- #1

ShayanJ

Gold Member

- 2,809

- 604

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

[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: