Vector Calculus: grad, div and curl

by AppliedMathematics.info

A scalar field is a value that is attached to every point in the domain, temperature is a simple example of this. For example T(x,y,z) can be used to represent the temperature at the point (x,y,z).

A vector field is also quantity that is attached to every point in the domain, but in this case it has both magnitude (size) and direction. Vectors are often written in bold type, to distinguish them from scalars. Velocity is an example of a vector quantity; the velocity at a point has both magnitude and direction.

Introduction to Vectors and Scalars

Time is another dimension in which scalar and vector quantities may vary.

In two dimensional space a vector may be represented by two scalar components, in three dimensions a vector may be represented by three scalar components. Most simply these are Cartesian coordinates. However in 2D vectors can be written in polar coordinates and in 3D they can be written in spherical or cylindrical coordinates.

The div, grad and curl of scalar and vector fields are defined by partial differentiation .

Printable Worksheet: Grad Div and Curl

Gradient of a scalar field

Let f(x,y,z) be a scalar field. The gradient is a vector

grad   f = ( f
x
, f
y
, f
z
) ,
it is the derivative of f in each direction. The gradient of a scalar field is a vector field. An alternative notation is to use the del or nabla operator, f = grad f.

Divergence of a vector field

Let F(x,y,z) be a vector field, continuously differentiable with respect to x,y and z. Then the divergence of F is defined by

div F = F1
x
+ F2
y
+ F3
z
 .
div F is a scalar field it can also be written as . F.

Curl of a vector field

Let F(x,y,z) be a vector field, continuously differentiable with respect to x,y and z. Then the divergence of F is defined by

curl F = [ F2
z
- F3
y
 ,  F3
x
- F1
z
 ,  F1
y
. F2
x
 ]
curl F is a vector field it can also be written as F.


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