- #1
Hypatio
- 151
- 1
I am dealing with an expression in a large amount of literature usually presented as:
[itex] \frac{\partial}{\partial \phi_i}\left(\nabla \phi_i \cdot \nabla \phi_j \right)[/itex]
I'm looking at tables of vector calculus identities and cannot seem to find one for the exact expression given, even if I remove the outside partial. Is it correct to expand this as:
[itex] \frac{\partial}{\partial \phi_i}\left[\frac{\partial \phi_i}{\partial x}\left(\frac{\partial \phi_j}{\partial x}\right)\right]+\frac{\partial}{\partial \phi_i}\left[\frac{\partial \phi_i}{\partial y}\left(\frac{\partial \phi_j}{\partial y}\right)\right][/itex]
or this:
[itex] \frac{\partial}{\partial \phi_i}\left[\frac{\partial}{\partial x}\left(\phi_i \frac{\partial \phi_j}{\partial x}\right)\right]+\frac{\partial}{\partial \phi_i}\left[\frac{\partial}{\partial y}\left(\phi_i\frac{\partial \phi_j}{\partial y}\right)\right][/itex]
Or are these the same?
I'm trying to construct the correct forward explicit, space centered, finite-difference of this expression but I can't find the correct form. Any help is appreciated.
EDIT: Looking at the wiki on vector calculus identities, it looks like this is a possible answer for the expression in parentheses:
[itex]\nabla^2(\phi_i \phi_j) = \phi_i\nabla^2\phi_j+2\nabla\phi_j\cdot\nabla\phi_j+\phi_j\nabla^2\phi_i[/itex]
rearranging:
[itex]\nabla\phi_j\cdot\nabla\phi_j = \frac{1}{2}\left(\nabla^2(\phi_i\phi_j)-\phi_i\nabla^2\phi_j-\phi_j\nabla^2\phi_i\right)[/itex]
Also, there is:
[itex]\nabla\cdot\left(\phi_i\nabla\phi_j\right) = \phi_i\nabla^2\phi_j + \nabla\phi_i\cdot \nabla\phi_j[/itex]
rearranging:
[itex]\nabla\phi_i\cdot \nabla\phi_j = \nabla\cdot\left(\phi_i\nabla\phi_j\right)- \phi_i\nabla^2\phi_j[/itex]
[itex] \frac{\partial}{\partial \phi_i}\left(\nabla \phi_i \cdot \nabla \phi_j \right)[/itex]
I'm looking at tables of vector calculus identities and cannot seem to find one for the exact expression given, even if I remove the outside partial. Is it correct to expand this as:
[itex] \frac{\partial}{\partial \phi_i}\left[\frac{\partial \phi_i}{\partial x}\left(\frac{\partial \phi_j}{\partial x}\right)\right]+\frac{\partial}{\partial \phi_i}\left[\frac{\partial \phi_i}{\partial y}\left(\frac{\partial \phi_j}{\partial y}\right)\right][/itex]
or this:
[itex] \frac{\partial}{\partial \phi_i}\left[\frac{\partial}{\partial x}\left(\phi_i \frac{\partial \phi_j}{\partial x}\right)\right]+\frac{\partial}{\partial \phi_i}\left[\frac{\partial}{\partial y}\left(\phi_i\frac{\partial \phi_j}{\partial y}\right)\right][/itex]
Or are these the same?
I'm trying to construct the correct forward explicit, space centered, finite-difference of this expression but I can't find the correct form. Any help is appreciated.
EDIT: Looking at the wiki on vector calculus identities, it looks like this is a possible answer for the expression in parentheses:
[itex]\nabla^2(\phi_i \phi_j) = \phi_i\nabla^2\phi_j+2\nabla\phi_j\cdot\nabla\phi_j+\phi_j\nabla^2\phi_i[/itex]
rearranging:
[itex]\nabla\phi_j\cdot\nabla\phi_j = \frac{1}{2}\left(\nabla^2(\phi_i\phi_j)-\phi_i\nabla^2\phi_j-\phi_j\nabla^2\phi_i\right)[/itex]
Also, there is:
[itex]\nabla\cdot\left(\phi_i\nabla\phi_j\right) = \phi_i\nabla^2\phi_j + \nabla\phi_i\cdot \nabla\phi_j[/itex]
rearranging:
[itex]\nabla\phi_i\cdot \nabla\phi_j = \nabla\cdot\left(\phi_i\nabla\phi_j\right)- \phi_i\nabla^2\phi_j[/itex]
Last edited: