Chapter 10: Multivariable Calculus
Section 10.1: Functions of Two or More Variables
Definition: A function of two variables is a rule that assigns a unique real number
f (x, y) to each ordered pair (x, y) ∈ D ⊂ R2 . The set D is called the domain of f and
{f (x, y)|(x, y) ∈ D} is called the range of f .
Example: Evaluate each function at the given point.
√
(a) f (x, y) = x + y at (4, 3)
(b) f (x, y) = 2x − 3y + ln(xy) at (1, 1)
Example: Find the largest possible domain of each function and the corresponding range.
p
(a) f (x, y) = 9 − x2 − y 2
(b) f (x, y) = e−(x
2 +y 2 )
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Example: Find and sketch the domain of the given functions:
p
(a) f (x, y) = y − x2
(b) f (x, y) =
(c) f (x, y) =
p
x2 + y 2 − 1 + ln(4 − x2 − y 2 )
√
x+
√
y+
p
x2 + y 2 − 16
p
9 − x2 − y 2
(d) f (x, y) =
2x + y
2
Definition: If f is a function of two variables with domain D, then the graph of f is the set
{(x, y, z) ∈ R3 |z = f (x, y), (x, y) ∈ D}.
Note: The graph of a function f of two variables is a surface with equation z = f (x, y). The
graph of f can be visualized as lying directly above or below the domain D in the xy-plane.
Example: Describe the graph of the following functions:
(a) f (x, y) = 2
(b) f (x, y) = 4 − 2x − y
(c) f (x, y) = x2 + y 2
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Definition: The level curves of a function of two variables are the curves defined by f (x, y) =
k, where k is a constant in the range of f . The level curves of f (x, y) are the horizontal
traces of the graph of f in the plane z = k projected onto the xy-plane. A graph of the level
curves is called a contour plot.
Note: Contour plots are commonly used in topographic maps.
Example: Sketch the level curves for the given functions:
(a) f (x, y) = x2 + 4y 2 for k = 0, 4, 16
(b) f (x, y) = y − x2 for k = −1, 0, 1, 2
(c) f (x, y) =
p
4 − x2 − y 2 for k = 0, 1, 2
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