Chapter 5: Functions of Several Variables
Introduction
Many real-world problems depend on multiple variables. Production depends on labor and capital. Temperature varies with position and time. This chapter extends single-variable calculus to multivariable functions, introducing partial derivatives, and important theorems for functions of multiple variables.
1. Functions of Two Variables
Definition
A function of two variables is f(x, y) where each pair (x, y) maps to a single value z = f(x, y). Example: f(x, y) = x^2 + y^2 (distance squared from origin). The domain is a region in the xy-plane; the range is a set of z-values.
Graphical Representation
A function of two variables can be graphed as a surface in 3D space. Contour lines (level curves) show where f(x, y) = c for constant c—useful for understanding the function without 3D visualization.
2. Partial Derivatives
Definition
The partial derivative of f with respect to x, denoted ∂f/∂x, is the derivative treating y as a constant. Similarly, ∂f/∂y treats x as constant. These measure the rate of change in each direction independently.
Computing Partial Derivatives
To find ∂f/∂x, differentiate f(x, y) with respect to x, treating y as a constant. Example: f(x, y) = x^2y + 3xy^2. Then ∂f/∂x = 2xy + 3y^2 and ∂f/∂y = x^2 + 6xy.
Higher-Order Partial Derivatives
Second partial derivatives: ∂²f/∂x², ∂²f/∂y², and mixed partials ∂²f/∂x∂y. For nice functions (continuous second derivatives), the mixed partials are equal: ∂²f/∂x∂y = ∂²f/∂y∂x.
3. Homogeneous Functions and Euler's Theorem
Homogeneous Functions
A function f is homogeneous of degree n if f(tx, ty) = t^n f(x, y) for all t > 0. Example: f(x, y) = x^2 + xy is homogeneous of degree 2 because f(tx, ty) = (tx)^2 + (tx)(ty) = t^2(x^2 + xy) = t^2 f(x, y).
Euler's Theorem
If f is homogeneous of degree n, then x(∂f/∂x) + y(∂f/∂y) = nf(x, y). This elegant theorem connects partial derivatives to the function value and its degree. For f(x, y) = x^2 + xy (degree 2): x(2x + y) + y(x + 0) = 2x^2 + 3xy... wait, let me recalculate. ∂f/∂x = 2x + y, ∂f/∂y = x. So x(2x + y) + y(x) = 2x^2 + 2xy = 2(x^2 + xy) = 2f(x, y). Correct.
4. Implicit Functions
Implicit Differentiation
An implicit function relates x and y without explicit solution for y. Example: x^2 + y^2 = 1 implicitly defines y as a function of x. To find dy/dx, differentiate both sides with respect to x, treating y as a function of x: 2x + 2y(dy/dx) = 0, so dy/dx = -x/y.
Partial Derivatives of Implicit Functions
For F(x, y) = 0 implicitly defining y = f(x), we have dy/dx = -(∂F/∂x)/(∂F/∂y). This formula avoids the tedious implicit differentiation step.
5. Extreme Values for Multivariable Functions
Critical Points
Critical points occur where ∂f/∂x = 0 and ∂f/∂y = 0 (or partial derivatives don't exist). These are candidates for local maxima, minima, or saddle points.
Second Derivative Test
Compute the Hessian matrix H = [∂²f/∂x², ∂²f/∂x∂y; ∂²f/∂y∂x, ∂²f/∂y²]. At a critical point (a, b): if det(H) > 0 and ∂²f/∂x² > 0, there's a local minimum. If det(H) > 0 and ∂²f/∂x² < 0, there's a local maximum. If det(H) < 0, it's a saddle point.
Conclusion
Multivariable calculus extends the powerful tools of single-variable calculus to real-world problems with multiple inputs. Partial derivatives, Euler's theorem, and optimization techniques are essential for applications in economics, physics, and engineering where many variables interact.