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Function of Several Variables
The function f is called a realvalued function of two variables if there are two independent variables, a realvalued 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). 
Examples
3. Let p(x, y) = x^{2} + xy  y^{2}. Complete the following table of values of p, and press "Check". 

Examples: Linear, Interaction, and Distance Functions
Linear Functions
Interaction Functions
Distance Functions
(Special case of above formula) The distance in the plane from the point (x, y) to the origin is given by
The distance in 3space from the point (x, y, z) to the point (a, b, c) is given by

Examples


ThreeDimensional Space and the Graph of a Function of Two Variables
Points in threedimensional space have three coordinates as shown in the following figure.
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 threedimensional 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).

Examples
The following figure shows where the point (1, 2, 3) is located in threedimensional space. The graph of f(x, y) = x^{2}  y^{2} 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 have Excel, the Excel Surface Graphing Utility. 

Partial Derivatives
The partial derivative of f with respect to x is the derivative of f with respect to x, treating all other variables as constant. 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
Higher Order Partial Derivatives
Similarly,
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 alwyas be equal to each other when all the first order partial derivatives are continuous. 
Examples


Geometric Interpretation of Partial Derivatives
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.)


Maxima and Minima
If f is a function of x and y, then f has a local maximum at (a, b) if f(a, b) „ f(x, y) for every (x, y) in a small neighborhood of (a, b). A local minimum is defined in a similar way. f has a saddle point at (a, b) if f has a local minimum there along one slice and a local 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 local extrema and saddle points not on the boundary of the domain of f occur as critical points, which are solutions to the equations
SecondDerivative Test for Functions of Two Variables
f has a local minimum at (a, b) if H > 0 and f_{xx}(a,b) > 0, f has a local 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. 
Examples
1. Let f(x, y) = x^{2}  (y1) ^{2}. Then f_{x}(x,y) = 2x; f_{y}(x,y) = 2(y1). To find the critical points, we solve the system
2(y1) = 0. To check which it is, first compute the second order derivtives:
f_{yy}(x, y) = 2 f_{xy}(x, y) = f_{yx}(x, y) = 0


Constrained Maxima and Minima and Applications
A constrained optimization problem has the form
Lagrange Multipliers To locate the candidates for local 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 l.


Double Integrals
Geometric Definition of the Double Integral
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 region a £ x £ b and c(x) £ y £ d(x) (see figure below) then we integrate over R according to the following equation.
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.

Examples
If R is the rectangle 1 £ x £ 2 and 1 £ y £ 3, then
Let R be the region described by 0 £ x £ 2, 0 £ y £ x (see figure)
Let R be the same region as above, but this time described by 0 £ y £ 2, y £ x £ 2 (see figure)
