huhu. mọi người cứu mình với. có ai giúp mình chứng minh các hằng đẳng thức nabla ko?
[tex]grad(\Psi .\phi )=\Psi grad\phi +\phi grad \Psi[/tex]
Ta có:
[tex]grad(\Psi .\phi )= \frac{\partial (\Psi .\phi )}{\partial x}\vec{i} + \frac{\partial (\Psi .\phi )}{\partial y}\vec{j} + \frac{\partial (\Psi .\phi )}{\partial z}\vec{k}[/tex]
Mà:
[tex]\frac{\partial (\Psi .\phi )}{\partial x}\vec{i} = \phi \frac{\partial (\Psi )}{\partial x}\vec{i} + \Psi \frac{\partial (\phi )}{\partial x}\vec{i}[/tex] (1)
[tex]\frac{\partial (\Psi .\phi )}{\partial y}\vec{j} = \phi \frac{\partial (\Psi )}{\partial y}\vec{j} + \Psi \frac{\partial (\phi )}{\partial y}\vec{j}[/tex] (2)
[tex]\frac{\partial (\Psi .\phi )}{\partial z}\vec{k} = \phi \frac{\partial (\Psi )}{\partial z}\vec{k} + \Psi \frac{\partial (\phi )}{\partial z}\vec{k}[/tex] (3)
~O) Cộng (1), (2) và (3) lại ta có:
[tex]grad(\Psi .\phi )= \left[ \phi \frac{\partial (\Psi )}{\partial x}\vec{i} + \phi \frac{\partial (\Psi )}{\partial y}\vec{j}+ \phi \frac{\partial (\Psi )}{\partial z}\vec{k}\right] + \left[ \Psi \frac{\partial (\phi )}{\partial x}\vec{i} + \Psi \frac{\partial (\phi )}{\partial y}\vec{j} + \Psi \frac{\partial (\phi )}{\partial z}\vec{k}\right][/tex]
[tex]\Leftrightarrow grad(\Psi .\phi )= \phi . grad \Psi+ \Psi. grad \phi[/tex]
