I am having a terrible hard time with the multivariable chain rule and its related stuff (I read my textbook many times, but it doesn't help that much because the explanations are very limited). I hope that someone can help me to withdraw from this darkness of confusion.(adsbygoogle = window.adsbygoogle || []).push({});

1) (Differentiation along curves) Find the rate of change of f(x,y) = 1 - x^2 - y^2 along the ellipse g(t) = (2cos(t),sin(t) ) at the point where t=pi/4.

Using chain rule, we can get that the answer is d(f(g(t))/dt [evaluated at t=pi/4] =3.

But what does this represent geometrically? I have no idea what is going on geometrically about differentiation along curves.

Is the ellipse evenlying onthe surface f(x,y) = 1 - x^2 - y^2?

If not, then why does it even make sense to talk about "the rate of change of f(x,y) = 1 - x^2 - y^2 along the ellipse g(t) = (2cos(t),sin(t) ) "?

2) From my textbook:

"Suppose w= f(x,y,t,s) and that x and y are themselves functions of the independent variables t and s.

[note: Let D=curly d, representing partial derivatives]

If we write Dw/Dt = (Dw/Dx)(Dx/Dt) + (Dw/Dy)(Dy/Dt) + (Dw/Dt) using the chain rule, this is nonsense because the Dw/Dt on the left and right denote different things. We definitely cannot cancel out the two Dw/Dt."

Now this is driving me crazy, when I use the chain rule, the above expression is what I get. How can this be wrong?

I absolutely don't understand why the Dw/Dt would be denoting different things.

And also, how can we cope with this problem? (i.e. what would be the correct way to write the expression?)

3) [note: Let D=curly d, representing partial derivatives]

If, for example, w=f(x,y,z), are Dw/Dx and Df/Dx always always equal?My textbook seems to use Dw/Dx and Df/Dx quite interchangably in most cases, but I am not sure whether they are ALWAYS equal. The variable-dependence thing is just driving me crazy...

Any help is greatly appreciated!

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# Homework Help: Multivariable Chain Rule & Variable-Dependence

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