Line, surface and volume integrals

Line integral

Recall: Riemann integral


Def: Line integral: We denote path joining A and B into N small line element , If a field is any point on the line element , then

Remark: If is closed, then

where is anticlockwise


  1. (could be) path independent

How to evaluate:


evaluate , where , along the curve from to .

Physics example:

Line integrals with respect to a scalar


where is arc length along the curve

How to solve:

where is the parameter of that can be represented by


evaluate , where is the semicircle from to for .

Green theorem in a plane

Green theorem:


Conservative fields and potentials

Def: Region is conservative if and only if

  1. is path independent
  2. where is a single valued function
  3. is exact differential


Find if

must satisfy

Surface integrals


where the direction of is parpendicular to the surface


How to evaluate:

find , where , .

Using spherical coordinate

Also, more generally,

convert to polar coordinate:


Vector areas of surfaces


Remark: for a closed surface,

is only depend on its boundary curve:

Volume integrals


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

  1. is a vector field which is continous and diffentiable in

  2. is divided into a large number of small volumes

  3. By defination of div.

    is surface of

  4. By summing over , we have

  5. Divergence theorem holds for simply and multiply connected regions


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