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Chapter 14: Q. 02 (page 1149)

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A simple closed surface that is smooth.

(b) A simple closed surface that is not smooth, but is piecewise smooth.

(c) A simple closed surface that is neither smooth nor piecewise smooth.

Short Answer

Expert verified

(a) The constructed example can be given as, $$x^{2}+y^{2}+z^{2}=16$$

(b) The constructed example can be given as, $$x^{\frac{2}{3}}+y^{\frac{2}{3}}+z^{\frac{2}{3}}=16^{\frac{2}{3}}$$

(c) The constructed example can be given as, $$x+y=16$$

Step by step solution

01

Step 1. Given information

A simple closed surface that is smooth.

02

Step 2. Explanation

A sphere is a simple, closed surface that is smooth.

Hence, the constructed example is a sphere, $$x^{2}+y^{2}+z^{2}=16$$

03

Step 3. Given Information

A simple closed surface that is not smooth, but is piecewise smooth.

04

Step 4. Explanation

The constructed example of a simple closed surface that is not smooth, but is piecewise smooth can be given as, $$x^{\frac{2}{3}}+y^{\frac{2}{3}}+z^{\frac{2}{3}}=16^{\frac{2}{3}}$$

05

Step 5. Given Information

A simple closed surface that is neither smooth nor piecewise smooth.

06

Step 6. Explanation

The constructed of a simple closed surface that is neither smooth nor piecewise smooth can be given as,

$$x+y=16$$

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