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Can you find out what's wrong ?

I'll present a proof that \(0 = 1\). Of-course there is an error in the proof. Can you find it?

\(\large \textbf{Proof}\)

Let us start with a vector field \(\overrightarrow{v} = \dfrac{1}{r^2} \hat{r}\).

Let us find the Divergence of this field. \[ \overrightarrow{\nabla} . \overrightarrow{v} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \dfrac{1}{r^2} \right) = \boxed{0}\]

Now let us calculate the flux of this field. \[\oint_{S}{ \overrightarrow{v}. \overrightarrow{da}} = 4\pi\]

Green’s Theorem state that \[\int_{V} {\left (\overrightarrow{\nabla} . \overrightarrow{v} \right ) d\tau} = \oint_{S}{ \overrightarrow{v}. \overrightarrow{da}} \]

Applying it we see L.H.S = 0 since \(\overrightarrow{\nabla} . \overrightarrow{v} = 0 \) and R.H.S = \(4\pi\).

This implies \( 0 = 4\pi\).
Dividing by \(4\pi\) we get
\[\boxed{ 0 = 1 }\]

Hence proved.

Note by Rajdeep Dhingra
7 months, 1 week ago

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Green's theorem does not hold as the divergence is not continuous inside a ball containing the origin. Abhishek Sinha · 7 months, 1 week ago

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Hint 1: There is some thing fishy with the divergence. Rajdeep Dhingra · 7 months, 1 week ago

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