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GATE 2015 EE - SET 2 - Question 3

Solution :

Stokes Theorem : 

Let S be an oriented smooth surface that is bounded by a simple close boundary curve C with a positive orientation . Also, let \vec{F} be a field vector ,

\oint _{c} Fdr = \iint_{s} curl F.ds

 \oint _{c} Fdr = \iint_{s}(\bigtriangledown * A ) .ds

\oint _{c}  Adl   =  \iint_{s}(\bigtriangledown * A ) .ds

Gauss theorem :

The total electric flux through any closed surface surrounding a charge Q is equal to the net positive charge enclosed by that surface .

\iint D.ds =  \sum Q
\oint \oint D.ds = Q

Divergence theorem :

Let V be a region in space with boundary dv .Then the volume integral of  the divergence (\bigtriangledown .F) over V and the surface integral of
F  over the boundary dv of v are related by

\int _{v}(\bigtriangledown .F)dv = \int _{dv}  F.da

\int \int \int (\bigtriangledown .A) dv = \oint \oint A.ds

Cauchy's Integral theorem :

If f(z) is analytic in some simple connected region R , then

\int f(z) dz = 0

Analytic \Rightarrow  f(z) satisfies Cauchy Riemann equations

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}

\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

 

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