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stunner5000pt
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Homework Statement
Prove the uniqueness of Laplace's equation
Note that V(x,y,z) = X(x) Y(y) Z(z))
Homework Equations
[tex] \frac{d^2 V}{dx^2} + \frac{d^2 V}{dy^2}+ \frac{d^2 V}{dz^2} = 0 [/tex]
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??