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.
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.