L24 Math 233 Lecture Notes - Lecture 36: Nissan L Engine
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L24 math 233 lecture 36- vector fields continued and line integrals. Not every vector field is a gradient field. Necessary condition: if then derivatives then by clairaut"s theorem. Thus, not all vector fields are gradient fields y x > (x, ) < , To be both, the vector field would have to tell you which way is up. Example: r = r (t) x(t), (t)) y be a parameter curve. If f(x,y) is a function, then the line integral of f with respect to arc length along the path r (from a to b) is f (x, )ds y b a. Ds is the arc length element = Example: f (x, ) y = 2 + x2 y. | d t gives distance traveled in a unit of time f s s c ds. Note:key steps in parameterizing c may require a piecewise function. 2 c1 f ds c2 f ds c3 f ds c4 f ds.
