Mathematical Methods of Physics/Gradient, Curl and Divergence

< Mathematical Methods of Physics

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.

Contents

Vector and Scalar FieldsEdit

DefinitionEdit

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,  

DefinitionEdit

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  

(ii) The velocity field in a fluid  

The GradientEdit

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 derivativeEdit

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.

DivergenceEdit

Let   be a vector field and let   be differentiable.

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

 

CurlEdit

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,