Help with Chain Rule: Solve Complex Math Problems

[hossam killua]

View image: IMG 20141102 00094

i know chain rule but it more complicated i can't go far with it please any help ??

 

[MarkFL]

Hello and welcome to MHB, hossam killua! :D

I have moved your thread to our Calculus sub-forum as it is a better fit since this is a calculus question. I have also embedded your image using the IMG tags so that people will not have to follow a link to see it.

 

[SamJohannes]

Hello hossam killua!

Here's how I'd approach it. First I'd expand the partial derivatives $\pd{z}{u}$ and $\pd{z}{v}$ using the chain rule. From there you can find ${\left(\pd{z}{v}\right)}^{2}$ and ${\left(\pd{z}{u}\right)}^{2}$.
When you add them together and sub in the known partials you should get some cancelling out and then the result.

Let me know how you go.

 

[hossam killua]

my solution

 

[SamJohannes]

I think there's a mistake in your expanding. You must remember to square the entire right side of the equation, not the individual terms.

eg.$\left(\pd{z}{u}\right)^2=\left(\pd{z}{x}\cos\left({\alpha}\right)+\pd{z}{y}\sin\left({\alpha}\right)\right)^2=\pd{z}{x}^2\cos^2{\alpha}+2\pd{z}{x}\pd{z}{y}\cos\alpha\sin\alpha+\pd{z}{y}^2\sin^2{\alpha} $

And then complete that for the other partial, add, and you should pretty much have it.

 

[hossam killua]

thank u