Let f: R2R where f(x, y) = 0 if (x, y) = (0,0) x². if (x, y) = (0,0). Use the e-6 definition of partial derivative to prove that Duf(0,0) = 0 for u= (1,0).
Let f: R2R where f(x, y) = 0 if (x, y) = (0,0) x². if (x, y) = (0,0). Use the e-6 definition of partial derivative to prove that Duf(0,0) = 0 for u= (1,0).
Let f: R2R where f(x, y) = 0 if (x, y) = (0,0) x². if (x, y) = (0,0). Use the e-6 definition of partial derivative to prove that Duf(0,0) = 0 for u= (1,0).
Real Analysis II
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Transcribed Image Text:2. Let \( f : \mathbb{R}^2 \to \mathbb{R} \) where
\[
f(x, y) =
\begin{cases}
0 & \text{if } (x, y) = (0, 0) \\
\frac{xy^2}{x^2+y^2} & \text{if } (x, y) \neq (0, 0)
\end{cases}
\]
Use the \(\epsilon\)-\(\delta\) definition of partial derivative to prove that \( D_u f(0,0) = 0 \) for \( u = (1,0) \).
Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
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![2. Let \( f : \mathbb{R}^2 \to \mathbb{R} \) where
\[
f(x, y) =
\begin{cases}
0 & \text{if } (x, y) = (0, 0) \\
\frac{xy^2}{x^2+y^2} & \text{if } (x, y) \neq (0, 0)
\end{cases}
\]
Use the \(\epsilon\)-\(\delta\) definition of partial derivative to prove that \( D_u f(0,0) = 0 \) for \( u = (1,0) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2F058c9d9d-471b-45c1-80ad-3e0d06e5a61b%2F3e5h89_processed.jpeg&w=3840&q=75)
