Smooth surface patch

Question

What is the meaning of the last equation and why does it make $\sigma$ smooth? Won’t the smoothness of $\sigma(u,v)$ be determined by the partial derivatives? What does smoothness has to do with the last equation?

Answer 1

Perhaps the text's definition of "curve" includes smoothness. (You don't cite your text, so there is no way for me to check its definitions.) Without that, this example is wrong because there is nothing explicitly here forcing $\gamma$ smooth. If $\gamma$ is not smooth (and $a$ is not chosen stupidly, explained in next paragraph), neither is $\sigma$.

The last equation is attempting to show that $\sigma$ is injective. (Contrast embedding and immersion. See also regular surface.) The last equation is also wrong. Consider a plane, let $\gamma$ be a circle in that plane, and $a$ point parallel to the plane. Then $\sigma$ is a solid strip in the plane with width the diameter of the circle and $\sigma$ is very far from injective. (This example is what I meant by "and $a$ is not chosen stupidly" above. If $\gamma$ is a non-smooth curve in the plane, we can get a smooth strip by this construction. $\sigma$ is still not injective.)

Even if we don't try to be pathological, the construction is missing much that is necessary to make it correct. Let $\gamma$ be the helix shown in the image.

Note that the projection chosen makes it appear as if this curve has self-intersections. It does not.

Choose $\alpha$ perpendicular to the plane of projection used in the image. (That is, choose $\alpha$ into or out of the screen.) $\sigma$ has self-intersections at each of the apparent self-intersections in the figure. This highlights why the example wants injectivity -- a self-intersection does not have the right tangent space for a smooth surface.