Functions of two variables II

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

Recall

A function of two variables assigns a unique real number to each ordered pair in its domain.

Typically, we express such functions as z = f(x, y), where x and y are the independent variables, and z is the dependent variable.

The domain of a function of two variables is the set of all possible input values for both variables x and y.

The graph of a two-variable function is typically a surface in a three-dimensional Cartesian space . This surface represents all points (x, y, z) such that z = f(x, y).

Sketching graphs

• $x^2+4y^2+z^2 = 4 ↭ \frac{x^2}{4}+ \frac{y^2}{1}+ \frac{z^2}{4} = 1$ is an ellipsoid with its center at the origin (0, 0, 0) where a = 2, b = 1, and c = 1 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 axis of length 2 (along the z-axis) and a horizontal 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), $\frac{x^2}{4}+ \frac{y^2}{1} = 1$.

The general ellipsoid is a quadratic surface which is defines in Cartesian coordinates as $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1$ where a, b, and c are the length of the semi-axes of the ellipsoid. The points (a, 0, 0), (0, b, 0), and (0, 0, c) lie on the surface.

• $x = y^2 + z^2$. To any fixed x-value, y² + x² 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 e.g., x = 0, y²+z² = 0 is just a point, the origin (0, 0, 0). x = 1, y²+z² = 1 is a circle in the yz-plane with radius 1. y²+z² = 4 is a circle in the yz-plane with radius 2 (Figure v).

The vertex of the paraboloid is at the origin (0, 0, 0). y = 0, x = z² is a parabola. z = 0, x = y² is a parabola, too.

• z = y²-x² 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).

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 $z = \frac{y^2}{b^2}-\frac{x^2}{a^2}$

Analyzing Traces

1. xy-Plane Trace: Set z = 0, y² = x²⇒y = ± x. In the xy-plane, the trace is a pair of lines y = x and y = −x (This trace is not specially helpful).
2. xz-Plane Trace: Set y = 0. z=−x². This is a downward-opening parabola in the xz-plane.
3. yz-Plane Trace: Set x = 0. z = y². This is an upward-opening parabola in the yz-plane.

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 = y² describes an upward-opening parabola. For both positive or negatives values of y, the z-values increase as y², forming a parabolic shape opening upwards. For fixed x-values, such as x₁, x₂, ··· 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 = −x² describes an downward-opening parabola. For both positive and negative values of x, the z-values decrease as -x², forming a parabolic shape opening downward. For fixed y-values, such as y = y₁, y₂,···, 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.

• z = 3 -x²-y². xy-trace: 0 = 3 -x² -y², it is a circle of radius $\sqrt{3}$. For any fixed z-value, the equation x² + y² = 3 -z represents a circle in the xy-plane with radius $\sqrt{3-z}$. For z = 3, x² + y² = 3 -3 = 0. This is just a point at (0, 0, 3). For z = 2, x² + y² = 3 -2. This is a circle in the xy-plane with radius 1. For z = 0, x² + y² = 3 -0 = 2. This is a circle in the xy-plane with radius $\sqrt{3}$ (Figure vi).

It is a paraboloid. It opens downwards along the z-axis because z is expressed as a decreasing function of the sum of the squares of x and y. The vertex of the paraboloid is (0, 0, 3). For x = 0, z = 3 -y² is a parabola. For y = 0, z = 3 -x² is a parabola, too.

• x²+y² = z². It is a double cone centered at the origin. It is symmetric with respect to the x-, y-, and z-axes (Figure A).

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

Analyzing Traces

1. xy-Plane, z = 0 x²+y² = 0 is just the origin (0, 0, 0). z = 1, x²+y² = 1 represents a circle of radius 1. z = 2, x² + y² = 2 is a circle of radius $\sqrt{2}$. In other words, moving away from the origin, in planes parallel to the xy-plane, the surface form circles of increasing radius.
2. yz-Plane. Set x = 0, y² = z², z = ±y, these are two lines passing through the origin with slopes of 1 and -1. For different values of x, say x = x₁, x = x₂,… 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, x² = z², z = ±x, which are two lines passing through the origin with slopes of 1 and -1. For different values of y, say y = y₁, y = y₂,… 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.

• x² + y² = z² -1. This equation is a specific case of a more general equation for a two-sheeted circular hyperboloid along the z-axis $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2} = -1$. In our case a = b = c = 1 (Figure C).

The surface consists of two separate parts, one for z ≥ 1 and one for z ≤ −1, which are symmetrical about the xy-plane.

Analyzing Traces

1. xy-Plane. Set z = 0, x²+y² = -1. This equation has no real solutions. Therefore, there are no intersections with the xy-plane. For z = ±1, x² + y² = 0. The only solutions are the points (0, 0, 1) and (by symmetry) (0, 0, -1). For z = ±2, x²+y² = 2²-1 = 3. These are two circles with radius $\sqrt{3}$ centered at (0, 0, 2) and (0, 0, -2). Similarity, for any z = k where |z| ≥ 1, x² + y² = k² -1, i.e., circles with radii $\sqrt{k^2-1}$.
2. yz-Plane. Set x = 0, y² - z² = -1. Rearranging gives: z² -y² = 1. This is a hyperbola opening upwards in the y- and z-directions.
3. xz-Plane. Set y = 0, x² = z² -1. Rearranging gives: z² -x² = 1. This is also a hyperbola opening upwards in the x- and z-directions.

This surface forms two separate, symmetrical sheets opening along the z-axis, creating the shape known as a two-sheeted hyperboloid.

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.