- #1

yungman

- 5,726

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Just want to verify Laplace equation in rectangular coordinates that:

[tex] \nabla ^2 \vec E = 0 [/tex]

[tex]\Rightarrow\; \nabla^2 \vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} +\frac {\partial^2}{\partial z^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) = 0 [/tex]

[tex]\hbox {(1)}\;\Rightarrow \;\frac {\partial^2 \vec E}{\partial x^2} = 0,\;\frac {\partial^2 \vec E}{\partial y^2} = 0 \;\hbox { and } \frac {\partial^2 \vec E}{\partial z^2} = 0 [/tex]

And

[tex]\hbox {(2)}\; \nabla ^2 \vec E = \nabla^2_{xy}\vec E + \frac {\partial^2 \vec E}{\partial z^2} = 0 \;\hbox { where }\; \nabla^2_{xy}\vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) [/tex]

[tex] \nabla ^2 \vec E = 0 [/tex]

[tex]\Rightarrow\; \nabla^2 \vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} +\frac {\partial^2}{\partial z^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) = 0 [/tex]

[tex]\hbox {(1)}\;\Rightarrow \;\frac {\partial^2 \vec E}{\partial x^2} = 0,\;\frac {\partial^2 \vec E}{\partial y^2} = 0 \;\hbox { and } \frac {\partial^2 \vec E}{\partial z^2} = 0 [/tex]

And

[tex]\hbox {(2)}\; \nabla ^2 \vec E = \nabla^2_{xy}\vec E + \frac {\partial^2 \vec E}{\partial z^2} = 0 \;\hbox { where }\; \nabla^2_{xy}\vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) [/tex]

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