A real-valued function, f, of x, y, z, ... is a rule for manufacturing a new number, written f(x, y, z, ...), from the values of a sequence of independent variables (x, y, z, ...).

The function f is called a real-valued function of two variables if there are two independent variables, a real-valued function of three variables if there are three independent variables, and so on.

As with functions of one variable, functions of several variables can be represented numerically (using a table of values), algebraically (using a formula), and sometimes graphically (using a graph).

1.	f(x, y) = x - y	Function of two variables
	f(1, 2) = 1 - 2 = -1	Substitute 1 for x and 2 for y
	f(2, -1) = 2 - (-1) = 3	Substitute 2 for x and -1 for y
	f(y, x) = y - x	Substitute y for x and x for y
2.	h(x, y, z) = x + y + xz	Function of three variables
	h(2, 2, -2) = 2 + 2 + 2(-2) = 0	Substitute 2 for x, 2 for y, and -2 for z.

3. Let P(x, y) = x² + xy - y². Complete the following table of values of p, and press "Check".

		x →
		-1	0	1
y	-1
↓	0
	1

 

Linear Functions
A linear function of the variables x₁, x₂, ... , x_n is a function of the form

f(x₁, x₂, ... , x_n) = a₀ + a₁x₁ + ... + a_nx_n

Interaction Functions
If we add to a linear function one or more terms of the form bx_ix_j (b constant), we get a second order interaction function.

Distance Functions
The distance in the plane from the point (x, y) to the point (a, b) can be expressed as a function of x and y:

d(x, y) = [(x - a)² + (y - b)²]^1/2.

(Special case of above formula) The distance in the plane from the point (x, y) to the origin is given by

d(x, y) = [x² + y²]^1/2.

The distance in 3-space from the point (x, y, z) to the point (a, b, c) is given by

d(x, y, z) = [(x - a)² + (y - b)² + (z - c)²]^1/2.

Points in three-dimensional space have three coordinates as shown in the following figure.

• The x-coordinate of a point is its distance in front of the yz-plane.
  (If the x-coordinate is negative, the point is behind the yz-plane.)
• The y-coordinate of a point is its distance to the right of the xz-plane.
  (If the y-coordinate is negative, the point is to the left of the xz-plane.)
• The z-coordinate of a point is its height above the xy-plane.
  (If the z-coordinate is negative, the point is below the xy-plane.)

Graph of a Function of Two Variables
The graph of the function f of two variables is the set of all points (x, y, f(x, y)) in three-dimensional space, where we restrict the values of (x, y) to lie in the domain of f. In other words, the graph is the set of all the points (x, y, z) with z = f(x, y).

The following figure shows where the point (1, 2, 3) is located in three-dimensional space.

The graph of f(x, y) = x² - y² is shown in the followig figure.

Many more can be found in the text book. If you wish to experiment by graphing surfaces on your computer, try the Surface Graphing Utility or, if you prefer Excel, the Excel Surface Graphing Utility.

Similarly, the partial derivative of f with respect to y is the derivative of f with respect to y, treating all other variables as constant, and so on for other variables. The partial derivatives are written as ∂f/∂x, ∂f/∂y, and so on. The symbol "∂" is used (instead of "d") to remind us that there is more than one variable, and that we are holding the other variables fixed.

Interpretation

∂f ∂x	is the rate at which f changes as x changes, for a fixed (constant) y.
∂f ∂y	is the rate at which f changes as y changes, for a fixed (constant) x.

Higher Order Partial Derivatives
If f is a function of x, y, and possibly other variables, then

∂2f ∂x2

is defined to be

∂ ∂x ∂f ∂x

Similarly,

∂2f ∂y2	is defined to be	∂ ∂y ∂f ∂y
∂2f ∂y∂x	is defined to be	∂ ∂y ∂f ∂x
∂2f ∂x∂y	is defined to be	∂ ∂x ∂f ∂y

The above second order partial derivatives can also be denoted by f_xx, f_yy, f_xy, and f_yx respectively.

The last two are called mixed derivatives and will always be equal to each other when all the first and second order partial derivatives are continuous.

If f is a function of x and y, taking the partial derivative ∂f/∂x and evaluating it at (a, b) amounts to holding y constant at y = b and finding the rate of change of f at x = a. Thus, the partial derivative is the slope of the tangent line to this curve at the point where x = a and y = b, along the plane y = b. (See the figure below.)

