Engineering Calculus Notes 356

Engineering Calculus Notes 356 - 344 CHAPTER 3. REAL-VALUED...

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Unformatted text preview: 344 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION and, if they are also differentiable, each has two partial derivatives, which are the second-order partials of f : ∂2f ∂2x ∂2f ∂y∂x ∂2f ∂x∂y ∂2f ∂2y ∂ ∂x ∂ = ∂y ∂ = ∂x ∂ = ∂y = ∂f ∂x ∂f ∂x ∂f ∂y ∂f . ∂y In subscript notation, the above would be written fxx = (fx )x fxy = (fx )y fyx = (fy )x fyy = (fy )y . Notice that in the “partial” notation, the order of differentiation is right-to-left, while in the subscript version it is left-to-right. (We shall see shortly that for C 2 functions, this is not an issue.) For example, the function f (x, y ) = x2 + 2xy + y − 1 + xy 3 has first-order partials ∂f = 2x + 2y + y 3 ∂x ∂f = 2x + 1 + 3xy 2 fy = ∂y fx = ...
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