I'm given this equation:
w= root(u^2 + v^2)
I'm to find all the second order partial derivatives.
Formulas are swirling in my mind and I need someone to show me at least one first and second order partial derivative step by step so I can make sense of the rest of them. I know how to differentiate but its hard to remember all the details. Do I have to use the Chain Rule?
And remember that this is "partial" differentiation so it is close to but not quite the same as ordinary differentiation.
Thanks for the help.
I use the quotient rule for the first derivative dw/du = 1/(2root(u^2+v^2)) x 2u or 2u/(2root(u^2+v^2)) and get the strange answer:
(4root(u^2 + v^2) - 8u^2))/8(u^2 + v^2)^3/2
while the book gives the answer: Wuu (or d^2w/du^2) = v^2/(u^2 + v^2)^3/2
I know I jammed up somewhere, but I need someone to show me where.
I'll do this one in baby-steps.
I hope you can use this as a template for the others.
Find all the second-order partial derivatives.
We have: .
. . Multiply top and bottom by
. . multiply top and bottom by
For the "mixed" partial, we have: .
. . and we differentiate with respect to : .
Verify that the function u = 1/root(x^2 + y^2 + z^2) is a solution of the three-dimensional Laplace equation Uxx + Uyy + Uzz = 0.
I used the methods that were stated in this thread however for Uxx I came up with (-2x^2 + y^2 + z^2) / (x^2 + y^2 + z^2)^-5/2
I guess I'll divide that (or multiply it) and see what I get for Uyy and Uzz..tell me if using the above method for the second partial derivative will be good here.