MH1811 Mathematics 2
Functions of Several Variables.
Dr Rachana Gupta
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
1 / 19
Outline
1
Functions of Several Variables
2
Function of 2 Variables.
3
Level Curves, Level Surfaces, Level Sets
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
2 / 19
Outline
1
Functions of Several Variables
2
Function of 2 Variables.
3
Level Curves, Level Surfaces, Level Sets
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
3 / 19
Introduction
We shall extend the basic ideas of single-variable calculus
to functions of several variables.
We shall discuss limit at a point and continuity.
As there are more than one variables, the derivatives
involved are more interesting and more varied. Applications
of derivatives will also be discussed.
Similarly, we will also discuss integrals involving several
variables.
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
4 / 19
The Set Rn
The set Rn is the set of n-tuples of real numbers
(x1 , x2 , x3 , . . . , xn ).
When n = 1, we have R1 = R represented by the real
number line.
When n = 2, we have R2 which is represented by the
infinite (x, y )-plane. We usually write (x, y ) instead of
(x1 , x2 ) for points in R2 .
When n = 3, we have R3 which is represented by the
3-dimensional space we live in. We usually write (x, y , z)
instead of (x1 , x2 , x3 ) for points in R3 .
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
5 / 19
Functions of Several Variables.
Definition
Let D ⊂ Rn . A function f of n variables is a rule that assigns to
each point (x1 , x2 , ..., xn ) ∈ D a unique real number denoted by
f (x1 , x2 , ..., xn ).
The set D is called the domain of f and the range is the set of
values f takes on.
The real number w = f (x1 , x2 , ..., xn ) is called the image of f
evaluated at the point (x1 , x2 , ..., xn ).
The variables x1 , x2 , x3 , . . . , xn are known as independent
variables. The variable w is known as a dependent variable
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
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Example: Functions of 2 Variables.
Many functions depend on more than one independent
variables.
Example
The power P (watts) of an electric circuit is related to the circuit’s
resistance R (ohms) and current I (amperes) by the equation
P = RI 2 .
This means that P is a function of two variables R and I.
We may express the function as
P(R, I) = RI 2 .
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
7 / 19
Example: Functions of Several Variables
Example
If a company uses n different ingredients in making a food
product, ci is the cost per unit of the i-th ingredient, then the
total cost T if xi units of i th ingredient (i = 1, 2, ..., n) is used is a
function of x1 , x2 , . . . , xn , where
T (x1 , x2 , ..., xn ) =
n
X
xi ci .
i=1
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
8 / 19
Outline
1
Functions of Several Variables
2
Function of 2 Variables.
3
Level Curves, Level Surfaces, Level Sets
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
9 / 19
Functions of 2 Variables.
For a function f of two variables, the domain D is a subset of R2 .
The function f is a rule that assigns to each point (x, y ) ∈ D a
unique real number which is denoted by f (x, y ).
We often write z = f (x, y ) to make explicit the value taken
on by f at the point (x, y ).
The variables x and y are independent variables and z is a
dependent variable.
NOTE When the domain of f is not specified, we take the largest
subset of R2 on which f (x, y ) is a real number, to be the domain
of f .
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
10 / 19
Example. Find and sketch the domain of
p
f (x, y ) = y − x 2 and evaluate f (1, 2).
p
[SOLUTION] Note that f (x, y ) = y − x 2 is defined when
y − x 2 ≥ 0.
Therefore the domain is D = (x, y ) : y ≥ x 2 and
√
f (1, 2) = 2 − 1 = 1.
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
11 / 19
Example. Find and sketch the domain of
2
f (x, y ) = x ln y − x and evaluate f (1, −2).
[SOLUTION] Note that f (x, y ) = x ln y 2 − x is defined when
y 2 − x > 0.
Therefore the domain is D = (x, y ) : y 2 > x and
f (1, −2) = 1 · ln (−2)2 − 1 = ln 3.
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
12 / 19
Outline
1
Functions of Several Variables
2
Function of 2 Variables.
3
Level Curves, Level Surfaces, Level Sets
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
13 / 19
Graph & Level Curves of f (x, y ).
The graph of the function 2-variable function f (x, y ) is a
3-dimensional graph z = f (x, y ). It is a challenge to sketch it on
paper.
(a) If f is a function of two variables with domain D, the graph of
f is the set
(x, y , z) ∈ R3 : z = f (x, y ) , (x, y ) ∈ D
(b) The set of points (x, y ) in R2 where f (x, y ) has the same
(constant) value f (x, y ) = c is called a level curve of f .
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
14 / 19
Example: Find the level curve of
f (x, y ) = 100 − x 2 − y 2 corresponds to (i)
f (x, y ) = 75, (ii) passes through (10, 0).
(i) To find the level curve corresponds to f (x, y ) = 75, we find
(x, y ) such that
100 − x 2 − y 2 = 75, i.e., x 2 + y 2 = 25.
This is a circle, with radius 5 and center (0, 0).
(ii) The level curve passes through (10, 0) corresponds to the
level curve corresponds to f (x, y ) = 0. This gives x 2 + y 2 = 102 ,
which is a circle, with radius 10 and center (0, 0).
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
15 / 19
Example: Graph & Level Curves of
f (x, y ) = 100 − x 2 − y 2.
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
16 / 19
Functions of 3 Variables & Level Surface.
For a function f of three variables, the set of points (x, y , z) in R3
where f (x, y , z) has a constant value f (x, y , z) = C is called a
level surface of f .
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
17 / 19
Functions of 3 Variables & Level Surfaces.
Example
Consider the function f (x, y , z) = (x − 1)2 + y 2 + z 2 .
(a) The level surface corresponds to f (x, y , z) = 9, i.e.,
(x − 1)2 + y 2 + z 2 = 9, is the sphere of radius 3 and center
at (1, 0, 0) .
(b) The level surface that passes through (1, −2, 3) is the set
(x, y , z) ∈ R3 | (x − 1)2 + y 2 + z 2 = 13
since f (1, −2, 3) = 02 + (−2)2 + 32 = 13. It is a sphere, with
√
radius 13 and center at (1, 0, 0).
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
18 / 19
Functions of n Variables & Level Sets.
For a function f of n variables, where n ≥ 4, the set of points
(x1 , x2 , . . . , xn ) in Rn where f (x1 , x2 , . . . , xn ) has a constant value
f (x1 , x2 , . . . , xn ) = C is called a level set of f .
Dr Rachana Gupta (Division of Mathematical Sciences School
MH1811
of Mathematics
Physical and Mathematical
2
Sciences Nanyang Technological University)
19 / 19