Math: Multivariable Functions

• Math 13.1: Chapter Overview
  • Chapter 13.1 introduces multivariable functions and their properties.
  • Multivariable functions depend on more than one independent variable.
  • The focus is on functions of two or three variables, denoted as f(x, y) or f(x, y, z).
  • Understanding domain, range, and level curves/surfaces is crucial.
  • Visualization is often achieved through graphs and contour plots.

Functions of Several Variables

• A function of several variables assigns a unique real number to each point in its domain.
• The domain of a function f(x, y) is the set of all points (x, y) in the xy-plane for which the function is defined; for f(x, y, z) the domain is a set of points in 3D space.
• The range is the set of all possible output values of the function.
• f(x, y) represents a function of two variables, and f(x, y, z) represents a function of three variables.
• Evaluating functions involves substituting the coordinates of a specific point into the function.
• If f(x, y) = x^2 + y, then f(2, 3) = 2^2 + 3 = 7.

Domain and Range

• Domain: Finding the domain involves identifying any restrictions on the input variables.
  • Common restrictions include division by zero, square roots of negative numbers, and logarithms of non-positive numbers.
  • For f(x, y) = sqrt(y - x^2), the domain is all (x, y) such that y >= x^2.
  • The domain can be visualized as a region in the xy-plane (for functions of two variables) or a region in 3D space.
• Range: Determining the range can be more complex, and is sometimes found by analyzing the function's behavior or by considering the possible values that the function can take.
  • For f(x, y) = x^2 + y^2, the range is [0, infinity) because squares are always non-negative.

Graphs of Functions of Two Variables

• The graph of a function f(x, y) is the set of all points (x, y, z) such that z = f(x, y).
• The graph is a surface in three-dimensional space.
• Software like Mathematica, Maple, or online graphing tools are commonly used for visualization.
• Analyzing cross-sections parallel to the xz-plane (y constant) or yz-plane (x constant) can help understand the shape.

Level Curves and Contour Plots

• A level curve of a function f(x, y) is a curve along which the function has a constant value, defined by f(x, y) = k, where k is a constant.
  • For f(x, y) = x^2 + y^2, the level curves are circles centered at the origin (x^2 + y^2 = k).
• A contour plot (or contour map) is a collection of level curves for different values of k.
• Contour plots provide a 2D representation of a 3D surface, showing function height.
• Closely spaced level curves indicate a steep slope; widely spaced curves indicate a gentle slope.

Level Surfaces

• A level surface of a function f(x, y, z) is a surface on which the function has a constant value, defined by f(x, y, z) = k, where k is a constant.
  • For f(x, y, z) = x^2 + y^2 + z^2, the level surfaces are spheres centered at origin.
• Visualizing level surfaces can be challenging but provides insight into the function's behavior in 3D space.

Examples of Multivariable Functions

• Linear Functions: Functions of the form f(x, y) = ax + by + c or f(x, y, z) = ax + by + cz + d.
• Quadratic Functions: Functions involving squared terms, such as f(x, y) = x^2 + y^2 or f(x, y) = xy.
• Polynomial Functions: Sums of terms involving non-negative integer powers of the variables.
• Rational Functions: Ratios of polynomials.
• Exponential and Logarithmic Functions: Functions involving exponential or logarithmic terms with multivariable inputs.
• Trigonometric Functions: Functions involving trigonometric functions with multivariable inputs.

Applications

• Physics: Describing physical quantities dependent on multiple spatial coordinates, like temperature distribution and electric potential.
• Engineering: Modeling systems with multiple design variables, such as optimizing airplane wing shape.
• Economics: Representing utility functions or production functions with multiple inputs.
• Computer Graphics: Defining surfaces and shapes in 3D space.

Key Concepts

• Multivariable functions extend the concept of single-variable functions to multiple inputs.
• Domain and range are essential for understanding the function's behavior and limitations.
• Graphs and level curves/surfaces are valuable tools for visualization.
• Applications are widespread in science, engineering, and economics.

Techniques and Skills

• Determining the domain and range of a multivariable function.
• Evaluating a multivariable function at a given point.
• Sketching or interpreting level curves and contour plots.
• Visualizing and analyzing surfaces in 3D space.
• Applying multivariable functions to model real-world phenomena.