18 02A Topic 37 Gradient and conservative fields Author Jeremy Orloff Read TB 21 2 to p 762 Z Z Review F hM N i F dr M dx N dy C C Z Z F dr F T ds Intrinsic formula C y C P1 x1 y1 Gradient fields F f hfx fy i Theorem Fundamental Theorem for gradient fields Z F dr f x y PP10 f x1 y1 f x0 y0 C where P0 x0 y0 and P1 x1 y1 are the endpoints of the curve C That is for gradient fields the line integral is independent of the path taken C P0 x0 y0 x Example 1 Let f x y xy 3 x2 F f hy 3 2x 3xy 2 i Z Let C be the curve shown and compute I F dr Do this both directly as in C the previous topic and using the above formula Method 1 parametrize C x x y 2x 0 x 1 Z Z 1 3 2 I y 2x dx 3xy dy 8x3 2x dx 12x3 2 dx 0 ZC 1 32x3 2x dx 9 0Z Method 2 f dr f 1 2 f 0 0 9 C y 1 2 C 0 0 Proof of the above theorem Z Z Z t1 dx dy fy x t y t dt f dr fx dx fy dy fx x t y t dt dt C t0 ZCt1 d f x t y t dt f x t y t tt10 f P1 f P0 dt t0 Z Significance F f F dr depends only on endpoints is path inependent C I F f F dr 0 see below for proof C We say F f is conservative Example 1 If F is the electric field of an electric charge it is conservative Example 2 A gravitational field of a mass is conservative 1 x 18 02A topic 37 2 Differentials Here we use differentials to rephrase what we ve done before First recall a f fx i fy j f dr fx dx fy dy Z b f dr f P1 f p0 C Using differentials we have df fx dx fy dy This is the same as f dr We say M dx N dy is an exact Z differential if MZ dx N dy df for some function f As in b above we have df f P1 f P0 M dx N dy C C R Path independence The line integral C F dr is called path independent if for any two points P and Q the integral yields the same value for every path between P and Q Z F dr is path independent for any curve C is equivalent to We show that C I F dr 0 for any closed path C We have to show i Path independence the line integral around any closed path is 0 ii The line integral around all closed paths is 0 path independence i Assume path independence and consider the closed path C shown I in figure i below F dr f P1 f P0 0 Since the starting point P0 is the same as the endpoint P1 we get C this proves i I I F dr 0 for any closed curve Then ii Assume F dr 0 C1 C2 C path independence y Z P1 C x Figure i End of topic 37 notes C1 P 0 C2 Figure ii F dr C1 y P0 P1 x Z F dr C2
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