21-356 Lecture 38

Then for every i 1 n a a a g f f x x dx

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Unformatted text preview: ives of f and g in R at all x ! U and they are continuous and bounded. Then for every i = 1, . . . , N , A A A #g #f f (x) (x) dx = ' g (x) (x) dx + f (x) g (x) 1i (x) dHN #1 (x) . # xi # xi U U "U Proof. Fix i ! {1, . . . , N }. We apply the divergence theorem to the function f : U * RN defined by & f (x) g (x) if j = i, fj (x) := 0 if j &= i. Then div f = N + # fj # (f g ) #g #f = =f +g , # xj # xi # xi # xi j =1 and so A, A A A #g #f f +g dx = div f dx = f · $ dHN #1 = f g 1i dHN #1 . # xi # xi U U "U "U 106...
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