2.1 The Geometry of Real-Valued Functions
In this section, we will develop methods for visualizing a function.
§Functions:
Let f be a function which assigns to each vector x = hx1, · · · , xni in a subset U
of Rn, a unique vector f (x) in Rm.
We call U is the domain of f . We denote this function f by
f : U ⊂ Rn → Rm
to indicate that f maps from U into Rm.
We denote the component functions of f (x) as follows:
f (x) = (f1(x), · · · , fm(x)).
• If m = 1, then we call f is a scalar-valued function.
• If m > 1, then we call f is a vector-valued function.
For example, we consider U is the campus. Let T : U → R1 as a temperature
function such that T (x) describes the temperature at every position x in campus,
U.
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We call the graph of f : U → Rm is the set of points in Rn+m of the form
(x1, · · · , xn, f1(x), · · · , fm(x)).
We also write
graph of f = {(x1, · · · , xn, f1(x), · · · , fm(x)) in Rn+m for x in U }.
Example 1. When n = 2, m = 1, f : R2 → R1.
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§Level Sets:
Level set is a subset of the domain of function f on which f is a constant.
That is,
Level set is {x : f (x) = c}
where c is constant
• n = 2: the level sets for f : R2 → R1 are curves.
We call level curves or level contours.
Example 2. f (x, y) = x2 + y 2. Describe the level sets of f .
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• n = 3: the level sets for f : R3 → R1 are surfaces.
We call level surfaces.
Example 3. f (x, y, z) = x2 + y 2 + z 2. Describe the level sets of f .
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§Sections:
Given a function f , a section of the graph of f is the intersection of the graph
with a vertical plane.
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Example 4. z = f (x, y) = x2 − y 2
1. Find the level curves c = 0, 1 for f .
2. Find the intersection of the graph of f and the plane x = −1. (Actually, it is
a section).
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