- #36
Delta2
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Something just doesn't look quite right to me, I guess because we change on the fly what is supposed to be kept constant, but anyway I guess that's how the chain rule is supposed to work.
In this problem, it is something of the form ## s=u_a v_a+u_b v_b ##. In this case ## \frac{\partial{s}}{\partial{s}}=u_a (\frac{\partial{v_a}}{\partial{s}})+v_a (\frac{ \partial{u_a}}{\partial{s}}) + (similarly \, for \, b)##. The partials on ## u ## and ## v ## are done with the chain rule. (We also have ## s=s(u_a, v_a, u_b,v_b) ##, and we apply the chain rule in taking the partial w.r.t. ##s ##, e.g. first taking the partial w.r.t. ## u_a ##, and treating the others as constants, etc., and then taking partial w.r.t. ## v_a ##, etc.).Delta2 said:Ok fine but something doesn't look quite right to me. The chain rule is
$$1=\frac{\partial s}{\partial s}=\frac{\partial s}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial s}{\partial y}\frac{\partial y}{\partial s}$$.
In the above when we calculate for example ##\frac{\partial s}{\partial x}## we keep constant y, but when we calculate ##\frac{\partial x}{\partial s}## we keep constant t. Isn't that contradicting or it's that how the chain rule is supposed to work?
Hm, sorry I just can't understand what are the ##u_a,v_a,u_b,v_b##. Don't we just take ##s=x^3+xy+y^3## and apply the chain rule as in post #34 to get 1 equation and then also do similar $$0=\frac{\partial t}{\partial s}=\frac{\partial t}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial t}{\partial y}\frac{\partial y}{\partial s}$$ to get the other equation with ##t=x^2y+xy^2##Charles Link said:In this problem, it is something of the form ## s=u_a v_a+u_b v_b ##. In this case ## \frac{\partial{s}}{\partial{s}}=u_a (\frac{\partial{v_a}}{\partial{s}})+v_a (\frac{ \partial{u_a}}{\partial{s}}) + (similarly \, for \, b)##. The partials on ## u ## and ## v ## are done with the chain rule. (We also have ## s=s(u_a, v_a, u_b,v_b) ##, and we apply the chain rule in taking the partial w.r.t. ##s ##, e.g. first taking the partial w.r.t. ## u_a ##, and treating the others as constants, etc., and then taking partial w.r.t. ## v_a ##, etc.).
Which post are you referring to?Office_Shredder said:I think the polar coordinates example demonstrating you can flip the partial derivatives is misleading
#17, though I now realize #18 is observing that it didn't work, so I guess my example was not helpful.haruspex said:Which post are you referring to?
Do you mean Cramer's rule?Charles Link said:I did it this way to verify the answer of post 28, and it's not too difficult if you use Kramer's rule type solution to the two linear equations.
Did I misspell it? I remember it from my second course in high school algebra, which was about 50 years ago. I googled it now, and yes, I misspelled it.haruspex said:Do you mean Cramer's rule?
Ok. A case of Kramer v. Cramer.Charles Link said:Did I misspell it? I remember it from my second course in high school algebra, which was about 50 years ago. I googled it now, and yes, I misspelled it.
In the Dustin Hoffman movie, "Kramer vs. Kramer", it is spelled with a "K". LOL :)haruspex said:Ok. A case of Kramer v. Cramer.
But what I hadn't noticed is that the movie title misspells versus as "vs.". That's the plural form; it should just be "v."Charles Link said:In the Dustin Hoffman movie, "Kramer vs. Kramer", it is spelled with a "K". LOL :)
Nice work indeed. With a little bit of algebra you get one equation for s that can give us both partials of w when differentiating with respect to s or t. Very clever shortcut, well done!.haruspex said:Ok. A case of Kramer v. Cramer.
I solved it this way:
s looks a bit like ( x+y)3, so try
##s=x^3+xy+y^3=w^3-3x^2y-3xy^2+xy##
##=w^3-3t+xy##
##t=xy(x+y)=xyw##
##xy=\frac tw##
##s=w^3-3t+\frac tw##
Now diff wrt s.