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I did the first part by the definition of total derivative, but can not think of the way to do the second part of the question. Any help please?

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- Oct 25th 2009, 05:35 AM6DOMTotal Derivative and Orthogonality
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I did the first part by the definition of total derivative, but can not think of the way to do the second part of the question. Any help please? - Oct 25th 2009, 07:29 AMOpalg
If $\displaystyle f(x)\in\partial B_r(0)$ then $\displaystyle (f_1(x))^2 + (f_2(x))^2 + \ldots + (f_n(x))^2 = r^2$ for all x in U. Differentiate that equation.

- Oct 25th 2009, 08:24 AM6DOM
- Oct 25th 2009, 11:10 AMOpalg
I meant differentiate with respect to x, using the chain rule. I suppose I should have used t rather than x, since the question says that f is a function of t. Then $\displaystyle \tfrac d{dt}(f_1(t))^2$ evaluated at t=a is equal to $\displaystyle 2f_1(a)\tfrac {df_1}{dt}(a)$, and similarly for the other coordinates. The result of differentiating both sides of the equation $\displaystyle (f_1(t))^2 + (f_2(t))^2 + \ldots + (f_n(t))^2 = r^2$ in that way ought to remind you of an inner product.

- Oct 25th 2009, 11:22 AM6DOM