Thursday, August 11, 2011

VECTOR CALCULUS : DEL OPRATOR(PART 1)

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THIS TEXT REQUIRE A PRIOR READING OF
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}

DEL
  • 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

0e80d641a82031c8bb18ad5d2b2d9998                                               Curl of gradient of any scalar is zero.

549dce8d08ec2a29d7f0f22140b10e4e
                                     Divergence of curl of any vector is zero.


24460c90e52738536a59c621e5830c72
Divergence of gradient is Laplacian of that scalar

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The 3 remaining vector derivatives are related by the equation