Q. 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 smooth surface with a smooth boundary.

(b) A surface that is not smooth, but that has a smooth boundary.

(c) A surface that is smooth, but does not have a smooth boundary

The objective is to construct an example of the thing described as follows.

"A smooth surface with a smooth boundary."

First, define the all terms of the thing in the above sentence:

A surface is smooth if it has a smooth parametrization, and a parametrization is smooth if it is differentiable.

If a curve which has derivatives of all orders at all points of the curve, this curve is called a smooth curve.

Consider a surface S is a part of $z=4-x^2-y^2$above the plane $z=0$.

The parametrization of this surface is,

Notice that, it is differentiable, so this surface is smooth.

The boundary curve of this surface is $4-x^2-y^2=0$or $x^2+y^2=4$, which is smooth.

Hence, an example for a smooth surface with a smooth boundary is as follows: surface S is a part of $z=4-x^2-y^2$above the plane $z=0$.

The objective is to construct an example of the thing described as follows:

"A surface that is not smooth, but that has a smooth boundary."

A cone with equation $z=\sqrt{x^2+y^2}$between $z=0$and $z=1$is not smooth or piece-wise smooth surface

At the vertex the normal vector is 0, so this surface is not smooth or piece-wise smooth surface, The boundary curve of this cone is$x^2+y^2=1$ which is smooth.

Hence, an example for a surface that is not smooth with a smooth boundary is as follows:

Surface S is a part of$z=\sqrt{x^2+y^2}$ between 0 and 1.

The objective is to construct an example of the thing described as follows:

"A surface that is smooth, but does not have a smooth boundary."

A plane with equation $x+y+z=1$that lies in the first octant is a smooth surface.

The boundary curve of this plane is bounded by the lines $x+y=1,x=0$, and $y=0$.

So this boundary is a triangle, and so it is not smooth. Hence, an example for a surface that is smooth, but does not have a smooth boundary is as follows:

Surface S is part of $x+y+z=1$that lies in the first octant.