4.1 Functions of Several Variables

4 min read•june 24, 2024

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. , contour maps, and help us grasp their behavior. We'll learn to identify , analyze , 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)f(x, y), the domain is a subset of the two-dimensional real coordinate space R2\mathbb{R}^2 ()
    • For functions of three variables, f(x,y,z)f(x, y, z), the domain is a subset of the three-dimensional real coordinate space R3\mathbb{R}^3 ()
  • 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 R\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)
  • and are important properties that affect the behavior of over their domains

3D plots and contour maps

  • 3D plots provide a visual representation of functions of two variables, f(x,y)f(x, y), in three-dimensional space
    • The graph is a surface in R3\mathbb{R}^3 with points (x,y,f(x,y))(x, y, f(x, y)) representing the function's output for each input pair (x,y)(x, y)
    • Allows for visualization of the function's behavior, including hills, valleys, and (peaks, troughs, passes)
  • Contour maps offer a two-dimensional representation of a function of two variables by showing
    • 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 (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)f(x, y), a level curve is the set of points (x,y)(x, y) satisfying the equation f(x,y)=cf(x, y) = c, where cc is a constant
    • Level curves represent the intersection of the graph of f(x,y)f(x, y) with a horizontal plane at height cc (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)f(x, y, z), a level surface is the set of points (x,y,z)(x, y, z) satisfying the equation f(x,y,z)=cf(x, y, z) = c, where cc is a constant
    • Level surfaces represent the intersection of the graph of f(x,y,z)f(x, y, z) with a hyperplane at height cc 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. : points where the function value is lower than its neighbors (valleys)
    2. : 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)f(-x) = f(x) (mirror symmetry)
    • Odd symmetry: f(−x)=−f(x)f(-x) = -f(x) (rotational symmetry)

Analysis of Multivariable Functions

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

Key Terms to Review (31)

