This document discusses functions of multiple variables and their properties. It introduces functions of two variables and their graphs, which can be interpreted as surfaces in space. It then discusses partial derivatives, which are derivatives of functions of multiple variables where one variable is held constant. Higher order partial derivatives and double integrals are also introduced. An example problem finds the volume of a solid bounded by a paraboloid and planes using iterated integrals.

1. Functions in Several
Variables

2.  The work done by a force (W = FD) and the volume of a right
circular cylinder (V =𝜋𝑟2
ℎ) are both functions of two variables.
 The volume of a rectangular solid (V = lwh) is a function of three
variables.
 The notation for a function of two or more variables is similar to that
for a function of a single variable. Here are two examples.

5.  For the function given by z = f(x, y), x and y are called the
independent variable and z is called the dependent variable.

6.  The graph of a function f of two variables is
the set of all points (x, y, z) for which z = f(x, y)
and (x, y) is in the domain of f.
 This graph can be interpreted geometrically
as a surface in space.
Note that the graph of z = f (x, y) is a surface
whose projection onto the xy-plane is the D, the
domain of f.

7. Partial Derivatives

8.  Functions of two or more variables do not have ordinary derivatives
of the type we studied for functions of one variable
 If 𝑓 is a function of two variables, say 𝑥 and 𝑦, then for each fixed
value of 𝑦, 𝑓 is a function of a single variable 𝑥. The derivative with
respect to 𝑥 (keeping 𝑦 fixed) is then called the partial derivative
with respect to 𝑥.
 If 𝑓 is a function of two variables, say 𝑥 and 𝑦, then for each fixed
value of 𝑥, 𝑓 is a function of a single variable 𝑦. The derivative with
respect to 𝑦 (keeping 𝑥 fixed) is then called the partial derivative
with respect to 𝑦.

9. Example 1:
 Find the partial derivatives 𝑓
𝑥 and 𝑓
𝑦 for the function

10. Example 1:
 Find the partial derivatives 𝑓
𝑥 and 𝑓
𝑦 for the function
Solution:
Consider 𝑦 as constant and differentiate with respect to 𝑥

11. Example 1:
 Find the partial derivatives 𝑓
𝑥 and 𝑓
𝑦 for the function
Solution:
Consider 𝑦 as constant and differentiate with respect to 𝑥

12. Example 1:
 Find the partial derivatives 𝑓
𝑥 and 𝑓
𝑦 for the function
Solution:
Consider 𝑦 as constant and differentiate with respect to 𝑥
Consider 𝑥 as constant and differentiate with respect to 𝑦

14.  The concept of a partial derivative can be extended naturally to
functions of three or more variables. For instance, if w = f(x, y, z),
there are three partial derivatives, each of which is formed by
holding two of the variables constant.
 That is, to define the partial derivative of w with respect to x,
consider y and z to be constant and differentiate with respect to x.
 A similar process is used to find the derivatives of w with respect to y
and with respect to z.

22. Higher-Order Partial Derivatives

29. Double Integrals

36. Iterated Integrals

38. Example 1 Evaluate the iterated
integrals

39. Example 1 Evaluate the iterated
integrals

40. Example 1 Evaluate the iterated
integrals

44. Example
 Find the volume of the solid S that is
enclosed by a paraboloid 𝑥2
+2𝑦2
+z=16, the
planes x=2 and y=2, and three coordinate
planes.
 We first observe that S is the solid that lines
under the surface z=16−𝑥2
−2𝑦2
and above
the square R= 𝑥, 𝑦 : 0 ≤ 𝑥 ≤ 2,0 ≤ 𝑦 ≤ 2 .