Im revising for an exam i have next week, and cannot for my life understand the solutions to this previous exam question:
http://i.imgur.com/hLGMwIp.jpg
Is this essentially the same question as:
z = f(x,y) , g = g(s,t), y = h(s,t)
and then find dz/ds and dz/dt ?
the solutions look like the employ implicit differentiation, but i am not sure. typically I am confused from the second line,
HOw is du/dr * dr/dt = -c * df/dr ?
My train of thought is dr/dt = -c
du/dr is another way of saying df/dr, and using that same deduction we get the second half of that equation
But then i am COMPLETELY lost on the third line
Any help would be appreciated.
Please explain to me, like I am an idiot though..
thank MHF, you're always a huge help
Typo: you mean x= g(s, t), right?
Yes, that's the chain rule. You are given that u(x, t)= f(x- ct)+ g(x+ ct) and they are setting r= x- ct, s(x+ ct) so that u(x, t)= f(r)+ g(s).and then find dz/ds and dz/dt ?
the solutions look like the employ implicit differentiation, but i am not sure. typically I am confused from the second line,
HOw is du/dr * dr/dt = -c * df/dr ?
Yes, that is true.My train of thought is dr/dt = -c
That's rather roughly what is happening. More specifically, since u= f(r)+ g(s) and r and s are functions or x and t,du/dr is another way of saying df/dr, and using that same deduction we get the second half of that equation
(The fact that f depends only on r and g only on s simplifies this. Otherwise we would have and as well.)
Do it again.But then i am COMPLETELY lost on the third line
And now use the fact that
where F is any differentialble function of x and t. Here,
Taking it one part at a time, and ./
Any help would be appreciated.
Please explain to me, like I am an idiot though..
thank MHF, you're always a huge help[/QUOTE]