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MATH115Functions of Several Variables

Paolo Lorenzo Bautista

De La Salle University

August 12, 2014

PLBautista (DLSU) MATH115 August 12, 2014 1 / 46

Functions of Several Variables

DefinitionThe set of all ordered n-tuples of real numbers is called then-dimensional number space, and is denoted by Rn. Each orderedn-tuple (x1, x2, . . . , xn) is called a point in the n-dimensional numberspace.



DefinitionA function of n variables is a set of ordered pairs of the form (P,w) inwhich no two distinct ordered pairs have the same first element. P is apoint in n-dimensional number space and w is a real number. The setof all admissible points P is called the domain of the function, and theset of all resulting values of w is called the range of the function.



ExampleDetermine the domain of the following functions:

1. f (x, y) =1

x2 + y2 − 12. f (x, y) =

√1− x2 − y2

3. f (x, y) =1√

1− x2 − y2



DefinitionIf a function f of n variables is defined by

w = f (x1, x2, . . . , xn)

then the variables x1, x2, . . . , xn are called the independent variables,and w is called the dependent variable.



DefinitionIf f is a function of a single variable and g is a function of twovariables, then the composite function f ◦ g is the function of twovariables defined by

(f ◦ g)(x, y) = f (g(x, y))

and the domain of f ◦ g is the set of all points (x, y) in the domain of gsuch that g(x, y) is in the domain of f .



DefinitionIf f is a function of a single variable and g is a function of n variables,then the composite function f ◦ g is the function of n variables definedby

(f ◦ g)(x1, x2, . . . , xn) = f (g(x1, x2, . . . , xn))

and the domain of f ◦ g is the set of all points (x1, x2, . . . , xn) in thedomain of g such that g(x1, x2, . . . , xn) is in the domain of f .



Example

Let f (t) = et and g(x, y) =√

x2 + y2. Find (f ◦ g)(x, y).



DefinitionA polynomial function of two variables x and y is a function f suchthat f (x, y) is the sum of the terms of the form cxnym, where c is a realnumber, and n and m are nonnegative integers. The degree of thepolynomial function is determined by the largest sum of the exponentsof x and y appearing in any one term.



DefinitionIf f is a function of two variables, then the graph of f is the set of allpoints (x, y, z) in R3 for which (x, y) is a point in the domain of f andz = f (x, y).



DefinitionIf f is a function of n variables, then the graph of f is the set of allpoints (x1, x2, . . . , xn,w) in Rn+1 for which (x1, x2, . . . , xn) is a point inthe domain of f and w = f (x1, x2, . . . , xn).


Limits and Continuity

DefinitionIf P(x1, x2, . . . , xn) and A(a1, a2, . . . , an) are two points in Rn, then thedistance between P and A is given by

||P− A|| =√

(x1 − a1)2 + (x2 − a2)2 + · · ·+ (xn − an)2.



DefinitionIf A is a point in Rn and r is a positive number, then the open ballB(A; r) is the set of all points P in Rn such that ||P− A|| < r.

DefinitionIf A is a point in Rn and r is a positive number, then the closed ballB[A; r] is the set of all points P in Rn such that ||P− A|| ≤ r.



DefinitionLet f be a function of n variables defined on some open ball B(A; r),except possibly at the point A itself. Then the limit of f (P) as Papproaches A is L, written as

limP→A

f (P) = L

if for any � > 0, however small, there exists a δ > 0 such that

if 0 < ||P− A|| < δ then |f (P)− L| < �.



DefinitionLet f be a function of two variables defined on some open diskB((x0, y0); r), except possibly at the point (x0, y0) itself. