Sketching graphs

Tell me and I’ll forget; show me and I may remember; involve me and I’ll understand, Chinese Proverb.

Functions of two variables: An Overview

Functions of two variables are natural generalizations or extensions of functions of one variable. They extend single-variable calculus into multi-dimensional space. They are mathematical functions $f: D \subseteq \mathbb{R}^2 \to \mathbb{R}$, meaning they map an ordered pair of real numbers $(x, y)$ in a domain $D$ to a unique real output $z = f(x, y)$.

They are fundamental in multivariable calculus and have numerous applications in science, engineering, and economics. Here, $x$ and $y$ are the independent variables, while $z$ is the dependent variable.

Definition. A function of two variables assigns a unique real number to each ordered pair in its domain. It allows us to model physical and mathematical quantities that depend on more than a single factor.

Sketching graphs

Sketching 3D quadric surfaces relies heavily on finding their traces—the 2D cross-sections created when the surface intersects the coordinate planes ($x=0$, $y=0$, $z=0$).

The Ellipsoid

The general ellipsoid is a closed quadratic surface defined by $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1$, where $a$, $b$, and $c$ are the lengths of the semi-axes. The points (a, 0, 0), (0, b, 0), and (0, 0, c) lie on the surface.

For the equation $x^2+4y^2+z^2 = 4$, we divide the entire equation by 4 to find the standard form: $\frac{x^2}{4}+ \frac{y^2}{1}+ \frac{z^2}{4} = 1$. This is an ellipsoid with its center at the origin (0, 0, 0) where a = 2, b = 1, and c = 2 are the length of the semi-axes (Figure iv).

Traces:

• In the yz-plane (x = 0), $\frac{y^2}{1}+ \frac{z^2}{4} = 1$ we see an ellipse with a vertical semi-axis of length 2 (along the z-axis) and a horizontal semi-axis of length 1 (along the y-axis).
• In the xz-plane (y = 0), we see a circle centered at the origin with radius 2, $\frac{x^2}{4}+ \frac{z^2}{4} = 1$.
• In the xy-plane (z = 0), we see an ellipse with a horizontal axis of length 2 (along the x-axis) and a vertical axis of length 1 (along the y-axis), $\frac{x^2}{4}+ \frac{y^2}{1} = 1$.

Elliptic Paraboloids

Paraboloids open along the axis of the linear variable (the variable not being squared).

Case A: Opening along the $x$-axis ($x = y^2 + z^2$).

To any fixed x-value, $y^2 + z^2$ represent a circle in the yz-plane with radius $\sqrt{x}$. As x increases, the radius of the circle increases. It is a paraboloid. It opens along the x-axis because x is expressed as a sum of the squares of y and z (x must be positive) e.g., $x =0, y^2 + z^2 = 0$ is just a point, the origin (0, 0, 0). $x = 1, y^2 + z^2 = 1$ is a circle in the yz-plane with radius 1. $x = 2, y^2 + z^2 = 2$ is a circle in the yz-plane with radius $\sqrt{2}$ (Figure v).

Case B: Opening along the $z$-axis ($z = 3 - x^2 - y^2$)

Because $z$ is a decreasing function of the squares of $x$ and $y$, the paraboloid opens downwards. Its vertex is shifted up to $(0, 0, 3)$. For any fixed z-value, the equation $x^2 + y^2 = 3 - z$ represents a circle in the xy-plane with radius $\sqrt{3-z}$. For z = 3, $x^2 + y^2 = 0$. This is just a point at (0, 0, 3). For z = 2, $x^2 + y^2 = 1$. This is a circle in the xy-plane with radius 1. For z = 0, $x^2 + y^2 = 3$. This is a circle in the xy-plane with radius $\sqrt{3}$ (Figure vi).

Hyperbolic Paraboloid (Saddle Surface)

A hyperbolic paraboloid is a type of quadratic surface in three-dimensional space. It is often referred to as a “saddle surface” because of its distinctive shape, which curves upwards in one direction and downwards in the perpendicular direction. The general equation for a hyperbolic paraboloid is $z = \frac{y^2}{b^2}-\frac{x^2}{a^2}$

• $z = y^2 - x^2$ is known as a hyperbolic paraboloid. It is a saddle surface because it curves upwards in one direction (along the y-axis) and downwards in another direction (along the x-axis) (Figure i).

Analyzing Traces:

1. xy-Plane Trace ($z=0$): Set z = 0, $y^2 = x^2 \implies y = \pm x$. In the xy-plane, the trace is a pair of straight lines y = x and y = −x crossing at the origin.
2. xz-Plane Trace ($y=0$): Set y = 0. $z = -x^2$. This is a downward-opening parabola in the xz-plane. Fixed $y$-values produce parallel parabolas shifted upward by $y^2$.
3. yz-Plane Trace ($x=0$): Set x = 0. $z = y^2$. This is an upward-opening parabola in the yz-plane. Fixed $x$-values produce parallel parabolas shifted downward by $x^2$.

