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What does it mean to evaluate a function on a space curve?

Eg for the following question

Consider the space curve defined by the following position vector: $$r(t) = \cos t \ i + \sin t \ j + t \ k$$ and the scalar valued function: $$V (x, y, z ) = (1/2)*(x^2 + y^2 + z^2) $$ Evaluate the function on the space curve, and then compute its derivative with respect to t.

I have no problem in computing the answer as $(1+t^2)/2$, then the derivative as t, however I don't understand the theory behind this.

Semsem
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user127700
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    You're doing composition of functions. Perhaps what is interesting to discover/remember is the chain rule, which will tell you that $$\frac d{dt} V(x(t),y(t),z(t)) = \nabla V(x(t),y(t),z(t))\cdot (x'(t),y'(t),z'(t)),.$$ – Ted Shifrin Feb 12 '14 at 14:20
  • see a picture [ http://www.wolframalpha.com/input/?i=parametricplot+%28cos+t+%2C+sin+t+%2Ct%2F5%29%2C%7Bt%2C-8%2C8%7D ] – janmarqz Feb 12 '14 at 15:49

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You have a composition of $r:{\Bbb{R}}\to{\Bbb{R}}^3$ with $V:{\Bbb{R}}^3\to{\Bbb{R}}$, i.e. you have a new function $$V\circ r:{\Bbb{R}}\to {\Bbb{R}},$$ which represents the effect of measuring $V$ along an helice's curve.

janmarqz
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  • the derivative satisfy $(V\circ r)'={\rm grad V}\bullet r'$ tells you how $V$ varies in the tangent direction $r'$. – janmarqz Feb 13 '14 at 01:03