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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 + (T We have then that f(r + Tdl) - f(r) = dl(T 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. Exercise 21.5 |
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