(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove the uniqueness of Laplace's equation

Note that V(x,y,z) = X(x) Y(y) Z(z))

2. Relevant equations

[tex] \frac{d^2 V}{dx^2} + \frac{d^2 V}{dy^2}+ \frac{d^2 V}{dz^2} = 0 [/tex]

3. The attempt at a solution

Suppose V is a solution of Lapalce's equation then let V1 also be a solution of Laplace's equation.

then V - V1 is also a solution of laplace's equation

[tex] \frac{d^2 (V-V_{1})}{dx^2} + \frac{d^2 (V-V_{1})}{dy^2}+ \frac{d^2 (V-V_{1})}{dz^2} = 0 [/tex]

Are we alowed to say that

[tex] \frac{d^2 (V-V_{1})}{dx^2} = \frac{d^2 V}{dx^2} - \frac{d^2 V_{1}}{dx^2} = 0 [/tex]

[tex] \frac{d^2 V}{dx^2} = \frac{d^2 V_{1}}{dx^2}[/tex]

because V is solved by separation of variables?

Since their derivatives are equal thus V must be the same as V1.

Is this a satisfactory solution??

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# Uniqueness of Laplace's equation

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