lec_week9

lec_week9 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 18.02 Lecture 21. Tue, Oct 30, 2007 Test for gradient fields. Observe: if F = M + N j is a gradient field then N x = M y . Indeed, if F = f then M = f x , N = f y , so N x = f yx = f xy = M y . Claim: Conversely, if F is defined and differentiable at every point of the plane, and N x = M y , then F = M + N j is a gradient field. Example: F = y + x j : N x = 1, M y = 1, so F is not a gradient field. Example: for which value(s) of a is F = (4 x 2 + axy ) + (3 y 2 + 4 x 2 ) j a gradient field? Answer: N x = 8 x , M y = ax , so a = 8. Finding the potential : if above test says F is a gradient field, we have 2 methods to find the potential function f . Illustrated for the above example (taking a = 8): Method 1: using line integrals (FTC backwards): We know that if C starts at (0 , 0) and ends at ( x 1 ,y 1 ) then f ( x 1 ,y 1 ) f (0 , 0) = F dr . Here C f (0 , 0) is just an integration constant (if f is a potential then so is f + c ). Can also choose the simplest curve C from (0 , 0) to ( x 1 ,y 1 ). Simplest choice: take C = portion of x-axis from (0 , 0) to ( x 1 , 0), then vertical segment from ( x 1 , 0) to ( x 1 ,y 1 ) (picture drawn). Then F dr = (4 x 2 + 8 xy ) dx + (3 y 2 + 4 x 2 ) dy : C C 1 + C 2 x 1 4 x 1 4 Over C 1 , 0 x x 1 , y = 0, dy = 0: = (4 x 2 + 8 x 0) dx = x 3 = x 1 3 . 3 3 C 1 y 1 Over C 2 , 0 y y 1 , x = x 1 , dx = 0: = (3 y 2 + 4 x 2 ) dy = y 3 + 4 x 1 2 y y 1 = y 3 + 4 x 1 2 y 1 . 1 1 C 2 4 So f ( x 1 ,y 1 ) = x 3 1 + y 1 3 + 4 x 1 2 y 1 (+constant). 3 Method 2: using antiderivatives: We want f ( x,y ) such that (1) f x = 4 x 2 + 8 xy , (2) f y = 3 y 2 + 4 x 2 . Taking antiderivative of (1) w.r.t. x (treating y as a constant), we get f ( x,y ) = 3 4 x 3 + 4 x 2 y + integration constant (independent of x ). The integration constant still depends on y , call it g ( y )....
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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lec_week9 - MIT OpenCourseWare http://ocw.mit.edu 18.02...

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