Quadric surface

A quadric surface is the graph of a second-degree equation in three variables x,y, and z. The most general such equation is

where $A,\ B,\ C,\ \ldots,\ J$ are constants, but by translation and rotation it can be brought into one of the two standard forms

For the form $Ax^2+By^2+Cz^2+J=0$

• The following diagram shows all possible results when we assume $J=0$.

graph TD
	J0["J is 0"]
	N0["A, B, and C are not zero"]
	C0["C is 0"]
	B0["B and C are zero"]
	3P0N["All positive"]
	P["The origin"]
	2P1N["Two positive, one negative"]
	cone["Cone"]
	2P0N["All positive"]
	z["z-axis"]
	1P1N["One positive, one negative"]
	NPP["Two non-parallel plains"]
	yz["yz-plain"]
	J0-->N0 & C0 & B0
	N0-->3P0N & 2P1N
	C0-->2P0N & 1P1N
	3P0N-->P
	2P1N-->cone
	2P0N-->z
	1P1N-->NPP
	B0-->yz

• The following diagram shows all possible cases if we assume $J\neq 0$, and, without loss of generality, we can assume $J<0.$

    graph TD
        J0["J>0"]
        N0["A, B, and C are not zero"]
        C0["C is 0"]
        B0["B and C are zero"]
        3P0N["All positive"]
        E["Ellipsoid"]
        2P1N["Two positive, one negative"]
        H1["Hyperboloid of one sheet"]
        1P2N["One positive, two negative"]
        H2["Hyperboloid of two sheets"]
        2P0N["All positive"]
        Cy["Cylinder"]
        1P1N["One positive, one negative"]
        HC["Hyperbolic cylinder"]
        1P["One positive"]
        PP["Two parallel plain"]
        J0-->N0 & C0 & B0
        N0-->3P0N & 2P1N & 1P2N
        C0-->2P0N & 1P1N
        3P0N-->E
        2P1N-->H1
        1P2N-->H2
        2P0N-->Cy
        1P1N-->HC
        B0-->1P-->PP
  	
  

For the form $Ax^2+By^2+Cz=0$

Without loss of generality, we can assume $C>0$, and we three cases:

• $AB\neq 0$
  • $AB>0$: Elliptic paraboloid
  • $AB<0$: Hyberbolic paraboloid

• $B=0$: Parabolic cylinder

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Functions of Several Variables

This semester, we will focus on the study of functions with several variables. In particular, we will examine two-variable functions $f:\Omega\subseteq\mathbb{R}^2\to \mathbb{R}$. We will also study three-variable functions. The reason being, two-variable functions and the level surfaces ($f(x,y,z)=c$) of three-variable functions can be visualized.

There are two major methods to visualize a two-variable function: cross-section and contour maps.

• To find cross-sections of a two-variable function, we can set $x=ay+b$ or $y=ax+b$ in the function. This will convert the function to a one-variable function. The cross-sections are useful for finding the range of a function.
• A contour map of a function $f(x,y)$ comprises its level curves. To create a contour map, solve the equation $f(x,y)=c$ for a specific set of numbers $c$. Nowadays, we use CAS to accomplish this task.

Domain and Range

Finding the domain of a multivariable function is similar to that of one-variable functions. However, determining the range of multivariable functions can be more complex. Often, analyzing the range of a multivariable function is easier when examining its cross-sections.

Relations between level curves/surface and higher dimensional function

A function with one variable $y=f(x)$ can be considered as the level curve of $F(x,y)=0$, where $F(x,y)=y-f(x)$. In the same way, a function with two variables $z=g(x,y)$ can be viewed as the level surface of $G(x,y,z)=0$, where $G=z-g(x,y)$.