∂z/∂x: ∂z/∂x, read as 'partial derivative of z with respect to x,' represents the rate of change of the function z with respect to the variable x, while holding all other variables constant. It is a fundamental concept in the study of functions of several variables.
∇f: The gradient of a function f(x,y,z) is a vector field that represents the direction and rate of change of the function at a given point. It is denoted by the symbol ∇f and is a fundamental concept in multivariable calculus.
3D plots: 3D plots are graphical representations that display data points in a three-dimensional space, allowing for the visualization of functions of several variables. They enhance understanding of how these functions behave in relation to two independent variables and one dependent variable, often representing complex surfaces or regions. By incorporating depth along with height and width, 3D plots provide a clearer picture of multivariable relationships that cannot be effectively captured in two dimensions.
Chain Rule for Partial Derivatives: The chain rule for partial derivatives is a method used to differentiate a composite function involving multiple independent variables. It allows for the calculation of the partial derivative of a function with respect to one variable, while treating the other variables as functions of that variable.
Continuity: Continuity is a fundamental concept in mathematics that describes the smoothness and connectedness of a function or curve. It is a crucial property that ensures a function behaves in a predictable and well-behaved manner, allowing for the application of various mathematical tools and techniques.
Contour Plot: A contour plot is a graphical representation of a three-dimensional surface, where the elevation or value of a function at a particular point is indicated by the distance of the contour line from a reference point. It is a way to visualize and analyze functions of two variables, such as those encountered in the topics of Functions of Several Variables and Partial Derivatives.
Critical Points: Critical points refer to the points in a function of several variables where the partial derivatives of the function vanish or become undefined. These points are of particular importance in the analysis of maxima, minima, and saddle points of a function, as well as in the application of Lagrange multipliers to constrained optimization problems.
Cylindrical Coordinates: Cylindrical coordinates are an alternative coordinate system used to describe the position of a point in three-dimensional space. Unlike the traditional Cartesian coordinate system, cylindrical coordinates use a radial distance, an angle, and a height to uniquely identify a location, providing a more natural way to represent certain geometric shapes and physical phenomena.
Differentiability: Differentiability is a fundamental concept in calculus that describes the smoothness and continuity of a function. It is a crucial property that determines the behavior and characteristics of a function, particularly in the context of functions of several variables and the construction of tangent planes.
Directional Derivative: The directional derivative is a measure of the rate of change of a function in a specific direction at a given point. It represents the slope of the function in a particular direction, providing information about how the function is changing along that direction.
Gradient: The gradient is a vector that represents the direction and rate of the fastest increase of a scalar function. It provides essential information about how a function changes in space, connecting to concepts such as optimizing functions, understanding the behavior of multi-variable functions, and exploring the properties of vector fields.
Implicit Differentiation: Implicit differentiation is a technique used to find the derivative of a function that is implicitly defined, meaning the function cannot be easily expressed in terms of a single dependent variable. This method allows for the differentiation of equations where the variables are not explicitly solved for.
Jacobian Matrix: The Jacobian matrix is a square matrix of all first-order partial derivatives of a vector-valued function. It represents the sensitivity of a set of functions with respect to changes in their input variables and is a crucial concept in multivariable calculus and change of variables for multiple integrals.
Level Curves: Level curves, also known as contour lines, are two-dimensional curves that represent the set of points in a function of two variables where the function has a constant value. They are used to visualize the behavior of a function over a region in the xy-plane.
Level Surfaces: Level surfaces are geometric constructs that represent points in space where a function of multiple variables maintains a constant value. They are essential in the study of multivariable calculus, particularly in the context of functions of several variables and the application of Lagrange multipliers.
Limits: Limits are a fundamental concept in calculus that describe the behavior of a function as it approaches a particular point or as the input values approach a specific value. They are crucial in understanding the properties and behavior of functions of several variables.
Local maxima: Local maxima are points in a function of several variables where the function value is higher than that of neighboring points in its immediate vicinity. These points represent relative peaks within a given area, contrasting with local minima, which indicate relative troughs. Understanding local maxima is essential for analyzing the behavior of functions and optimizing processes, particularly in applications involving multiple variables.
Local Minima: A local minimum is a point on a function where the function value is less than or equal to the function values in the immediate surrounding area, but not necessarily the absolute lowest value of the function. It represents a point where the function temporarily reaches a minimum before potentially increasing again.
Monotonicity: Monotonicity is a property of functions that describes their behavior as the input variable increases or decreases. A function is considered monotonic if it either consistently increases or consistently decreases over its domain, without any local maxima or minima.
Multivariable Functions: A multivariable function is a function that depends on more than one independent variable. These functions are used to model complex real-world phenomena that cannot be adequately described by a single variable. They are central to the study of calculus of several variables, which extends the concepts of limits, continuity, differentiation, and optimization from functions of a single variable to functions of multiple variables.
Partial Derivative: A partial derivative is a derivative of a function of multiple variables taken with respect to one variable while keeping the other variables constant. This concept is essential for understanding how functions change in multiple dimensions, allowing for analysis of surfaces and their behaviors. Partial derivatives help us explore the gradients and directional changes in a multivariable setting, connecting directly to optimization, differential equations, and surface analysis.
Partial Differentiation: Partial differentiation is the process of taking the derivative of a function with respect to one of its variables, while treating the other variables as constants. It allows for the analysis of how a function changes with respect to a specific variable, independent of the other variables.
Partial Order: A partial order is a binary relation on a set that satisfies the properties of reflexivity, antisymmetry, and transitivity. It is a way of comparing and ordering elements within a set based on a specific criterion, while allowing for the possibility that some elements may not be comparable to each other.
Quadric Surfaces: Quadric surfaces are a class of three-dimensional geometric shapes that are defined by quadratic equations in three variables. These surfaces are fundamental in the study of multivariable calculus and play a crucial role in understanding the behavior of functions of several variables.
Saddle Points: A saddle point is a special type of critical point that occurs in functions of several variables. It represents a point where the function has a local maximum in one direction and a local minimum in another direction, resembling the shape of a saddle.
Scalar field: A scalar field is a mathematical function that assigns a single scalar value to every point in a space. It helps describe various physical phenomena, such as temperature or pressure, which can vary from point to point. Scalar fields are essential in understanding how certain quantities change in multiple dimensions, allowing for better analysis of complex systems.
Spherical Coordinates: Spherical coordinates are a three-dimensional coordinate system that uses three values, $r$, $\theta$, and $\phi$, to specify the location of a point in space. This system provides a natural way to describe positions on the surface of a sphere or within a spherical volume, and is widely used in various fields of mathematics, physics, and engineering.
Tangent plane: A tangent plane is a flat surface that touches a curved surface at a single point and represents the best linear approximation of that surface near that point. It provides insight into how functions of several variables behave in the vicinity of a specific point, and it relies on the concept of partial derivatives to define its slope and orientation. By understanding tangent planes, one can also explore linear approximations and the behavior of functions in different directions, especially in relation to gradients and directional derivatives.
Vector Field: A vector field is a function that assigns a vector to every point in a given space, such as a plane or three-dimensional space. It describes the magnitude and direction of a quantity, such as a force or a flow, at every point in that space.
Xy-plane: The xy-plane, also known as the horizontal plane, is a fundamental concept in three-dimensional Cartesian coordinate systems. It is one of the three principal planes that intersect at the origin, forming a three-dimensional coordinate system. The xy-plane is defined as the plane that is perpendicular to the z-axis and contains the x and y axes.
Xyz-Space: xyz-Space, also known as three-dimensional Cartesian space, is a mathematical construct that represents the physical world in which we live. It is a three-dimensional coordinate system where points are defined by their x, y, and z coordinates, allowing for the representation and analysis of objects, functions, and relationships in a three-dimensional space.
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