172 4 The Derivative Ty(y²-x²) x² +y² if (x, y) # (0,0), if (x, y) = (0,0). f(x, y) = Away from (0, 0), ƒ is rational, and so it is continuous and all its partial derivatives of all orders exist and are continuous. Show: (a) ƒ is continuous at (0,0), (b) D1f and D2f exist and are continuous at (0,0), (c) D12f(0,0) = 17 -1 = D21f(0, 0).
172 4 The Derivative Ty(y²-x²) x² +y² if (x, y) # (0,0), if (x, y) = (0,0). f(x, y) = Away from (0, 0), ƒ is rational, and so it is continuous and all its partial derivatives of all orders exist and are continuous. Show: (a) ƒ is continuous at (0,0), (b) D1f and D2f exist and are continuous at (0,0), (c) D12f(0,0) = 17 -1 = D21f(0, 0).
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
Prove Problem 4.6.1
![4.6.1. This exercise shows that continuity is necessary for the equality of mixed partial derivatives. Let
\[ f(x, y) = \begin{cases}
\frac{xy(y^2-x^2)}{x^2+y^2} & \text{if } (x, y) \neq (0, 0), \\
0 & \text{if } (x, y) = (0, 0).
\end{cases} \]
Away from (0,0), \( f \) is rational, and so it is continuous and all its partial derivatives of all orders exist and are continuous. Show: (a) \( f \) is continuous at (0,0), (b) \( D_1f \) and \( D_2f \) exist and are continuous at (0,0), (c) \( D_{12}f(0,0) = 1 \neq -1 = D_{21}f(0,0) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f41cb94-d8e0-4009-bc66-df82df6fba5b%2F2609af91-35ee-408f-9bd5-6d25b58ed08f%2Fguetx7h_processed.png&w=3840&q=75)
Transcribed Image Text:4.6.1. This exercise shows that continuity is necessary for the equality of mixed partial derivatives. Let
\[ f(x, y) = \begin{cases}
\frac{xy(y^2-x^2)}{x^2+y^2} & \text{if } (x, y) \neq (0, 0), \\
0 & \text{if } (x, y) = (0, 0).
\end{cases} \]
Away from (0,0), \( f \) is rational, and so it is continuous and all its partial derivatives of all orders exist and are continuous. Show: (a) \( f \) is continuous at (0,0), (b) \( D_1f \) and \( D_2f \) exist and are continuous at (0,0), (c) \( D_{12}f(0,0) = 1 \neq -1 = D_{21}f(0,0) \).
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