Functions of two variables: An Overview

By failing to prepare, you are preparing to fail, Benjamin Franklin.

Recall: Vector Basics

Definition. A vector $\vec{AB}$ is a geometric object or quantity that has both magnitude (or length) and direction. Vectors in an n-dimensional Euclidean space can be represented as coordinates vectors in a Cartesian coordinate system.

Definition. The magnitude of the vector $\vec{A}$, also known as length or norm, is given by the square root of the sum of its components squared, $|\vec{A}|~ or~ ||\vec{A}|| = \sqrt{a_1^2+a_2^2+a_3^2}$, e.g., $\vert{}\vert{}\langle 3, 2, 1 \rangle\vert{}\vert{} = \sqrt{3^2+2^2+1^2} = \sqrt{14}$, $\vert{}\vert{}\langle 3, -4, 5 \rangle\vert{}\vert{} = \sqrt{3^2+(-4)^2+5^2} = \sqrt{50} = 5\sqrt{2}$, or $\vert{}\vert{}\langle 1, 0, 0 \rangle\vert{}\vert{} = \sqrt{1^2+0^2+0^2} = \sqrt{1} = 1$.

Basic Vector Operations

• Vector Addition: The sum of two vectors is the sum of their components, $\vec{A}+ \vec{B} = (a_1+b_1)\vec{i}+(a_2+b_2)\vec{j}+(a_3+b_3)\vec{k} = \langle a_1+b_1, a_2+b_2, a_3+b_3 \rangle$.

• Vector Subtraction: The subtraction of two vectors is similar to addition and is also done component-wise, it is given by simply subtracting their corresponding components (x, y, and z in 3D), $\vec{A} - \vec{B} = (a_1-b_1)\vec{i}+(a_2-b_2)\vec{j}+(a_3-b_3)\vec{k} = \langle a_1-b_1, a_2-b_2, a_3-b_3 \rangle$.

• Scalar multiplication is the multiplication of a vector by a scalar, a real number, changing its magnitude without altering its direction (unless the scalar is negative, which reverses the direction). It is effectively multiplying each component of the vector by the scalar value, $c\vec{A} = (ca_1)\vec{i}+(ca_2)\vec{j}+(ca_3)\vec{k} = \langle ca_1, ca_2, ca_3 \rangle$.

• The dot or scalar product is a fundamental operation between two vectors. It produces a scalar quantity that represents the projection of one vector onto another. The dot product is the sum of the products of their corresponding components: $\vec{A}·\vec{B} = \sum a_ib_i = a_1b_1 + a_2b_2 + a_3b_3.$, e.g. $\vec{A}·\vec{B} = \sum a_ib_i = \langle 2, 2, -1 \rangle \cdot \langle 5, -3, 2 \rangle = (2 \cdot 5) + (2 \cdot -3) + (-1 \cdot 2) = 10 - 6 - 2 = 2.$
  Geometrically, it is the product of their magnitudes multiplied by the cosine of the angle $\theta$ between them, $\boxed{\vec{A} \cdot \vec{B} = \vert{}\vert{}\vec{A}\vert{}\vert{} \cdot \vert{}\vert{}\vec{B}\vert{}\vert{} \cos(\theta)}$

• The projection of $\vec{a}$ onto $\vec{b}$ is $proj_{\vec{b}}\vec{a} = \frac{\vec{a}·\vec{b}}{||\vec{b}||^2}\vec{b}$.

