346 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION then the order of diFerentiation in any mixed partial derivative of order up to r does not aFect its value. This reduces the number of diferent partial derivatives of a given order tremendously. ProoF. We shall give the proof for a function of two variables; after ±nishing the proof, we shall note how this actually gives the same conclusion for three variables. The proof is based on looking at second-order diferences : given two points ( x0 ,y0 ) and ( x 1 ,y 1 ) = ( x0 + △ x,y0 + △ y ), we can go from the ±rst to the second in two steps: increase one of the variables, holding the other ±xed, then increase the other variable. This can be done in two ways, depending on which variable we change ±rst; the two paths form the sides of a rectangle with ( x i ,y i ), i = 1 , 2 at opposite corners (²igure 3.29 ). Let (+ f ) ( x0 ,y0 ) ( − f ) ( x0 + △ x,y0 ) ( − f ) ( x0 ,y0 + △ y ) (+ f ) ( x0 + △ x,y0 + △ y ) ²igure 3.29: Second order diFerences us now consider the diFerence between the values of
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