Stewart: Concepts and Context 11.5#42

 

If u = f(x,y), where x = e^scos t  and y = e^ssin t,  show that  

 

Solution.  It’s important to note that we don’t know what f is, only that it must be twice differentiable with respect to x, y, s and t  otherwise the equation wouldn’t make any sense.  

 

Now, where to start?  How about the observations that 
, ,  and  . 
These are key in simplifying the first order partials:

 and   

What next?  What else?  Second order partials!  Don’t forget that, since we don’t have a specific formula for u, we simply write u_x for u_y for those partials.

 and  

Adding these together leads to some cancellation and you can factor a bit if you like:

 

Now we can safely assume that if u is well-behaved enough to have all these second order partials than its first order partials must be continuous and so ,   , and  

Substituting these into the previous result we have  

Now since , the desired (pant, pant) result follows.  QED