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mathmari

Gold Member

MHB

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Hey!

With appropriate conditions, I want to show that $$\iiint_{\Omega}(\nabla \phi)\cdot \textbf{f}\ dV=\iint_{\Sigma}\phi\textbf{f}\cdot \textbf{N}\ dA-\iiint_{\Omega}\phi\nabla\cdot \textbf{f}\ dV$$ With appropriate conditions, I want to prove Green's identities: $$\iint_{\Sigma}\phi\nabla\psi\cdot \textbf{N}\ dA=\iiint_{\Omega}\left (\phi \Delta\psi+\nabla\phi\cdot \nabla\psi \right )\ dV\\ \iint_{\Sigma}\left (\phi\nabla \psi-\psi\nabla\phi \right )\cdot \textbf{N}\ dA=\iiint_{\Omega}\left (\phi\Delta\psi-\psi\Delta\phi \right )\ dV$$ Does $\phi\textbf{f}$ mean that $\phi$ is scalar and $\textbf{f}$ a vector? (Wondering) For the first equality we have:

Using Gauss theorem for $\phi\textbf{f}$ we get $$\iint_{\Sigma}\phi\textbf{f}\cdot \textbf{N}\ dA=\iiint_{\Omega}\nabla \cdot (\phi\textbf{f})\ dV$$ It holds that $\nabla \cdot (\phi\textbf{f})=\phi (\nabla \cdot \textbf{f})+\textbf{f}\cdot (\nabla \phi)$, right? How could we prove this? (Wondering)

Then we get $$\iint_{\Sigma}\phi\textbf{f}\cdot \textbf{N}\ dA=\iiint_{\Omega}\left [\phi (\nabla \cdot \textbf{f})+\textbf{f}\cdot (\nabla \phi)\right ]\ dV=\iiint_{\Omega}\phi (\nabla \cdot \textbf{f})\ dV+\iiint_{\Omega}\textbf{f}\cdot (\nabla \phi)\ dV \\ \Rightarrow \iiint_{\Omega}\textbf{f}\cdot (\nabla \phi)\ dV=\iint_{\Sigma}\phi\textbf{f}\cdot \textbf{N}\ dA-\iiint_{\Omega}\phi (\nabla \cdot \textbf{f})\ dV$$

For the second equality:

Using Gauss theorem for $\phi\nabla\psi$ we get $$\iint_{\Sigma}\phi\nabla\psi\cdot \textbf{N}\ dA=\iiint_{\Omega}\nabla \cdot (\phi\nabla\psi)\ dV$$ Does it hold that $\nabla \cdot (\phi\nabla\psi)=\nabla \phi\cdot \nabla\psi+\phi \nabla \cdot \nabla\psi$ ? If yes, how could we prove that? (Wondering)

If this is correct we get $\nabla \cdot (\phi\nabla\psi)=\nabla \phi\cdot \nabla\psi+\phi \Delta\psi$ and so $$\iint_{\Sigma}\phi\nabla\psi\cdot \textbf{N}\ dA=\iiint_{\Omega}\left (\nabla \phi\cdot \nabla\psi+\phi \Delta\psi\right ) dV$$

For the third equality:

Using Gauss theorem for $\phi\nabla \psi-\psi\nabla\phi $ we get $$\iint_{\Sigma}\left (\phi\nabla \psi-\psi\nabla\phi \right )\cdot \textbf{N}\ dA=\iiint_{\Omega}\nabla \cdot (\phi\nabla \psi-\psi\nabla\phi )\ dV=\iiint_{\Omega}\left [\nabla \cdot (\phi\nabla \psi)-\nabla \cdot(\psi\nabla\phi )\right ]\ dV$$ If the equality that I used previously was correct we get $\nabla \cdot (\phi\nabla\psi)=\nabla \phi\cdot \nabla\psi+\phi \nabla \cdot \nabla\psi=\nabla \phi\cdot \nabla\psi+\phi \Delta\psi$ and $\nabla \cdot (\psi\nabla\phi)=\nabla \psi\cdot \nabla\phi+\psi \nabla \cdot \nabla\phi=\nabla \psi\cdot \nabla\phi+\psi \Delta\phi$.

Then $\nabla \cdot (\phi\nabla \psi)-\nabla \cdot(\psi\nabla\phi )=(\nabla \phi\cdot \nabla\psi+\phi \Delta\psi)-(\nabla \psi\cdot \nabla\phi+\psi \Delta\phi)=\nabla \phi\cdot \nabla\psi+\phi \Delta\psi-\nabla \psi\cdot \nabla\phi-\psi \Delta\phi=\phi \Delta\psi-\psi \Delta\phi$.

