Partial Differentiation Question

[Vernon]

Hello there ,

I'm currently a bit stuck as to where the start on the following question,

r = f(s,t) , a = s + pt; b = s - pt and r is twice differentiable.Find \(\displaystyle \frac{\delta^2 r}{\delta a \delta b}\)

My thoughts are that

\(\displaystyle \frac{\delta r}{\delta s} =\frac{\delta r}{\delta a}\frac{\delta a}{\delta s} + \frac{\delta r}{\delta b}\frac{\delta b}{\delta s} = \frac{\delta r}{\delta a} + \frac{\delta r}{\delta b}\)

and

\(\displaystyle \frac{\delta r}{\delta t} =\frac{\delta r}{\delta a}\frac{\delta a}{\delta t} + \frac{\delta r}{\delta b}\frac{\delta b}{\delta t} = p \frac{\delta r}{\delta a} - p \frac{\delta r}{\delta b} = p(\frac{\delta r}{\delta a} - \frac{\delta r}{\delta b})\)

But not sure if this is going in the right direction and if so where to go.

Any help would be most appreciated

 

[sgtpepper]

I think you first rearrange your equations so that they are functions of s and t:

s = a - pt
t = (s-b)/p

Then you take the partial derivative of r with respect to b:

partial(r)/partial(b) = [partial(r)/partial(s)]*[partial(s)/partial(b)] + [partial(r)/partial(t)]*[partial(t)/partial(b)]

then you take the partial derivative of whatever you get for that with respect to a but I'm not entirely sure...

 

[daon]

You haven't supplied a function to differentiate. What is f(s,t)?

 

[Vernon]

Thats the bit that is making me struggle , as the 1st reply says it's probably a rearranging of the two equations a and b. Going to give it a try in a bit and see where it leads

*EDIT , nope still a bit stuck *

 

[sgtpepper]

I asked my teacher today (he has a PhD in some sort of math - so I trust his word) and he said that you would solve for s and t and then take it from there, (as I said earlier) but I also might have written the problem in very small handwriting and he might not have been able to see it properly - I tried working it and I couldn't figure it out either. I'll ask again tomorrow.

 

[Vernon]

Great! Thanks for the help.