Question

If u=f(x,y) where x=(e^s)(cos(t)) and y=(e^s)(sin(t)), show that (du/dx)^2 + (du/dy)^2 = (e^(-2s))((du/ds)^2+(du/dt)^2))

Answer 1

pretty complicated chain rule problem, i'd really appreciate some help on this one

Answer 2

what do you mean (du/dx)^2
is that the second derivative ?

Answer 3

no, its the square of du/dx

Answer 4

no, its the square of du/dx

Answer 5

Its kinda confusing since usually f(x,y) is something like X^2 + y^3
something like that. Another confusing thing about the problem is that you have the derivative of u in respect to x and (du/dx) and derivative of u in respect to y (du/dy) which doesnt make sense.

Answer 6

i mean you can say u= x+ y then the question would make more sense but in that case you have to find (dx/dt) or (dx/ds) or vice versa with y

Answer 7

you mean have x+y be the function? f(x,y)=x+y?

Answer 8

no usually the problem is set up this way
f(x,y)= z
if z= w/e ( it has an x and a y)
where y= (something that has s and t)
and x= ( something that has s and t)
but your question does not make sense to me. Are you sure you wrote it right?

Answer 9

yes, i copied it exactly.

Answer 10

what does it mean to find du/dx?

Answer 11

To find the derivative of the function u in respect to x
for example lets say u= x^2
if you want to find du/dx = 2* x

Answer 12

ok well thanks anyway for your help

Answer 13

Still there?

Answer 14

YES I AM. you really are a hero! the only thing i dont understand is what du/ds and du/dt are equal to, and how you got that.

Answer 15

when you say u sub x times c, what is c?

Answer 16

Oh, sorry, I use "c" and "s" for cos(t), sin(t) here.

Answer 17

Where I write "NTS", that means, "need to show".

Answer 18

It's messy.

Answer 19

yes, but you have no idea how messy mine is after working on it for over an hour now. thank you so much!

Answer 20

du/ds and du/dt...that comes from a result in calculus (chain rule for functions of many variables).

Answer 21

You have u = f(x(s,t), y(s,t)). The parameters that do all the 'controlling' in the function are s and t, so you take the partial derivative with respect to them.

Answer 22

Is that okay?

Answer 23

yes, thank you.

Answer 24

You're welcome :)