Therefore we get $$\iint_{\Sigma}\left (\phi\nabla \psi-\psi\nabla\phi \right )\cdot \textbf{N}\ dA\iiint_{\Omega}\left [\phi \Delta\psi-\psi \Delta\phi\right ]\ dV$$ Is everything correct? (Wondering)

With appropriate conditions, I want to show that $$\iiint_{\Omega}(\nabla \phi)\cdot \textbf{f}\ dV=\iint_{\Sigma}\phi\textbf{f}\cdot \textbf{N}\ dA-\iiint_{\Omega}\phi\nabla\cdot \textbf{f}\ dV$$ With appropriate conditions, I want to prove Green's identities: $$\iint_{\Sigma}\phi\nabla\psi\cdot \textbf{N}\ dA=\iiint_{\Omega}\left (\phi \Delta\psi+\nabla\phi\cdot \nabla\psi \right )\ dV\\ \iint_{\Sigma}\left (\phi\nabla \psi-\psi\nabla\phi \right )\cdot \textbf{N}\ dA=\iiint_{\Omega}\left (\phi\Delta\psi-\psi\Delta\phi \right )\ dV$$ Does $\phi\textbf{f}$ mean that $\phi$ is scalar and $\textbf{f}$ a vector? (Wondering) For the first equality we have:

Using Gauss theorem for $\phi\textbf{f}$ we get $$\iint_{\Sigma}\phi\textbf{f}\cdot \textbf{N}\ dA=\iiint_{\Omega}\nabla \cdot (\phi\textbf{f})\ dV$$ It holds that $\nabla \cdot (\phi\textbf{f})=\phi (\nabla \cdot \textbf{f})+\textbf{f}\cdot (\nabla \phi)$, right? How could we prove this? (Wondering)

Then we get $$\iint_{\Sigma}\phi\textbf{f}\cdot \textbf{N}\ dA=\iiint_{\Omega}\left [\phi (\nabla \cdot \textbf{f})+\textbf{f}\cdot (\nabla \phi)\right ]\ dV=\iiint_{\Omega}\phi (\nabla \cdot \textbf{f})\ dV+\iiint_{\Omega}\textbf{f}\cdot (\nabla \phi)\ dV \\ \Rightarrow \iiint_{\Omega}\textbf{f}\cdot (\nabla \phi)\ dV=\iint_{\Sigma}\phi\textbf{f}\cdot \textbf{N}\ dA-\iiint_{\Omega}\phi (\nabla \cdot \textbf{f})\ dV$$

For the second equality:

Using Gauss theorem for $\phi\nabla\psi$ we get $$\iint_{\Sigma}\phi\nabla\psi\cdot \textbf{N}\ dA=\iiint_{\Omega}\nabla \cdot (\phi\nabla\psi)\ dV$$ Does it hold that $\nabla \cdot (\phi\nabla\psi)=\nabla \phi\cdot \nabla\psi+\phi \nabla \cdot \nabla\psi$ ? If yes, how could we prove that? (Wondering)

If this is correct we get $\nabla \cdot (\phi\nabla\psi)=\nabla \phi\cdot \nabla\psi+\phi \Delta\psi$ and so $$\iint_{\Sigma}\phi\nabla\psi\cdot \textbf{N}\ dA=\iiint_{\Omega}\left (\nabla \phi\cdot \nabla\psi+\phi \Delta\psi\right ) dV$$

For the third equality:

Using Gauss theorem for $\phi\nabla \psi-\psi\nabla\phi $ we get $$\iint_{\Sigma}\left (\phi\nabla \psi-\psi\nabla\phi \right )\cdot \textbf{N}\ dA=\iiint_{\Omega}\nabla \cdot (\phi\nabla \psi-\psi\nabla\phi )\ dV=\iiint_{\Omega}\left [\nabla \cdot (\phi\nabla \psi)-\nabla \cdot(\psi\nabla\phi )\right ]\ dV$$ If the equality that I used previously was correct we get $\nabla \cdot (\phi\nabla\psi)=\nabla \phi\cdot \nabla\psi+\phi \nabla \cdot \nabla\psi=\nabla \phi\cdot \nabla\psi+\phi \Delta\psi$ and $\nabla \cdot (\psi\nabla\phi)=\nabla \psi\cdot \nabla\phi+\psi \nabla \cdot \nabla\phi=\nabla \psi\cdot \nabla\phi+\psi \Delta\phi$.

Then $\nabla \cdot (\phi\nabla \psi)-\nabla \cdot(\psi\nabla\phi )=(\nabla \phi\cdot \nabla\psi+\phi \Delta\psi)-(\nabla \psi\cdot \nabla\phi+\psi \Delta\phi)=\nabla \phi\cdot \nabla\psi+\phi \Delta\psi-\nabla \psi\cdot \nabla\phi-\psi \Delta\phi=\phi \Delta\psi-\psi \Delta\phi$.

Therefore we get $$\iint_{\Sigma}\left (\phi\nabla \psi-\psi\nabla\phi \right )\cdot \textbf{N}\ dA\iiint_{\Omega}\left [\phi \Delta\psi-\psi \Delta\phi\right ]\ dV$$ Is everything correct? (Wondering)

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