Question

(Jacobian question)

Question: http://f.imgtmp.com/Vuvz3.jpg
Solution: http://f.imgtmp.com/LOQdv.jpg

For (∂x/∂y)|_z and (∂w/∂z)|_x (or anything else), what's the pattern for computing the Jacobians?

For instance when computing (∂x/∂y)|_z, why is F and G at the top and why is y and w at the bottom on the numerator and why is F and G at the top and x and w at the bottom in the denominator?

Answer 1

gah, i need my textbook. if no one else answers within 15 i'll find it and explain it for you.

Answer 2

but basically the jacobian will not include any partial derivatives with respect to z, since that is the variable you are examining.

Answer 3

since y is in denominator it goes to numerator
since x is in numerator it goes to denominator.
w is in top and bottom because it was not included at all, it's always the second variable

Answer 4

the actual theory behind it, i'd have to check.

Answer 5

that's just how i remembered it.

Answer 6

you know how to get the actual matrix though right?

Answer 7

2 secs, let me read (it might take a while since i need to process this and i am sick, etc)

Answer 8

calculus takes time, but it's time well spent!

Answer 9

yes, yes, i find this stuff useful and fun except school sucks all the fun out of it :( anyways i'll answer if i get it or if not ask a few more questions soon

Answer 10

i know how to compute it, yes. what i am confused with though is that what you said doesn't seem to apply for my second example (posted here in the one question).

Answer 11

or wait, maybe i just misunderstood something. specifically, why is there an x in the numerator and denominator of the second jacobian?

Answer 12

Oh, I think I get it now thanks to you (mattt9) as well as my text book. The question said find (∂w/∂z)|_y and not (∂w/∂z)|_x now that I look at it and the answer assumes it was (∂w/∂z)|_y but writes (∂w/∂z)|_x = (what (∂w/∂z)|_y should be equal to). Am I correct in thinking it's just a mini-typo in the solution? I still would appreciate confirmation if I am right.

Answer 13

I figured it out!

Answer 14

that method is the same except the one I did was the positive version. so you don't have to include thenegative at the beginning and then the jacobian gives the same result as the way the showed in your solution

Answer 15

and yes it was just a miny typo in the solution

Answer 16

had to be