partial - HOMEWORK: 13.3: 10, 14. 13.4: 20, 32, 50 PARTIAL...

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Unformatted text preview: HOMEWORK: 13.3: 10, 14. 13.4: 20, 32, 50 PARTIAL DERIVATIVE. If f ( x, y ) is a function of two variables, then x f ( x, y, z ) is defined as the deriva- tive of the function g ( x ) = f ( x, y, z ), where y is fixed. The partial derivative with respect to y is defined similarly. NOTATION. One also writes f x ( x, y ) = x f ( x, y ) etc. For iterated derivatives the notation is similar: for example f xy = x y f . REMARK. The notation for partial derivatives x f, y f were introduced by Jacobi. Lagrange had used the term partial differences. Partial derivatives measure the rate of change of the function in the x or y directions. For functions of more variables, the partial derivatives are defined in a similar way. EXAMPLE. f ( x, y ) = x 4 6 x 2 y 2 + y 4 . We have f x ( x, y ) = 4 x 3 12 xy 2 , f xx = 12 x 2 12 y 2 , f y ( x, y ) = 12 x 2 y +4 y 3 , f yy = 12 x 2 +12 y 2 . We see that f xx + f yy = 0. A function which satisfies this equation is called harmonic . The equation f xx + f yy = 0 is an example of a...
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