Fundamental Theorem of Line Integrals
- Author:
- Katherine Hall, Andreas Lindner
Introduction
The Fundamental Theorem of Line Integrals say that if is a smooth curve parametrized by with , and is a differentiable function of two or more variables with which is continuous along then
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Below, we can see the graph of a function and its gradient vector field. When we integrate the gradient vector field either along the blue curve or the orange curve, we get the difference between the function values (z values) between the pink and blue points in the surface. Note that the sign depends on the direction of the curve.
Try plugging in different functions and moving the beginning and end points of the curves around.