THIS IS MY EFFORT:
I just wanted to know if there are mistakes in my work...
#1
#2
THANKS FOR THOSE WHO WILL HELP ^_^
Those are correct. However, I notice that in using chain rule for the second problem, you have
$\displaystyle \frac{\partial u}{\partial t}= 2e^{\frac{y}{x}}sec^2 t$
while using direct substitution, you get
$\displaystyle \frac{\partial u}{\partial t}= 2e^{2tan t}sec^2 t$
Those are, of course, the same because $\displaystyle x= 2r cos t$ and $\displaystyle y= 4r cos t$ so that $\displaystyle \frac{y}{x}= \frac{4r cos t}{2r cos t}= 2 tan t$
but the second, with only the variable t is the better expression.
oh yeah!! since it is with respect to t..thanks a lot for that one...
however, i am just wondering in my number 1 answer...try to take a look in Chain Rule the y=re^-s, are the derivatives under it are correct??