## Line integral

Recall: Riemann integral

**Notation:**

**Def:** Line integral: We denote path

**Remark:** If

where

**Properties:**

(could be) path independent

**How to evaluate: **

Example:

evaluate

, where , along the curve from to .

Physics example:

### Line integrals with respect to a scalar

**Notion:**

where

How to solve:

where

Example:

evaluate

, where is the semicircle from to for .

## Green theorem in a plane

**Green theorem:**

where

## Conservative fields and potentials

**Def:** Region **conservative** if and only if

is path independent where is a single valued function is exact differential

Example:

Find

if

must satisfy

## Surface integrals

**Notion:**

where the direction of

**Def:**

How to evaluate:

find

, where , . Using spherical coordinate

Also, more generally,

convert to polar coordinate:

where

### Vector areas of surfaces

**Def:**

**Remark:** for a closed surface,

## Volume integrals

**Notion:**

### Volumes of three-dimensional regions

## Integral forms for grad., div. and curl

where V is a small volume enclosing P and S is its bounding surface.

it can be shown in Cartisian, Cylindrical and Spherical Coordinates.

## Divergence theorem and related theorem

is a vector field which is continous and diffentiable in is divided into a large number of small volumes By defination of div.

is surface of By summing over

, we have Divergence theorem holds for simply and multiply connected regions

example:

evaluate

, where and is a open surface .

### Green theorems

### Other related integral transforms

## Stokes' theorem and related theorems

### related integral theorems

Last Updated on 2 years by Yichen Liu