Vectors/Vector Calculus

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Vector Calculus[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

Some frequently used identities from vector calculus are listed below.

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

Green-Gauss Divergence Theorem[edit]

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