95 - The Line Integral - Arc Length

We recall that the line integral is the infinite sum of infinitesimally small changes in the graph of a function, We now introduce the line integral, which is much like a nomal integral of a function, except that this line can go anywhere in space, curve in many directions.

What we do is we recognize that in such case a tiny movement along two directions can always be connected with two sides of a triangle.
Using Pythagora's Theorem, we can unify this into the hypotenuse, a tiny change along the x and y directions, and can be written as a change in one direction, that of the curve itself at that point, known as the change in arc length.

Suppose we have some scalar valued function , where the x and y components are functions, parametrized (taking is an input) by some variable , then the infinitesimal change in arc length, is defined

Notation for Integrating along some curve C

The arc length integral is als ofound written in the form , for which the start and end points are also given, where the little means that we are taking the integral along the chosen curve , with respect to the infinitesimal change of the arc length of this curve,

Vector Notation

Suppose the parametric curve , parametrized by some variable , which is a point , is given by some vector valued function , between some points and , so that , then the arc length is defined in vector form.

Just like we considered the hypotenuse of the triangle where two sides are the derivatives, the infinitesimal steps along some direction, our vector already has its direction, we need only take that in terms of its magnitude, also the square root of the sum of the squares of its component, and this works in as many dimensions as we want,

96 - Path Parametrization and Independence