• The cross product, denoted by $\vec{A} \times \vec{B}$, is a binary operation on two vectors in three-dimensional space. It yields a new vector that is perpendicular (orthogonal, at a right angle) to both original vectors and has a magnitude equal to the area of the parallelogram formed by the two input vectors.
  The cross product $\vec{A}x\vec{B}$ can be computed using the determinant of a $3 \times 3$ matrix formed by the standard basis vectors ($\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}$) and the components of the two vectors: $\vec{A} \times \vec{B} = det(\begin{smallmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\\ a_1 & a_2 & a_3\\\ b_1 & b_2 & b_3\end{smallmatrix}) =|\begin{smallmatrix}a_2 & a_3\\\ b_2 & b_3\end{smallmatrix}|\vec{i}-|\begin{smallmatrix}a_1 & a_3\\\ b_1 & b_3\end{smallmatrix}|\vec{j}+|\begin{smallmatrix}a_1 & a_2\\\ b_1 & b_2\end{smallmatrix}|\vec{k}$.
  Example: If $\vec{A} = \langle 1, 2, 3 \rangle$ and $\vec{B} = \langle 4, 5, 6 \rangle$: $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix} = (2\cdot 6 - 3\cdot 5)\hat{\mathbf{i}} - (1\cdot 6 - 3\cdot 4)\hat{\mathbf{j}} + (1\cdot 5 - 2\cdot 4)\hat{\mathbf{k}} = -3\hat{\mathbf{i}} + 6\hat{\mathbf{j}} - 3\hat{\mathbf{k}} = \langle -3, 6, -3 \rangle$

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.

Examples

• Geometry: Area of a triangle with base $b$ and height $h$: $A(b, h) = \frac{1}{2}bh$.
• Physics & Chemistry: Ideal Gas Law expressed for pressure $P(V, T)$: $P(V, T) = \frac{nRT}{V}$
• Simple linear functions. Flat surfaces in space: z = f(x, y) = 2x + 3y; z = f(x, y) = 3x + 7y -4.
• Quadratic Functions: Bowl-shaped surfaces: $f(x, y) = x^2 + y^2$.
• Exponential Models: Joint growth models: $f(x, y) = e^{x+y}$

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

Domain of Functions

The domain of a function of two variables $f(x, y)$ is the set of all ordered pairs $(x, y)$ in the $xy$-plane for which the expression $f(x, y)$ is defined as a real number. Here are some examples:

Functions naturally generalize to three or more variables, $f: \mathbb{R}^n \to \mathbb{R}$, where the domain of a three-variable function exists as a region in 3D space $\mathbb{R}^3$.

Visualizing 3D Surface Graphs

While single-variable functions $y = f(x)$ produce 2D curves, the graph of a two-variable function is typically a 2D surface in a three-dimensional Cartesian space $\mathbb{R}^3$.(Figure v). This surface represents all points (x, y, z) such that z = f(x, y).

To illustrate, consider the function $f(x, y) = x^2 + y^2$.

Here’s how you can visualize it:

1. Create a Grid: Imagine a grid on the xy-plane.
2. Calculate z: For each point (x, y) on the grid, calculate $z = x^2 + y^2$.
3. Plot the Points: Plot each point (x, y, z) in a three-dimensional space.
4. Connect the Points: The set of all these points forms a smooth surface (Illustration 1).

The graph is a paraboloid, opening upwards. At (0, 0), z is at its minimum value, 0. As you move away from the origin in any direction on the xy-plane, z increases quadratically.

Common Quadric Surfaces and Planes

Planes ($z = ax + by + c$): Flat surfaces extending infinitely.

• $f(x, y) = -y$, Figure vi. Because the variable x does not appear in the equation, the value of z depends entirely on y. The graph is a plane tilted along the $y$-axis (passing through $(0, 1, -1)$ and $(0, -1, 1)$).. The graph looks exactly like the line z = -y. This shape is then extended infinitely along the x-axis.

• f(x, y) = $1-x-\frac{1}{2}y$ is a plane, a flat surface that extends infinitely in all directions. The easiest way to plot a plane is to find the x-, y-, and z-intercepts, namely $x$-intercept: $(1, 0, 0)$, $y$-intercept: $(0, 2, 0)$, and $z$-intercept: $(0, 0, 1)$ (Illustration 3).

Paraboloids

• Upward Circular Paraboloid: $z = x^2 + y^2$ has a global minimum at $(0, 0, 0)$ and increases quadratically outward in all directions.