∂z ∂x

(a, b)

is the slope of the tangent line at the point P(a, b, f(a, b)) along the slice through y = b

If f is a function of x and y, then f has a relative maximum at (a, b) if f(a, b) ³ f(x, y) for every (x, y) in a small neighborhood of (a, b). A relative minimum is defined in a similar way. f has a saddle point at (a, b) if f has a relative minimum there along one slice and a relative maximum along another slice.

The function illustrated below has has a relative minimum at (0, 0), a relative maximum at (1, 1), and saddle points at (1, 0) and (0, 1).

In the cases we study, all relative extrema and saddle points not on the boundary of the domain of f occur as critical points, which are solutions to the equations

f_x(x,y) = 0

f_y(x,y) = 0.

Second-Derivative Test for Functions of Two Variables
Suppose f(x, y) is a function of two variables and that (a, b) is a critical point of f. (That is, f_x(a, b) = 0 and f_y(a, b) = 0.) Assume also that the second order partial derivatives exist and are continuous, so that, by theorems in calculus f_xy is the same as f_yx. Let

H = f_xx(a, b)f_yy(a, b) -[f_xy(a, b)]².

f has a relative minimum at (a, b) if H > 0 and f_xx(a,b) > 0,

f has a relative maximum at (a, b) if H > 0 and f_xx(a,b) < 0, and

f has a saddle point at (a, b) if H < 0.

If H = 0 the test tells us nothing, so we need to look at the graph to see what is going on.

1. Let f(x, y) = x² - (y-1) ². Then f_x(x,y) = 2x; f_y(x, y) = -2(y-1). To find the critical points, we solve the system

2x = 0
-2(y-1) = 0.

To check which it is, first compute the second order derivtives:

f_xx(x, y) = 2
f_yy(x, y) = -2
f_xy(x, y) = f_yx(x, y) = 0

H	=	fxx(0, 1)fyy(0, 1) -[fxy(0, 1)]2
	=	(2)(-2) -02 =- 4

A constrained optimization problem has the form

Maximize (or minimize) f(x, y,. . . ) subject to constraints.

Lagrange Multipliers
To locate the candidates for relative extrema of a function f(x, y, . . .) subject to the constraint g(x, y, ...) = 0, solve the following system of equations for x, y, ... and λ.

f_x = λg_x

f_y = λg_y

    ...

g = 0

Geometric Definition of the Double Integral
The double integral of f(x, y) over the region R in the xy-plane is defined as

R

f(x, y) dx dy

= Volume above R and under the graph of f       - Volume below R and above the graph of f.

The following figure illustrates this volume (in the case that the graph of f is above the region R).

Computing Double Integrals
If R is the rectangle a ≤ x ≤ b and c ≤ y ≤ d (see figure below) then

	R	f(x, y) dx dy =		d c			b a	f(x, y) dx		dy
		=		b a			d c	f(x, y) dy		dx

If R is the region a ≤ x ≤ b and c(x) ≤ y ≤ d(x) (see figure below) then we integrate over R according to the following equation.

R

f(x, y) dx dy =

b a

d(x) c(x)

f(x, y) dy

dx

If R is the region c ≤ y ≤ d and a(y) ≤ x ≤ b(y) (see figure below) then we integrate over R according to the following equation.

R

f(x, y) dx dy =

d b

b(y) a(y)

f(x, y) dx

dy

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If R is the rectangle 1 ≤ x ≤ 2 and 1 ≤ y ≤ 3, then

	R	x dx dy =	3 1 2 1 x dx dy
		=	3 1 3 2 dy	= 3

Let R be the region described by 0 ≤ x ≤ 2, 0 ≤ y ≤ x (see figure)

	R	x dx dy =	2 0 x 0 x dy dx
		=	2 0 xy x y=0 dx
		=	2 0 x2 dx = 8 3

Let R be the same region as above, but this time described by 0 ≤ y ≤ 2, y ≤ x ≤ 2 (see figure)

	R	x dx dy =	2 0 2 y x dx dy
		=	2 0 x2 2 2 x=y dy
		=	2 0 2 - y2 2 dy = 8 3