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Calculus

Found in: Page 930

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Problem Zero: Read the section and make your own summary of the material.

Summary of the section 12.2 includes Open sets and closed sets, Limit of a function and Continuity of a function.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Open sets and closed sets

Section 12.2 covers the concepts of open sets, closed sets, the limit of a function, and the continuity of a function.

To understand the concept of open sets, we first need to understand what an open disk or ball is. An open disk in $$\mathbb{R}^ 2 $$ is a subset of the form $$\{ (x,y) | (x-x_{0})^2+(y-y_{0})^2 < \epsilon \}$$A open ball in $$\mathbb{R}^ 3 $$ is a subset of the form $$\{ (x,y,z) | (x-x_{0})^2+(y-y_{0})^2 +(z-z_{0})^2 < \epsilon \}$$A subset $$S$$ of $$\mathbb{R}^ 2 $$ or $$\mathbb{R}^ 3 $$ is said to be open if, for every point $$(x,y)$$ or $$(x,y,z)$$ in $$S$$there exists a open disk $$D$$ or ball $$B$$ such that $$D \in S$$ or $$B \in S$$A closed set is a set whose complement is open. i.e for $$A$$ to be a closed set, $$A^{c}$$ must be open.

Let $$S$$ be a subset of $$\mathbb{R}^ 2 $$ or $$\mathbb{R}^ 3 $$ A point $$(x,y)$$ or $$(x,y,z)$$ in $$S$$ is said to be a boundary point if every open disk or ball containing it intersects both $$S$$ and $$S^{c}$$The set of all boundary points of $$S$$ is called the boundary of $$S$$A subset of $$\mathbb{R}^ 2 $$ or $$\mathbb{R}^ 3 $$ is said to be bounded if it is a subset of some open disk or ball in $$\mathbb{R}^ 2 $$ or $$\mathbb{R}^ 3$$ A subset which is not bounded is said to be unbounded.

Step 2. Limit of a function

Let $$f$$ be a function of two or more variables. The limit of $$f$$ at $$a$$ is $$L$$ if, for every $$ \epsilon > 0$$there exists $$ \delta >0$$such that $$ | f-L|<\epsilon$$ whenever $$0 < |x-a|<\delta$$Then we can write \[ \lim_{x\to a} f(x) = L \]In the case where $$f$$ is a function is two variables, $$ x= \langle (x,y) \rangle $$ and $$ a= \langle (a,b) \rangle $$Therefore \[ \lim_{x\to a} f(x) = L \] means $$ | f(x,y)-L|<\epsilon$$ whenever $$0 < \sqrt{(x-a)^2+(y-b)^2}<\delta$$. In the case where $$f$$ is a function is three variables, $$ x= \langle (x,y,z) \rangle $$ and $$ a= \langle (a,b,c) \rangle $$. Therefore \[ \lim_{x\to a} f(x) = L \] means $$ | f(x,y,z)-L|<\epsilon$$ whenever $$0 < \sqrt{(x-a)^2+(y-b)^2+(z-c)^2}<\delta$$

Algebra of limits of function in one variable can be applied directly to the functions of two or more variables.

The limit of a function in two or more variables exists if and only if the the limit is same for every path containing the given point. i.e. the limit \[ \lim_{x\to a} f(x) = L \] exists if and only if \[ \lim_{x\to a} f(x) = L \] for every path $$C$$ containing $$(a,b)$$ in open set $$S$$.

Step 3. Continuity of a function

A function $$f$$ in two or more variables defined over an open set $$S$$ is said to be continuous at a point $$a$$ in $$S$$ if \[ \lim_{x\to a} f(x) = f(a) \]. Also, $$f$$ is continuous on $$S$$ if it is continuous on every point in $$S$$ A function is said to be continuous everywhere if it is continuous at every point of it's domain.

Most popular questions for Math Textbooks

Construct examples of the thing(s) described in

the following.

Try to find examples that are different than

any in the reading.

(a) A function z = f(x, y) for which ∇f(0, 0) = 0 but f is

not differentiable at (0, 0).

(b) A function z = f(x, y) for which ∇f(0, 0) = 0 for every

point in R2.

(c) A function z = f(x, y) and a unit vector u such that

Du f(0, 0) = ∇f(0, 0) · u.

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:

$f (x, y, z) = x z^{2} e^{y}$

Describe the meanings of each of the following mathematical expressions:

$\nabla f (x, y, z)$

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y, z) \to (1, 0, - 1)} \frac{\sin (x y)}{x^{2} - y^{2} + z^{2}}$

Gradients: Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

$f (x, y, z) = \frac{x^{2} y}{z}, P = (0, 3, - 1)$

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