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

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

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