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2: Functions of Several Variables

• • Contributed by Michael Corral
• Professor (Mathematics) at Schoolcraft College

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In single-variable calculus, we were concerned with functions that map the real numbers to real numbers, sometimes called "real functions of one variable'', meaning the "input'' is a single real number and the "output'' is likewise a single real number. In the last chapter we considered functions taking a real number to a vector, which may also be viewed as functions, that is, for each input value we get a position in space. Now we turn to functions of several variables, meaning several input variables, functions. We will deal primarily $$\mathbb{R}^2$$ and $$\mathbb{R}^3$$ spaces, however, many of the techniques we discuss can be applied to larger dimensions spaces as well.

• 2.1: Functions of Two or Three Variables
We will now examine real-valued functions of a point (or vector) in $$\mathbb{R}^2$$ or $$\mathbb{R}^3$$ . For the most part these functions will be defined on sets of points in  $$\mathbb{R}^2$$ , but there will be times when we will use points in $$\mathbb{R}^3$$ , and there will also be times when it will be convenient to think of the points as vectors (or terminal points of vectors).
• 2.2: Partial Derivatives
Now that we have an idea of what functions of several variables are, and what a limit of such a function is, we can start to develop an idea of a derivative of a function of two or more variables.
• 2.3: Tangent Plane to a Surface
Since the derivative dy/dx of a function y=f(x) is used to find the tangent line to the graph of f (which is a curve in R2), you might expect that partial derivatives can be used to define a tangent plane to the graph of a surface z=f(x,y). This indeed turns out to be the case. First, we need a definition of a tangent plane. The intuitive idea is that a tangent plane “just touches” a surface at a point. The formal definition mimics the intuitive notion of a tangent line to a curve.
• 2.4: Directional Derivatives and the Gradient
For a function z=f(x,y), we learned that the partial derivatives ∂f/∂x and∂f/∂y represent the (instantaneous) rate of change of f in the positive x and y directions, respectively. What about other directions? It turns out that we can find the rate of change in any direction using a more general type of derivative called a directional derivative.
• 2.5: Maxima and Minima
The gradient can be used to find extreme points of real-valued functions of several variables, that is, points where the function has a local maximum or local minimum. We will consider only functions of two variables; functions of three or more variables require methods using linear algebra.
• 2.6: Unconstrained Optimization: Numerical Methods
The types of problems that we solved previously were examples of unconstrained optimization problems.  If the equations involve polynomials in x and y  of degree three or higher, or complicated expressions involving trigonometric, exponential, or logarithmic functions, then solving  could be impossible by elementary means and the only choice may be to find a solution using some numerical method which gives a sequence of numbers which converge to the actual solution (e.g., Newton’s method).
• 2.7: Constrained Optimization - Lagrange Multipliers
In this section we will use a general method, called the Lagrange multiplier method, for solving constrained optimization problems. Points (x,y) which are maxima or minima of f(x,y) with the condition that they satisfy the constraint equation g(x,y)=c  are called constrained maximum or constrained minimum points, respectively. Similar definitions hold for functions of three variables.  The Lagrange multiplier method for solving such problems.
• 2.E: Functions of Several Variables (Exercises)
Problems and select solutions for the chapter.

Thumbnail: Real function of two real variables. Image used with permission (Public Domain; Maschen).