Real Analysis IIPlease solve 1b,c&d **Title: Analysis of a Multivariable Function**---**Problem Statement:**Let $ f : \mathbb{R}^2 \to \mathbb{R} $ where \[ f(x, y) = 0 \quad \text{if } (x, y) = (0, 0) \]and \[ f(x, y) = \frac{xy}{x^2 + y^2} \quad \text{if } (x, y) \neq (0, 0). \]1. Prove the following: (a) The directional derivative $ D_{\mathbf{u}} f(0, 0) = 0 $ if $ \mathbf{u} = (1, 0) $. (b) The directional derivative $ D_{\mathbf{u}} f(0, 0) = 0 $ if $ \mathbf{u} = (0, 1) $. (c) The directional derivative $ D_{\mathbf{u}} f(0, 0) $ does not exist if $ \mathbf{u} = (a, b) $ where $ ab \neq 0 $. (d) The derivative $ Df(0, 0) $ does not exist, shown by proving $ f $ is not continuous at $ (0, 0) $.---**Solution Outline:**- **Part (a):** For $\mathbf{u} = (1,0)$, calculate the limit: \[ \lim_{h \to 0} \frac{f(h, 0) - f(0, 0)}{h} = \lim_{h \to 0} \frac{0}{h} = 0. \]- **Part (b):** For $\mathbf{u} = (0,1)$, calculate the limit: \[ \lim_{k \to 0} \frac{f(0, k) - f(0, 0)}{k} = \lim_{k \to 0} \frac{0}{k} = 0. \]- **Part (c):** For $\mathbf{u} = (a, b)$ with $ab \neq 0$, evaluate:

Given: Let f:ℝ2→ℝ where fx,y=0 if…

xy + Let f: R2 → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = 72²² if (x, y) = (0,0). (a) Prove that Duf (0,0) = 0 if u = (1,0). (b) Prove that Duf(0, 0) = 0 if u = (0, 1). (c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0. (d) Prove that Df(0, 0) does not exist (this can be done by proving that f is not continuous at (0, 0)).

xy + Let f: R2 → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = 72²² if (x, y) = (0,0). (a) Prove that Duf (0,0) = 0 if u = (1,0). (b) Prove that Duf(0, 0) = 0 if u = (0, 1). (c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0. (d) Prove that Df(0, 0) does not exist (this can be done by proving that f is not continuous at (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

Real Analysis II Please solve 1b,c&d

**Title: Analysis of a Multivariable Function**

---

**Problem Statement:**

Let $ f : \mathbb{R}^2 \to \mathbb{R} $ where

\[ f(x, y) = 0 \quad \text{if } (x, y) = (0, 0) \]

and

\[ f(x, y) = \frac{xy}{x^2 + y^2} \quad \text{if } (x, y) \neq (0, 0). \]

1. Prove the following:

(a) The directional derivative $ D_{\mathbf{u}} f(0, 0) = 0 $ if $ \mathbf{u} = (1, 0) $.

(b) The directional derivative $ D_{\mathbf{u}} f(0, 0) = 0 $ if $ \mathbf{u} = (0, 1) $.

(c) The directional derivative $ D_{\mathbf{u}} f(0, 0) $ does not exist if $ \mathbf{u} = (a, b) $ where $ ab \neq 0 $.

(d) The derivative $ Df(0, 0) $ does not exist, shown by proving $ f $ is not continuous at $ (0, 0) $.

---

**Solution Outline:**

- **Part (a):** For $\mathbf{u} = (1,0)$, calculate the limit:

\[
\lim_{h \to 0} \frac{f(h, 0) - f(0, 0)}{h} = \lim_{h \to 0} \frac{0}{h} = 0.
\]

- **Part (b):** For $\mathbf{u} = (0,1)$, calculate the limit:

\[
\lim_{k \to 0} \frac{f(0, k) - f(0, 0)}{k} = \lim_{k \to 0} \frac{0}{k} = 0.
\]

- **Part (c):** For $\mathbf{u} = (a, b)$ with $ab \neq 0$, evaluate:

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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