GATE 2015 EE – SET 2 – Question 3
GATE 2015 EE – SET 2 – Question 3,Answer Key & Full Solutions to GATE 2017 Questions Papers for All branches .Full Solutions to GATE 2017 Questions Papers
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 GATE 2015 EE – SET 2 – Question 3
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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
$$\frac{\partial u}{\partial x} $$= $$\frac{\partial v}{\partial y}$$
$$\frac{\partial u}{\partial y} $$= $$\frac{\partial v}{\partial x}$$