Hi, new here. I have a question of the fundamental solutions to the Laplace equation?

kruegger100

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Hi, new here. I have a question of the fundamental solutions to the Laplace equation?

How is equation #4 equal to zero? can someone explain the steps?

 

HallsofIvy

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The left side of equation 1 is \(\displaystyle \frac{\partial^2 f}{\partial x^2}+ \frac{\partial^2 f}{\partial^2 y}\). All the calculation is to show that, with new variables, \(\displaystyle \eta\) and \(\displaystyle \xi\), that becomes \(\displaystyle \frac{\partial^2f}{\partial \eta \partial \xi}\).

The right side hasn't changed. Since the right side of equation 1 is "0", so is the right side or equation 4.
 
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kruegger100

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Why is equation #4 equal to zero? Can you compute the second partial derivative

2f/∂η∂ξ with the variables η and ξ. Also how equation #5 is the solution for equation #4?
 
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HallsofIvy

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My post, just before yours, answers that question.
Look at it!

Equation 4 is \(\displaystyle 4\frac{\partial}{\partial \eta}\frac{\partial f}{\partial \xi}= 0\).

First, obviously, you can divide both sides by 4 to get
\(\displaystyle \frac{\partial}{\partial \eta}\frac{\partial f}{\partial \xi}= 0\).

Now, to make it easier to see, let \(\displaystyle u= \frac{\partial f}{\partial \xi}\) so that we can write the equation is
\(\displaystyle \frac{\partial u}{\partial \eta}= 0 \) ..................edited

The integral of 0 is, of course, a constant so integrating this equation with respect to \(\displaystyle \eta\) that "constant" might actually be a function of the other variable, \(\displaystyle \xi\). We can write \(\displaystyle u=f(\eta)\) where f can be any differentiable function of \(\displaystyle \eta\).

Since \(\displaystyle u= \frac{\partial f}{\partial \xi}= f(\xi)\). Integrating that \(\displaystyle f= F(\xi)+ G(\eta)\) where F is the integral of f and G is the "constant of integration" which might be a function of the other variable \(\displaystyle \eta\).[/tex]
 
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HallsofIvy

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Yes, after setting \(\displaystyle u= \frac{\partial f}{\partial \xi}\), I carelessly wrote \(\displaystyle \frac{\partial f}{\partial \eta}= 0\) when it should have been \(\displaystyle \frac{\partial u}{\partial \eta}= 0\). ............................response #4 has been edited to reflect the change
 
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kruegger100

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Thanks for the clarification. Can you explain how do you get equation #5 from the integration of ∂f/∂η=0 laplace question 002.jpg
 
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