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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:
<br /> \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} = <br /> \frac {\frac { \partial ^ {2} \varphi } {\partial x ^ {2} } } <br /> {\sqrt { 1+ (\frac{\frac {\partial \varphi} {\partial y} } {\frac {\partial \varphi } {\partial x} } )^{2} } }<br /> <br />
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 \frac {\partial \varphi} {\partial y} as constant:
\frac{\frac {\partial \varphi } {\partial y} }{\frac {\partial \varphi } {\partial x} }=\tan{\theta}<br />
<br /> \frac{\partial ^{2} \varphi }{\partial x ^{2} }=- \frac {\partial \varphi } {\partial y} \csc ^{2}{\theta} d \theta<br />
so I arrived at:
<br /> <br /> \int \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =<br /> \frac {\partial \varphi} {\partial y} \csc {\theta}<br /> <br />
I reverted and got the following:
<br /> <br /> \int \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =<br /> \left| \nabla \varphi \right|<br /> <br />
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
<br /> \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} = <br /> \frac {\frac { \partial ^ {2} \varphi } {\partial x ^ {2} } } <br /> {\sqrt { 1+ (\frac{\frac {\partial \varphi} {\partial y} } {\frac {\partial \varphi } {\partial x} } )^{2} } }<br /> <br />
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 \frac {\partial \varphi} {\partial y} as constant:
\frac{\frac {\partial \varphi } {\partial y} }{\frac {\partial \varphi } {\partial x} }=\tan{\theta}<br />
<br /> \frac{\partial ^{2} \varphi }{\partial x ^{2} }=- \frac {\partial \varphi } {\partial y} \csc ^{2}{\theta} d \theta<br />
so I arrived at:
<br /> <br /> \int \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =<br /> \frac {\partial \varphi} {\partial y} \csc {\theta}<br /> <br />
I reverted and got the following:
<br /> <br /> \int \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =<br /> \left| \nabla \varphi \right|<br /> <br />
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
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