Similar to parametric surfaces, paths are typically defined by using a parametric function mapping a single input, to any number of dimensions through a vector valued function. However, instead of visualizing as time, we instead visualize it by tracing out the entire path along the domain of
One question we might have about a path is how long it is. We can use integrals to work out how long a path is by breaking the problem into smaller sub-problems.
Say for instance we take a very small section of that path like . You could imagine approximating that path using a vector, or a series of vectors in their place.
Because we can reduce the problem into a sum of vector lengths over a defined interval, we can also express the problem as an integral! To do this we integrate over the length of each vector for . We determine what each vector is by taking the gradient vector of the path at each point . Then measure the length.
So for instance, on our function above, we would have:
Now considering that a negative squared is a positive, we can rewrite as follows:
And remembering that we can simplify: