4.1 Functions of Several Variables

Functions of several variables expand our mathematical toolkit beyond single-variable calculus. They allow us to model complex relationships in multiple dimensions, opening doors to real-world applications in physics, engineering, and economics.

Visualizing these functions becomes crucial. 3D plots, contour maps, and level surfaces help us grasp their behavior. We'll learn to identify critical points, analyze monotonicity, and understand symmetry, building a solid foundation for multivariable calculus.

Functions of Several Variables

Domains and ranges of multivariable functions

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• Domain of a multivariable function represents the set of all possible input values for which the function is defined
  • For functions of two variables, $f(x, y)$, the domain is a subset of the two-dimensional real coordinate space $\mathbb{R}^2$ (xy-plane)
  • For functions of three variables, $f(x, y, z)$, the domain is a subset of the three-dimensional real coordinate space $\mathbb{R}^3$ (xyz-space)
• Range of a multivariable function represents the set of all possible output values that the function can produce
  • For functions of two or more variables, the range is always a subset of the one-dimensional real number line $\mathbb{R}$ (y-axis)
• Restrictions on the domain are determined by the context of the problem or the function's definition to ensure the function remains well-defined and real-valued
  • Avoid division by zero (rational functions)
  • Avoid square roots of negative numbers (even-powered roots)
  • Avoid logarithms of non-positive numbers (logarithmic functions)
• Continuity and differentiability are important properties that affect the behavior of multivariable functions over their domains

3D plots and contour maps

• 3D plots provide a visual representation of functions of two variables, $f(x, y)$, in three-dimensional space
  • The graph is a surface in $\mathbb{R}^3$ with points $(x, y, f(x, y))$ representing the function's output for each input pair $(x, y)$
  • Allows for visualization of the function's behavior, including hills, valleys, and saddle points (peaks, troughs, passes)
• Contour maps offer a two-dimensional representation of a function of two variables by showing level curves
  • Level curves are curves along which the function has a constant value, labeled with the corresponding function value (elevation)
  • Provides information about the function's behavior and gradient (steepness) in different regions
  • Contour lines close together indicate steep gradients, while lines far apart indicate shallow gradients (cliffs vs. plains)

Level curves and surfaces

• For a function $f(x, y)$, a level curve is the set of points $(x, y)$ satisfying the equation $f(x, y) = c$, where $c$ is a constant
  • Level curves represent the intersection of the graph of $f(x, y)$ with a horizontal plane at height $c$ (slicing the surface)
  • Provide information about the function's behavior and gradient in different regions of the domain (terrain features)
• For a function $f(x, y, z)$, a level surface is the set of points $(x, y, z)$ satisfying the equation $f(x, y, z) = c$, where $c$ is a constant
  • Level surfaces represent the intersection of the graph of $f(x, y, z)$ with a hyperplane at height $c$ in three-dimensional space (slicing the hypersurface)
  • Provide information about the function's behavior and gradient in different regions of the domain in three-dimensional space (3D terrain features)

Geometric properties from graphs

• Critical points are points where the gradient of the function is zero or undefined, indicating potential local extrema or saddle points
  1. Local minima: points where the function value is lower than its neighbors (valleys)
  2. Local maxima: points where the function value is higher than its neighbors (peaks)
  3. Saddle points: points where the function value is a minimum in one direction and a maximum in another (passes)
• Monotonicity describes the increasing or decreasing behavior of the function along a particular direction, determined by analyzing the function's partial derivatives (slopes)
  • Increasing: function values grow larger as inputs increase (uphill)
  • Decreasing: function values grow smaller as inputs increase (downhill)
• Concavity describes whether the function curves upward (concave up) or downward (concave down) in a particular direction, determined by analyzing the function's second partial derivatives (curvature)
  • Concave up: function curves upward, like a cup (valley)
  • Concave down: function curves downward, like a dome (hill)
• Symmetry in functions may exist with respect to the coordinate axes, the origin, or other lines or planes, simplifying the analysis and understanding of the function's behavior (reflections)
  • Even symmetry: $f(-x) = f(x)$ (mirror symmetry)
  • Odd symmetry: $f(-x) = -f(x)$ (rotational symmetry)

Analysis of Multivariable Functions

• Limits of multivariable functions describe the behavior of the function as it approaches a specific point or region in its domain
• The Jacobian matrix, containing all first-order partial derivatives, is used to analyze the local behavior of multivariable functions
• Partial order relations can be used to compare vectors in multidimensional spaces, extending the concept of inequality to higher dimensions