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RJLiberator
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Homework Statement
Write the chain rule for the following composition using a tree diagram:
z =g(x,y)
where x=x(r,theta) and y=y(r,theta). Write formulas for the partial derivatives dz/dr and dz/dtheta. Use them to answer: Find first partial derivatives of the function z=e^x+yx^2, in polar coordinates, that is find dz/dr and dz/dtheta as a function of polar coordinates (r, theta).
Homework Equations
The Attempt at a Solution
Tree diagram was relatively simple
g extends to x and y. x and y both extend to r and theta.
**all "d's" symbolize partial derivatives**
dz/dr = dg/dx*dx/dr+ dg/dy*dy/dr
dz/dtheta = dg/dx*dx/dtheta+dg/dy*dy/dtheta
Ok.
So
partial x = e^x+2yx
partial y = x^2
Now the problem I'm having trouble with is taking the partial derivative of the polar coordinate functions.
If I am right, then:
x=rcos(theta)
y=rsin(theta)
Taking the partial derivatives dx/dr and dy/dr the answers would be cos(theta) for x and sin(theta) for y, as we treat these as 'constants'.
From here it is merely a plug-n-chug.
Is this correct? Do you have any opinions? Thank you.
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