The surface is symmetric with respect to both the xz-plane and the yz-plane. At the origin (0, 0, 0), the function equals zero. Along the y-axis (x = 0), $z = y^2$ describes an upward-opening parabola. For both positive or negative values of y, the z-values increase as $y^2$, forming a parabolic shape opening upwards. For fixed x-values, such as $x_1, x_2, \cdots $ the equation $z=y^2-x_i^2$ represents a series of upward-opening parabolas shifted downward by the constant $x_i^2$. These parabolas have their vertices at $z = -x_i^2$ on the z-axis.

Along the x-axis (y = 0), $z = -x^2$ describes a downward-opening parabola. For both positive and negative values of x, the z-values decrease as $-x^2$, forming a parabolic shape opening downward. For fixed y-values, such as $y = y_1, y_2, \cdots$ the equation $z = y_i^2-x^2$ represents a series of downward-opening parabolas shifted upward by the constant $y_i^2$. These parabolas have their vertices at $z =y_i^2$ on the z-axis. This combination of upward and downward curvatures gives the surface its characteristic saddle shape.

The Double Cone

A double cone consists of two cones placed apex to apex (the vertex at the top or bottom). The double cone is given by the algebraic equation $\frac{z^2}{c^2}= \frac{x^2+y^2}{a^2}$.

For the specific equation $x^2 + y^2 = z^2$, the surface is a double cone centered at the origin. It is symmetric with respect to the x-, y-, and z-axes (Figure A).

Analyzing Traces

1. xy-Plane, z = 0, $x^2 + y^2 = 0$ is just the origin (0, 0, 0). z = 1, $x^2 + y^2 = 1$ represents a circle of radius 1. z = 2, $x^2 + y^2 = 2$ is a circle of radius $\sqrt{2}$. In other words, moving away from the origin, in planes parallel to the xy-plane (z = k), the surface form circles of increasing radius.
2. yz-Plane. Set x = 0, $y^2 = z^2 \implies z = \pm y$, these are two lines passing through the origin with slopes of 1 and -1. For different values of x, say $x = x_1, x_2, \cdots$ the equation becomes $y^2 = z^2 -x_i^2$. These represent hyperbolas opening in the y- and z-directions (Figure B).
3. xz-Plane. Set y = 0, $x^2 = z^2 \implies z = \pm x$, which are two lines passing through the origin with slopes of 1 and -1. For different values of y, say $y = y_1, y_2, \cdots $ the equation becomes $x^2 = z^2 -y_i^2$. These represent hyperbolas opening in the x- and z-directions.

In the xz- and yz-planes, the surface forms intersecting lines that pass through the origin and create a cone shape. Combining these traces, we see that the surface forms two identical cones, one opening upwards and the other downwards, meeting at the origin.

Two-Sheeted Hyperboloid

The equation $x^2 + y^2 = z^2 - 1$ is a specific case of the general two-sheeted circular hyperboloid $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2} = -1$. Rearranging our equation gives $z^2 - x^2 - y^2 = 1$, where $a = b = c = 1$ (Figure C).

The surface consists of two separate parts, one for $z \ge 1$ and one for $z \le -1$, which are symmetrical about the xy-plane.

Analyzing Traces

1. 📌 xy-Plane. Set z = 0, $x^2 + y^2 = -1$. This equation has no real solutions. Therefore, there are no intersections with the xy-plane.
   For z = ±1, $x^2 + y^2 = 0$. The only solutions are the points (0, 0, 1) and (by symmetry) (0, 0, -1).
   For z = ±2, $x^2 + y^2 = 3$. These are two circles with radius $\sqrt{3}$ centered at (0, 0, 2) and (0, 0, -2).
   Similarly, for any z = k where $|z| \gt 1, x^2 + y^2 = k^2 -1$, i.e., circles with radii $\sqrt{k^2-1}$.
   As ∣𝑧∣ increases, the circles grow. This is the hallmark of a two‑sheeted hyperboloid: each sheet expands outward as you move away from the “tip” at $z = \pm 1$.
2. 📌 yz-Plane. Set x = 0, $y^2 - z^2 = -1$. Rearranging gives: $z^2 - y^2 = 1$. This is a hyperbola opening upward and downward along the z‑axis. Vertices at $z = \pm 1$. Symmetric in 𝑦.
3. 📌 xz-Plane. Set y = 0, $x^2 = z^2 -1$. Rearranging gives:$z^2 - x^2 = 1$. Same hyperbola as before, but now symmetric in 𝑥.

🧠 This surface forms two separate, symmetrical sheets opening along the z-axis, creating the shape known as a two-sheeted hyperboloid. Each is shaped like a “flared bowl” opening along the z‑axis.

Bibliography

1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, Andrew Misseldine, and MathMajor, YouTube’s channels.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. MIT OpenCourseWare, 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007, YouTube.
8. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.