Vectors/Vector Calculus

Vector Calculus


So far we have dealt with constant vectors. It also helps if the vectors are allowed to vary in space. Then we can define derivatives and integrals and deal with vector fields. Some basic ideas of vector calculus are discussed below.

Derivative of a vector valued function


Let   be a vector function that can be represented as


where   is a scalar.

Then the derivative of   with respect to   is


Note: In the above equation, the unit vectors   (i=1,2,3) are assumed constant.
If   and   are two vector functions, then from the chain rule we get


Scalar and vector fields


Let   be the position vector of any point in space. Suppose that there is a scalar function ( ) that assigns a value to each point in space. Then


represents a scalar field. An example of a scalar field is the temperature. See Figure4(a).

If there is a vector function ( ) that assigns a vector to each point in space, then


represents a vector field. An example is the displacement field. See Figure 4(b).

Gradient of a scalar field


Let   be a scalar function. Assume that the partial derivatives of the function are continuous in some region of space. If the point   has coordinates ( ) with respect to the basis ( ), the gradient of   is defined as


In index notation,


The gradient is obviously a vector and has a direction. We can think of the gradient at a point being the vector perpendicular to the level contour at that point.

It is often useful to think of the symbol   as an operator of the form


Divergence of a vector field


If we form a scalar product of a vector field   with the   operator, we get a scalar quantity called the divergence of the vector field. Thus,


In index notation,


If  , then   is called a divergence-free field.

The physical significance of the divergence of a vector field is the rate at which some density exits a given region of space. In the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region.

Curl of a vector field


The curl of a vector field   is a vector defined as


The physical significance of the curl of a vector field is the amount of rotation or angular momentum of the contents of a region of space.

Laplacian of a scalar or vector field


The Laplacian of a scalar field   is a scalar defined as


The Laplacian of a vector field   is a vector defined as


Identities in vector calculus


Some frequently used identities from vector calculus are listed below.

  1.  ~.
  2.  ~.
  3.  ~.
  4.  ~.
  5.  ~.

Green-Gauss Divergence Theorem


Let   be a continuous and differentiable vector field on a body   with boundary  . The divergence theorem states that


where   is the outward unit normal to the surface (see Figure 5).

In index notation,


For more details on the topics of this chapter, see Vector calculus in the wikibook on Calculus.

Vector Algebra