18.02A Topic 37: Gradient and conservative fields.Author: Jeremy OrloffRead: TB: 21.2 to p.762Review: F = hM, Ni,ZCF · dr =ZCM dx + N dy.Intrinsic formula:ZCF · dr =ZCF · T ds.Gradient fields: F = ∇f = hfx, fyi.Theorem (Fundamental Theorem for gradient fields):ZCF · dr = f(x, y)|P1P0= f(x1, y1) − f(x0, y0)where P0= (x0, y0) and P1= (x1, y1) are the endpointsof the curve C. That is, for gradient fields the line integralis independent of the path taken.xyCP0= (x0, y0)P1= (x1, y1)Example 1: Let f(x, y) = xy3+ x2⇒ F = ∇f = hy3+ 2x, 3xy2iLet C be the curve shown and compute I =ZCF · dr. Do this both directly (as inthe previous topic) and using the above formula.Method 1: parametrize C: x = x, y = 2x, 0 ≤ x ≤ 1.⇒ I =ZC(y3+ 2x) dx + 3xy2dy =Z10(8x3+ 2x) dx + 12x32 dx=Z1032x3+ 2x dx = 9.Method 2:ZC∇f · dr = f (1, 2) − f (0, 0) = 9.xyC(0, 0)(1, 2)Proof of the above theoremZC∇f · dr =ZCfxdx + fydy =Zt1t0fx(x(t), y(t))dxdt+ fy(x(t), y(t))dydtdt=Zt1t0ddtf(x(t), y(t)) dt = f (x(t), y(t))|t1t0= f(P1) − f(P0)Significance: F = ∇f ⇒ZCF · dr depends only on endpoints (is path inependent).F = ∇f ⇒ICF · dr = 0 (see below for proof)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.118.02A topic 37 2Differentials: Here we use differentials to rephrase what we’ve done before. Firstrecall:a) ∇f = fxi + fyj ⇒ ∇f · dr = fxdx + fydy.b)ZC∇f · dr = f (P1) − f(p0).Using differentials we have df = fxdx + fydy. (This is the same as ∇f · dr.) Wesay M dx + N dy is an exact differential if M dx + N dy = df for some function f.As in (b) above we haveZCM dx + N dy =ZCdf = f(P1) − f(P0).Path independence: The line integralRCF · dr is called path independent if forany two points P and Q the integral yields the same value for every path between Pand Q.We show thatZCF · dr is path independent for any curve C is equivalent toICF · dr = 0 for any closed path.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 in figure (i) below.Since the starting point P0is the same as the endpoint P1we getICF · dr = f(P1) − f(P0) = 0(this proves (i)).ii) AssumeICF · dr = 0 for any closed curve. ThenIC1−C2F · dr = 0 ⇒ZC1F · dr =ZC2F · dr⇒ path independence.xyCP0= P1Figure (i)xyC2C1P0P1Figure (ii)End of topic 37
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