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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This note was uploaded on 03/29/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.
 Spring '08
 Skrypka
 Calculus, Derivative

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