We will get to know some basic quadric surfaces.

As we have seen, if we look at the set of points that satisfy an equation
where , we obtain a surface in . A basic class of surfaces are the *quadric
surfaces*.

**quadric surface**in is a surface of the form where , , , , , , , , , and are constants and at least one of , , , , , or are nonzero.

We will be interested in a special class of quadric surfaces, those that arise naturally when computing the Taylor polynomial of a surface at a point where: When these first partials are zero, the quadric is of the form:

Why are we doing this?

**Understanding quadric surfaces will help us find extrema
of surfaces.**

In what follows, we will study each shape by considering various *cross-sections* of the
surface.

Now that we have the basic tool of using a cross-section, we will explore our quadric surfaces.

### Elliptic paraboloids

An **elliptic paraboloid** is a surface with graph:

- , in this case we now have , a parabola.
- , in this case we now have , a parabola.
- , in this case we now have , an ellipse.

### Hyperbolic paraboloids

A **hyperbolic paraboloid** is a surface with graph:

- , in this case we now have , a parabola.
- , in this case we now have , a parabola that opens the
*opposite*direction as the previous one. - , in this case we now have , a hyperbola.

### Identifying quadric surfaces

Let’s start by working a specific example.

This parabola will open “downward” when we can find such that is negative. The expression is zero when

Let’s draw a sign-chart:

Later in this course, we will be looking at quadric surfaces of the form

and trying to identify them as either elliptic paraboloids, or as hyperbolic paraboloids. In what follows, let This will aid in our analysis of the quadric surfaces.

#### The pure partials have opposite signs

If then we can examine the following cross-sections: If then the surface

becomes and this is a parabola that opens in the -direction of the sign of .

If then the surface becomes and this is a parabola that opens in the -direction of the sign of . Since we see that when the pure partials have opposite signs, then the quadric surface is a hyperbolic paraboloid.

#### The pure partials have the same sign

If then we start by examining the cross-section: Substituting this into the surface above, we find

Factoring and rearranging, set and now

This is a parabola that opens in the -direction of the sign of when has the same sign as . The parabola opens in the opposite direction, when has the opposite sign as . We can find a that produces this opposite sign when the quadratic equation, in the variable , has two real solutions. Let’s investigate using the quadratic formula: We see that there are two real solutions for when or equivalently when So, we see that when the pure partials have the same sign, the quadric surface is a hyperbolic paraboloid when and an elliptic paraboloid when When the anaylsis above is insufficient to make any conclusions.

### The second derivative test

Given a function , and a point where our work above allows us to identify what a surface looks like locally. Specifically we get what is known as the second derivative test:

- If , then locally looks like an elliptic paraboloid.
- If , then locally looks like a hyperbolic paraboloid.
- If , the test is inconclusive.

Try your hand at identifying local behavior of a surface.