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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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