L24 Math 233 Lecture Notes - Lecture 40: Nissan L Engine, Antiderivative, C More Entertainment
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Set up: let c be a smooth curve with a differentiable, 1-1 parametrization r(t), so. Let f(x,y) be a continuous function on an open domain d that contains c. suppose that gradient f exists and it continuous on d. * d = f r f (r(b)) This is the fundamental theorem for line integrals c. F (r(a)) b a d dt (f(r(t))dt. Application: solve line integrals of vector fields and then, evaluating f(r(b))-f(r(a)) c f dr by finding f such that f=gradient f. Note: not all f are gradient f. call such f conservative with potential function f. Compute the work done from (3,4,12) to (2,2,0) along some smooth path c. Note: f is not continuous at (0,0,0), so c must avoid that point. For such c, apply the fundamental theorem of line integrals to get. Definition: sat that line integral two smooth paths c1 and c2 with the same starting point and ending point give.
