# 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
5 years, 5 months ago

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@Rajdeep Dhingra is nishant abhangi better than u??

- 4 years, 4 months ago

Green's theorem does not hold as the divergence is not continuous inside a ball containing the origin.

- 5 years, 5 months ago

Yes , correct. To correct this error we have to use a Diract Delta Function.

- 4 years, 4 months ago

Hint 1: There is some thing fishy with the divergence.

- 5 years, 5 months ago