Vector Calculus

Differentiation of vectors

Def: vector that is a function of a scalar ,

Differentiation of compose vector expression

Let be vector function of , be scalar function of


Differentiation of a vector

We use a small change in a vector function resulting from a small change in its argument. When ,

For example, for velocity ,

Intergration of a vector

  1. If , then

Vector fuctions of several argument

  1. Chain rule: If , then

  2. Special case:

  3. Differential:

Scalar and vector field

Focus on

  1. Some region
  2. 3-dimension only
  3. continous and diffentiable
  4. line, surface, volume

Vector Operators





  1. Infinitesimal change in from to is

    where is the position in the field.

  2. If represents a curve in space, then total deraviative of respected to along the curve is

  3. The change rate of w.r.t (was respected to) distance along is given by

    is called directional dedrivative

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

Chain rule:




Def: Laplacian operator

for a scalar field,




Remark: Vector operator formaula: scalar, vector

Combination of grad. div. curl

All posibilities:


Cylindrical and spherical polar coordiates

Cylindrical polar coordinates


The position vector:

The basis vectors:

where ,


The vector operators:

the volume of the parallelpiped:

Spherical polar coordinates


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 be the position vector then be the vector tangent to curve at . The unit vector

where is the length (scale factors)

infinitesimal vector displacement:

The linear operators are given by

Last Updated on 2 years by Yichen Liu