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

#### kruegger100

##### New member
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

##### Elite Member
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

##### New member
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

##### Elite Member
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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#### kruegger100

##### New member
Thanks for your help. Just some questions: attachments

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#### HallsofIvy

##### Elite Member
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

##### New member
Thanks for the clarification. Can you explain how do you get equation #5 from the integration of ∂f/∂η=0

#### HallsofIvy

##### Elite Member
Thanks for the clarification. Can you explain how do you get equation #5 from the integration of ∂f/∂η=0 View attachment 26592
You start with $$\displaystyle \frac{\partial f}{\partial \eta}= 0$$ and immediately say
"Means for integrating that
$$\displaystyle \partial f= \partial \eta$$"

NO! It doesn't! That would be true the equation were $$\displaystyle \frac{\partial f}{\partial \eta}= 1$$

but $$\displaystyle \frac{\partial f}{\partial \eta}= 0$$ gives only $$\displaystyle \partial f= 0$$.

More simply if the derivative of f is 0 with respect to a variable, then f is a constant with respect to that variable.
You learned that in Calculus I!