2nd year Calculus: partial derivatives

[gcfve]

Homework Statement

See attatched image.

Homework Equations

I just don't know where to start...

The Attempt at a Solution

Any help would be appreciated! :)

 

[lanedance]

images don't clear for a while, try writing the problem for a quicker reply

there's tips on using tex floating around or just click on other etex to see how its written

 

[gcfve]

If u=f(x,y), where x=e^scos(t) and y=e^ssin(t), show that:
second derivative of u wrt x + second derivative of u wrt y = e^-2s(second derivative of u wrt s + second derivative of u wrt t)

 

[Mark44]

Is this the question?
$$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2}$$
$$=~e^{-2s}(\frac{\partial^2u}{\partial s^2} + \frac{\partial^2u}{\partial t^2})$$

Do you know the form of the chain rule for multivariable functions? If so, start by taking the partials of u w. r. t. x and y, and then take the partials of the first with respect to x and of the second with respect to y, then add them together.

 

[gcfve]

Ok, well I got :
$$\frac{\partial^2u}{\partial x^2}$$ =-2e^ssin(t) and

$$\frac{\partial^2u}{\partial y^2}$$ = 2e^scos(t)

 

[Mark44]

So what do you get for
$$\frac{\partial^2u}{\partial s^2}$$
and
$$\frac{\partial^2u}{\partial t^2}$$

 

[gcfve]

well, from the formula,
$$(\frac{\partial^2u}{\partial s^2} + \frac{\partial^2u}{\partial t^2})$$ should = 2e^3s(cos(t) - sin(t))
But I still don't understant how to get to the partails wrt s and t
I have u=f(x,y) so u=f(e^scost,e^ssin t)

 

[Mark44]

So u is a function of x and y, and x and y are each functions of s and t. Here's the form of the chain rule for one of the partials you need.

$$\frac{\partial u}{\partial s}~=~\frac{\partial u}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial s}$$

The other partial derivative is very similar, but any partial with respect to s should be with respect to t.

 

[gcfve]

Ok, well when I tried to do that I got answerd without sin and cos in them.. is what i did so far right?
and is there a formula for finding the second partials wrt s and t?