chain rule:

### Divergence

**Def:**

**Remark:**

**Def:** Laplacian operator

for a scalar field

### Curl

**Def:**

**Remark:**

**Remark:** Vector operator formaula:

### Combination of grad. div. curl

All posibilities:

**Remark:**

## Cylindrical and spherical polar coordiates

### Cylindrical polar coordinates

**Def:**

The position vector:

The basis vectors:

where

therefore,

The vector operators:

the volume

### 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 **vector tangent** to **unit** vector

where

infinitesimal vector displacement:

The linear operators are given by

Last Updated on 2 years by Yichen Liu