» Required Reading » Table of Contents » Chapter 21 21.3 The Fundamental Theorem in Integration on a Path in Euclidean Space
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We take a path in the P that can be divided into a finite number of pieces each of which resembles a straight line at small distances, and choose a differentiable scalar field f defined on and around it. The difference in values of f between the front and back of a small line like segment of P having unit tangent vector T and length dl starting at the point (x, y, z) and ending at (x + (Ti)dl, y + (Tj)dl, z + (Tk)dl) will be the directional derivative of f in the direction of T multiplied by dl; which is dl times the component of the gradient of f in the direction of T: namely dl(Tf).

We have then that f(r + Tdl) - f(r) = dl(Tf).

The sum of the change in f over each segment of the path that starts at a and ends at b will be the change in f over the entire path or f(b) - f(a). If we sum the previous statement over the entire path P, we get

This means that the integral of the component of the gradient of f along the path will give the change in f or difference between its values at the front and back ends of P.
And this is the form that the fundamental theorem of calculus takes for line or path integrals in several dimensions.

Exercise 21.5 Find the integral of over a path from (0, 0, 0) to (1, 2, 3) that goes from the origin up the x axis to x = 1, then parallels the y axis to y =2 and then parallel to the z axis to z = 3.
(Hint: ignore the path and use the fundamental theorem)