Module 15 : Vector fields, Gradient, Divergence and Curl
Lecture 44 : Gradient Divergence and Curl [Section 44.1]
Here, is treated as a vector with components and is treated as the cross product. We also write
44.1.9
Note:
Once again, through the definition of is in terms of components of which depend upon the choice of a coordinate system, one can show that the definition of does not depend upon the choice of the coordinate system.
We give an example to illustrate the importance for curl operator.
44.1.10
Example:
We saw in example 43.12 (ii), that for the rotation of a rigid body about an axis in space,its velocity vector at a point is given by
,
where is a vector along the axis of rotation and is the position vector of In case we choose the coordinate system to be right handed cartesian coordinates with axis along the axis of rotation with where is the angular speed, then
is treated as a vector with components 
and 
is treated as the cross product. We also write 
is in terms of components of 
which depend upon the choice of a coordinate system, one can show that the definition of 
does not depend upon the choice of the coordinate system. 
is given by 
,
is a vector along the axis of rotation and 
is the position vector of 
In case we choose the coordinate system to be right handed cartesian coordinates with 
axis along the axis of rotation with 
where 
is the angular speed, then 
