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03.13.08_Page_1 - More on partial derivations 14 3-4 Second...

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Unformatted text preview: More on partial derivations 14 3-4 Second partial derivatives f (x, y) 52f 5 6f 5x =E[E] 52f _ 6 6f 5x5y—E[E] 62f _ 6 5f 5y5x—5—yig] 52f_ 5 _ fl (3)22 5y 532 EX f (x; y) = sin(xy) i: E: 5x ycos(agi),5y xcos(xy) 52 f 5 5x2 =5—x(yCOSDCy) = -y2 sinW) 52 f _ 5 ——xcos 2x2 5x )0; = cos xy — xy sin(y) 2 ‘3 f = iycosocy) 5y5x 5y = cos(xy)— yx sin(xy) 52f 5 2 = —xcos xy 5y 5y = —x2 sin(xy) 62f _ 52f 5x5}; 5y5x 2 2 5f and 5f 51:5}; 53.251: Notice that Theorem Suppose exist and are continuous functions of (x, y) hear (a,b) 2 2 Then 5 f (51,19): 5 f 5x5}; 5y5x Question: Is there a function f(x, y) such that i—f = xy,:— = y2 x y ...
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