What have I done?

  • Thread starter ShayanJ
  • Start date
  • #1
2,793
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I know my question is strange(and maybe stupid) but I'm really curious about it.I once tried to calculate the gradient of the magnitude of the gradient of a scalar field which for its x coordinate,I found:

[itex]
\left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =
\frac {\frac { \partial ^ {2} \varphi } {\partial x ^ {2} } }
{\sqrt { 1+ (\frac{\frac {\partial \varphi} {\partial y} } {\frac {\partial \varphi } {\partial x} } )^{2} } }

[/itex]

then I tried to write it in a simpler form and it just came into my mind that I can integrate it and write it as the derivative of a function so I made substitutions below and treated [itex] \frac {\partial \varphi} {\partial y} [/itex] as constant:

[itex]\frac{\frac {\partial \varphi } {\partial y} }{\frac {\partial \varphi } {\partial x} }=\tan{\theta}
[/itex]
[itex]
\frac{\partial ^{2} \varphi }{\partial x ^{2} }=- \frac {\partial \varphi } {\partial y} \csc ^{2}{\theta} d \theta
[/itex]

so I arrived at:

[itex]

\int \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =
\frac {\partial \varphi} {\partial y} \csc {\theta}

[/itex]

I reverted and got the following:

[itex]

\int \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =
\left| \nabla \varphi \right|

[/itex]

Now comes my question:
When I check the things I've done,I just can tell bullsh*t.But the result tells that was really an inegration but I really don't understand how can that be an integration.

Sorry for such a mess and thanks in advance
 
Last edited:

Answers and Replies

  • #2
166
0
I'm not really sure what you did (I'm not really good at that stuff), but I don't think you have to treat [itex]\frac {\partial \varphi} {\partial y}[/itex] as constant since [itex]\theta[/itex] can be a variable (I'm not really sure if I'm right, but I think it works that way).
 

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