Mathematical Methods of Physics/Gradient, Curl and Divergence

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In this section we shall consider the vector space over reals with the basis .

We now wish to deal with some of the introductory concepts of vector calculus.

Vector and Scalar Fields[edit | edit source]

Definition[edit | edit source]

Let , where is a field. We say that is a scalar field

In the physical world, examples of scalar fields are

(i) The electrostatic potential in space

(ii) The distribution of temperature in a solid body,

Definition[edit | edit source]

Let be a vector space. Let , we say that is a vector field; it associates a vector from with every point of .

In the physical world, examples of vector fields are

(i) The electric and magnetic fields in space

(ii) The velocity field in a fluid

The Gradient[edit | edit source]

Let be a scalar field. We define the gradient as an "operator" mapping the field to a vector in such that

, or as is commonly denoted

We shall encounter the physicist's notion of "operator" before defining it formally in the chapter Hilbert Spaces. It can be loosely thought of as "a function of functions"

Gradient and the total derivative[edit | edit source]

Recall from multivariable calculus that the total derivative of a function at is defined as the linear transformation that satisfies

In the usual basis, we can express as the row matrix

It is customary to denote vectors as column matrices. Thus we may write

The transpose of a matrix given by constituents is the matrix with constituents

Thus, the gradient is the transpose of the total derivative.

Divergence[edit | edit source]

Let be a vector field and let be differentiable.

We define the divergence as the operator mapping to a scalar such that

Curl[edit | edit source]

Let be a vector field and let be differentiable.

We define the curl as the operator mapping to a linear transformation from onto itself such that the linear transformation can be expressed as the matrix

written in short as . Here, denote and so on.

the curl can be explicitly given by the matrix:

this notation is also sometimes used to denote the vector exterior or cross product,