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The Mixed Derivative Theorem You may have noticed that the "mixed" second-order partial derivatives a²f and dy dx a²f dx dy in Example 9 are equal. This is not a coincidence. They must be equal whenever f. fe fy fy and for are continuous, as stated in the following theorem. However, the mixed derivatives can be different when the continuity conditions are not satisfied (see Exercise 82). THEOREM 2-The Mixed Derivative Theorem If f(x, y) and its partial derivatives fx. fy fay, and fy are defined throughout an open region containing a point (a, b) and are all continuous at (a, b), then Lala, b) f(a, b).

The Mixed Derivative Theorem You may have noticed that the "mixed" second-order partial derivatives a²f and dy dx a²f dx dy in Example 9 are equal. This is not a coincidence. They must be equal whenever f. fe fy fy and for are continuous, as stated in the following theorem. However, the mixed derivatives can be different when the continuity conditions are not satisfied (see Exercise 82). THEOREM 2-The Mixed Derivative Theorem If f(x, y) and its partial derivatives fx. fy fay, and fy are defined throughout an open region containing a point (a, b) and are all continuous at (a, b), then Lala, b) f(a, b).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Calculus 3 Partial Derivatives .

Question 2: Read the subsection “The Mixed Derivative Theorem” (p. 830) including Example 10. 
Explain what the Mixed Derivative Theorem (Theorem 2) states and how it can be used to make 
the computation of a mixed second derivative easier.

As noted, we can take third-order and higher partial derivatives, but we will stick with the first and
second order partial derivatives for this class.

EXAMPLE 10 Find a²w ax ay if w = xy + dw ax Solution The symbol aw/ax dy tells us to differentiate first with respect to y and then with respect to x. However, if we interchange the order of differentiation and differentiate first with respect to x we get the answer more quickly. In two steps, ² + y and 1. If we differentiate first with respect to y, we obtain aw/ax ay = 1 as well. We can differen- tiate in either order because the conditions of Theorem 2 hold for w at all points (xo.). a³f ax ay² a²w ay ax Partial Derivatives of Still Higher Order Although we will deal mostly with first- and second-order partial derivatives, because these appear the most frequently in applications, there is no theoretical limit to how many times we can differentiate a function as long as the derivatives involved exist. Thus, we get third- and fourth-order derivatives denoted by symbols like fyyx+ aff ax² ay² fymes and so on. As with second-order derivatives, the order of differentiation is immaterial as long as all the derivatives through the order in question are continuous.
Transcribed Image Text:EXAMPLE 10 Find a²w ax ay if w = xy + dw ax Solution The symbol aw/ax dy tells us to differentiate first with respect to y and then with respect to x. However, if we interchange the order of differentiation and differentiate first with respect to x we get the answer more quickly. In two steps, ² + y and 1. If we differentiate first with respect to y, we obtain aw/ax ay = 1 as well. We can differen- tiate in either order because the conditions of Theorem 2 hold for w at all points (xo.). a³f ax ay² a²w ay ax Partial Derivatives of Still Higher Order Although we will deal mostly with first- and second-order partial derivatives, because these appear the most frequently in applications, there is no theoretical limit to how many times we can differentiate a function as long as the derivatives involved exist. Thus, we get third- and fourth-order derivatives denoted by symbols like fyyx+ aff ax² ay² fymes and so on. As with second-order derivatives, the order of differentiation is immaterial as long as all the derivatives through the order in question are continuous.
830 Chapter 14 Partial Derivatives HISTORICAL BIOGRAPHY Alexis Clairaut (1713-1765) www.goo.gl/MYhzm3 Chapter 14 Partial Derivatives The Mixed Derivative Theorem You may have noticed that the "mixed" second-order partial derivatives and a²f ar by dy ax in Example 9 are equal. This is not a coincidence. They must be equal whenever f. fr. fy fy, and for are continuous, as stated in the following theorem. However, the mixed derivatives can be different when the continuity conditions are not satisfied (see Exercise 82). THEOREM 2-The Mixed Derivative Theorem If f(x, y) and its partial derivatives f. Sy+ fxy, and fx are defined throughout an open region containing a point (a, b) and are all continuous at (a, b), then f(a, b) = f(a, b). Theorem 2 is also known as Clairaut's Theorem, named after the French mathemati- cian Alexis Clairaut, who discovered it. A proof is given in Appendix 9. Theorem 2 says that to calculate a mixed second-order derivative, we may differentiate in either order, provided the continuity conditions are satisfied. This ability to proceed in different order sometimes simplifies our calculations.
Transcribed Image Text:830 Chapter 14 Partial Derivatives HISTORICAL BIOGRAPHY Alexis Clairaut (1713-1765) www.goo.gl/MYhzm3 Chapter 14 Partial Derivatives The Mixed Derivative Theorem You may have noticed that the "mixed" second-order partial derivatives and a²f ar by dy ax in Example 9 are equal. This is not a coincidence. They must be equal whenever f. fr. fy fy, and for are continuous, as stated in the following theorem. However, the mixed derivatives can be different when the continuity conditions are not satisfied (see Exercise 82). THEOREM 2-The Mixed Derivative Theorem If f(x, y) and its partial derivatives f. Sy+ fxy, and fx are defined throughout an open region containing a point (a, b) and are all continuous at (a, b), then f(a, b) = f(a, b). Theorem 2 is also known as Clairaut's Theorem, named after the French mathemati- cian Alexis Clairaut, who discovered it. A proof is given in Appendix 9. Theorem 2 says that to calculate a mixed second-order derivative, we may differentiate in either order, provided the continuity conditions are satisfied. This ability to proceed in different order sometimes simplifies our calculations.
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