Curves and vector valued functions

A vector valued function is a function from to $\ensuremath{\mathbb{R} } ^n$ where n is usually 2 or 3 in this class. Such a function $\v{p}(t)$ can be thought of as giving the position of a particle at time t. The first derivative $\v{p'}(t)$ then gives the velocity of the particle and the second derivative $\v{p''}(t)$ gives the acceleration.

Such a function can also be thought of as a parametrization of a curve. The curve itself is the image of the function-telling which points were hit but losing the time information. Properties of the curve should be obtainable from the function which gives a parametrization of the curve. A curve will be smooth if it can be traced using differentiable components withour stopping; that is, if each p_i(t) is differentiable and there is no time when all of the components have $\v{p'}_i(t)=0$.

Analysis of the difference quotient leads to the conclusion that $\v{p'}(t)$ gives a tangent vector to the curve at time t.

The arc length of the curve from time t=t₀ to time t is given by

$$s(t)=\int_{t_0}^t \sqrt{\left(\frac{dx_1}{du}\right)^2 + \ldots + \left(\frac{dx_n}{du}\right)^2 }\ du$$

thus by the Fundamental Theorem of Integral Calculus we get $\vert\v{p}'(t)\vert = \frac{ds}{dt}$.

A standard unit tangent vector can then be given by

$$\v{\bf T}(t) = \frac{\v{p'}(t)}{\frac{ds}{dt}}$$