THOUGHT CRACKERS: 08/01/2011

Thursday, August 11, 2011

VECTOR CALCULUS : DEL OPRATOR(PART 1)

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1.COORDINATE SYSTEMS

DEL (∇) is always used to know the spatial variations of any quantity i.e. changing in space.
When operated on any quantity it specifies the rate of change of quantity along with the direction of this change. This is called a directional derivative.

$\nabla = \mathbf{\hat{x}} {\partial \over \partial x} + \mathbf{\hat{y}} {\partial \over \partial y} + \mathbf{\hat{z}} {\partial \over \partial z}$

Above figure explains the different operation of del (∇) operator on the scalar and vector fields.
The spatial variation of any scalar quality results in a vector quantity which is called "Gradient of Scalar”.
The spatial variation of any vector quantity can be analyzed in two ways:

A) First is Dot Product i.e. Divergence Of Vector that gives a scalar quantity.

           B) Second is Cross Product i.e. Curl Of a Vector that gives a vector quantity.
VECTOR IDENTITIES

                                               Curl of gradient of any scalar is zero.

                                     Divergence of curl of any vector is zero.