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.

#Calculus

Easy Math Editor

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Comments

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

Abhishek Sinha - 5 years, 3 months ago

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

Rajdeep Dhingra - 4 years, 2 months ago

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

Rajdeep Dhingra - 5 years, 3 months ago

@Rajdeep Dhingra is nishant abhangi better than u??

Meera Somani - 4 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Can you find out what's wrong ?

Proof\large \textbf{Proof}Proof

Comments

$\large \textbf{Proof}$