• Downward Paraboloid: $z = 1 - x^2 - y^2$ has a vertex at $(0, 0, 1)$ (Figure vii). Notice that in the yz plane, where x = 0, the graph is a parabola $z = 1 -y^2$. Similarly, in the xz plane, where y = 0, the graph is a parabola $z = 1 -x^2$ and in the xy plane, where z = 0, $1 = x^2 + y^2$ forms the unit circle. The graph is a paraboloid opening downward.

• Upper Hemisphere: f(x, y) = $\sqrt{4-x^2-y^2}↭[\text{Rewrite in Terms of z}] z^2 = 4 -x^2 -y^2 ↭ x^2 +y^2 + z^2 = 4$. This equation represents a sphere centered at the origin with a radius of 2. However, since $z = f(x, y) = \sqrt{4-x^2-y^2}$ is non-negative, the graph represents the upper hemisphere (Illustration 2) centered at the origin with a radius of 2.

Cylinders (The Missing Variable Rule)

In $\mathbb{R}^3$, an equation that contains only two of the three variables is called a cylinder. The cross-sectional curve defined by the equation is swept infinitely along the axis of the missing variable.

• Circular Cylinder ($x^2 + y^2 = 4$). The missing variable is $z$. It is a cylinder with radius 2 whose axis runs parallel to the z-axis. In the xy-plane, the equation $x^2 + y^2 = 4$ is a circle centered at the origin (0, 0) with radius 2. This circle extends infinitely upward and downward parallel to the z-axis (Figure i).

• Parabolic Cylinder ($y = z^2$): The missing variable is $x$. It describes a parabola in the yz plane. Extending this parabola along the x-axis forms a parabolic cylinder that extends infinitely along the x-axis (Figure ii).

• Sinusoidal Cylinder $y = \cos(x)$, describes a cosine wave in the xy-plane. Extending this wave along the z-axis forms a surface that oscillates with the cosine function (Figure iii) like a series of waves running parallel to the z-axis.

Contour Plots and Level Curves

Definition. Contour plots are used to represent three-dimensional data on a two-dimensional surface. They are similar to topographical maps, where instead of plotting longitude, latitude, and elevation, we plot x-, y-, and z-values.

A contour plot displays level curves of a function z = f(x, y), i.e., it shows all the points where z = f(x, y) equals some fixed constant, aiding in the visualization of the function’s gradient and overall shape, e.g., $z = x^2 + y^2$.

A level curve is simply a cross section of the graph of z = f(x,y) taken at a constant value, say z = c. A level curve can be interpreted as a contour line on a map, where each line represents points with the same elevation.

Creating a Contour Plot:

1. Choose Intervals: Select regular intervals for the constant values of z.
2. Plot Contour Lines: For each interval, plot the set of (x, y) points that satisfy $z = f(x, y) = c.$

Key Properties of Level Curves

• Steepness: When level curves are packed tightly together, $z$ changes rapidly over a short distance in the $xy$-plane, indicating a steep slope/gradient. When level curves are far apart, the surface is relatively flat.

• Concentric Patterns: Concentric closed loops typically indicate local mountains/peaks (maxima) or pits (minima).

Applications of Level Curves

Level curves translate complex 3D surface data into actionable 2D maps across multiple fields:

• Meteorology:
  Isobars: Lines connecting points of equal atmospheric pressure on a weather map. Closely spaced isobars signal high wind speeds.
  Isotherms: Lines connecting points of equal temperature.
• Topographical Maps: Contour lines represent constant ground elevation above sea level. Points on the same line have the same altitude. Clustered lines highlight cliffs or steep mountainsides.
• Economics:
  Indifference Curves: Combinations of two goods $(x, y)$ that yield the exact same utility or satisfaction to a consumer.
  Isoquants: Combinations of inputs (e.g., labor $L$, capital $K$) that produce a constant output quantity.

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.