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Suppose S is a subset of R 2 , and that f is a continuous real-valued function on S . If both partial derivatives of f exist at each point of the interior S 0 of S , and both are continuous on S 0 , then f is said to belong to C 1 ( S ) . If all k th order mixed partial derivatives exist at each point of S 0 , and all of them are continuous on S 0 , then f is said to belong to C k ( S ) . Finally, if all mixed partial derivatives, of arbitrary orders, exist and are continuous on S 0 , then f is said to belong to C ( S ) .

  1. Suppose f is a real-valued function of two real variables and that it is differentiable, as a function of two real variables, at the point ( a , b ) . Show that the numbers L 1 and L 2 in the definition are exactly the partial derivatives of f at ( a , b ) . That is,
    L 1 = t i a l f t i a l x ( a , b ) = lim h 0 f ( a + h , b ) - f ( a , b ) h
    and
    L 2 = t i a l f t i a l y ( a , b ) = lim h 0 f ( a , b + h ) - f ( a , b ) h .
  2. Define f on R 2 as follows: f ( 0 , 0 ) = 0 , and if ( x , y ) ( 0 , 0 ) , then f ( x , y ) = x y / ( x 2 + y 2 ) . Show that both partial derivatives of f at ( 0 , 0 ) exist and are 0. Show also that f is not , as a function of two real variables, differentiable at ( 0 , 0 ) . HINT: Let h and k run through the numbers 1 / n .
  3. What do parts (a) and (b) tell about the relationship between a function of two real variables being differentiable at a point ( a , b ) and its having both partial derivatives exist at ( a , b ) ?
  4. Suppose f = u + i v is a complex-valued function of a complex variable, and assume that f is differentiable, as a function of a complex variable, at a point c = a + b i ( a , b ) . Prove that the real and imaginary parts u and v of f are differentiable, as functions of two real variables. Relate the five quantities
    t i a l u t i a l x ( a , b ) , t i a l u t i a l y ( a , b ) , t i a l v t i a l x ( a , b ) , t i a l v t i a l y ( a , b ) , and f ' ( c ) .

Perhaps the most interesting theorem about partial derivatives is the “mixed partials are equal” theorem. That is, f x y = f y x . The point is that this is not always the case. An extra hypothesis is necessary.

Theorem on mixed partials

Let f : S R be such that both second order partials derivatives f x y and f y x exist at a point ( a , b ) of the interior of S , and assume in addition that one of these second order partials exists at every point in a disk B r ( a , b ) around ( a , b ) and that it is continuous at the point ( a , b ) . Then f x y ( a , b ) = f y x ( a , b ) .

Suppose that it is f y x that is continuous at ( a , b ) . Let ϵ > 0 be given, and let δ 1 > 0 be such that if | ( c , d ) - ( a , b ) | < δ 1 then | f y x ( c , d ) - f y x ( a , b ) | < ϵ . Next, choose a δ 2 such that if 0 < | k | < δ 2 , then

| f x y ( a , b ) - f x ( a , b + k ) - f x ( a , b ) k | < ϵ ,

and fix such a k . We may also assume that | k | < δ 1 / 2 . Finally, choose a δ 3 > 0 such that if 0 < | h | < δ 3 , then

| f x ( a , b + k ) - f ( a + h , b + k ) - f ( a , b + k ) h | < | k | ϵ ,

and

| f x ( a , b ) - f ( a + h , b ) - f ( a , b ) h | < | k | ϵ ,

and fix such an h . Again, we may also assume that | h | < δ 1 / 2 .

In the following calculation we will use the Mean Value Theorem twice.

0 | f x y ( a , b ) - f y x ( a , b ) | | f x y ( a , b ) - f x ( a , b + k ) - f x ( a , b ) k | + | f x ( a , b + k ) - f x ( a , b ) k - f y x ( a , b ) | ϵ + | f x ( a , b + k ) - f ( a + h , b + k ) - f ( a , b + k ) h k | + | f ( a + h , b ) - f ( a , b ) h - f x ( a , b ) k | + | f ( a + h , b + k ) - f ( a , b + k ) + ( f ( a + h , b ) - f ( a , b ) ) h k - f y x ( a , b ) | < 3 ϵ + | f ( a + h , b + k ) - f ( a , b + k ) + ( f ( a + h , b ) - f ( a , b ) ) h k - f y x ( a , b ) | = 3 ϵ + | f y ( a + h , b ' ) - f y ( a , b ' ) h - f y x ( a , b ) | = 3 ϵ + | f y x ( a ' , b ' ) - f y x ( a , b ) | < 4 ϵ ,

because b ' is between b and b + k , and a ' is between a and a + h , so that | ( a ' , b ' ) - ( a , b ) | < δ 1 / 2 < δ 1 . Hence, | f x y ( a , b ) - f y x ( a , b ) < 4 ϵ , for an arbitrary ϵ , and so the theorem is proved.

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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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