Question

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

(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

Step 1. (a) Given Information

A simple closed surface that is smooth.

Step 2. Explanation

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

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

Step 3. (b) Given Information

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

Step 4. Explanation

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

Step 5. (c) Given Information

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

Step 6. Explanation

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

$$x+y=16$$