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 andsecond order partial derivatives for this class. EXAMPLE 10Finda²wax ayifw = xy +dwaxSolution The symbol aw/ax dy tells us to differentiate first with respect to y and thenwith respect to x. However, if we interchange the order of differentiation and differentiatefirst with respect to x we get the answer more quickly. In two steps,² +y and1.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.).a³fax ay²a²way axPartial Derivatives of Still Higher OrderAlthough we will deal mostly with first- and second-order partial derivatives, becausethese appear the most frequently in applications, there is no theoretical limit to how manytimes we can differentiate a function as long as the derivatives involved exist. Thus, we getthird- and fourth-order derivatives denoted by symbols likefyyx+affax² ay²fymesand so on. As with second-order derivatives, the order of differentiation is immaterial aslong as all the derivatives through the order in question are continuous. 830Chapter 14 Partial DerivativesHISTORICAL BIOGRAPHYAlexis Clairaut(1713-1765)www.goo.gl/MYhzm3Chapter 14 Partial DerivativesThe Mixed Derivative TheoremYou may have noticed that the "mixed" second-order partial derivativesanda²far bydy axin Example 9 are equal. This is not a coincidence. They must be equal wheneverf. fr. fy fy, and for are continuous, as stated in the following theorem. However, themixed derivatives can be different when the continuity conditions are not satisfied (seeExercise 82).THEOREM 2-The Mixed Derivative TheoremIf f(x, y) and its partial derivatives f. Sy+ fxy, and fx are defined throughout anopen region containing a point (a, b) and are all continuous at (a, b), thenf(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 saysthat 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 ordersometimes simplifies our calculations.

The Mixed Derivative Theorem also known as Clairaut's Theorem, states that for a function of two…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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.

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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