Lecture 21 Vector eld F F M N Two questions if F f then F r f B f A This is the fundamental theorem of calculus for line integrals 1 How do we know F is a gradient of the vector eld If F is path independent then it is a gradient of the vector eld F r F r But we cannot check all of the paths so we need a different method If F M N is a gradient of the vector eld Theorem if F is differential everywhere then F is a gradient of the vector end when M N 2 If it is how do we nd the potential f The fundamental theorem of calculus says f x y f 0 0 F r for a curve from 0 0 to x y Example F y x therefore F is not the gradient of the vector eld Example For what value of a is 4x axy 3y 4x therefore a 8 Given that a 8 nd the potential M 4x 8xy N 3y 4x Method 1 The fundamental theorem of calculus says that f x y f 0 0 F r F r c t 0 0 t x c x t 0 t y f x y f 0 0 and we can choose f 0 0 to equal zero f Method 2 f 4x 8xy f 3y 4x Integrate f f where c does not matter Therefore we get f Example F 3x y x De nition curlF N M Theorem if F is differential everywhere then F is a gradient of the vector eld when curlF equals zero Curl measures rotation how much spin is being imparted onto a particle Example F y x curl y x x y 1 1 2 Example F 0 x curl 0 x 2x
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