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Functions of two variables.

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

Recall

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., $||< 3, 2, 1 >|| = \sqrt{3^2+2^2+1^2}=\sqrt{14}$, $||< 3, -4, 5 >|| = \sqrt{3^2+(-4)^2+5^2}=\sqrt{50}=5\sqrt{2}$, or $||< 1, 0, 0 >|| = \sqrt{1^2+0^2+0^2}=\sqrt{1}=1$.

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Functions of two variables: An Overview

Functions of two variables are natural generalizations or extensions of functions of one variable. They are mathematical functions f: ℝ x ℝ → ℝ, meaning they take two real input (x, y) and produce one real number output f(x, y). Essentially, they map each pair of real numbers (x, y) to a third real number f(x,y).

They are fundamental in multivariable calculus and have numerous applications in science, engineering, and economics.

Definition

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

Examples

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

Domain of Functions

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

  1. f(x, y) = $\sqrt{x+y}$, Domain(f) = {x, y ∈ ℝ x ℝ : x + y ≥ 0}.
  2. f(x, y) = $\frac{1}{x+y}$, Domain(f) = {x, y ∈ ℝ x ℝ : x + y ≠ 0}.
  3. f(x, y) = ln(x·y), Domain(f) = {x, y ∈ ℝ x ℝ : x·y > 0}.
  4. f(x, y, z) = $\frac{x}{\sqrt{9-x^2-y^2-z^2}}, 9-x^2-y^2-z^2 > 0 ↭ 9 > x^2 + y^2 + z^2, Domain(f) = {x, y, z ∈ ℝ x ℝ x ℝ : x^2 + y^2 + z^2 < 9}$. Example: f(2, 1, 1) = $\frac{2}{\sqrt{3}}$

Graphs of Two-Variable Functions

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

To illustrate, consider the function f(x, y) = x2 + y2.

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 = x2 + y2.
  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.

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Contour plots

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 = x2 + y2 (Illustration 4), Figure via, Figure ia, Figure viia.

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.

If the level curves are getting closer and closer, the function is getting steeper and steeper (picture it like a mountain)

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Examples

Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].
  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.
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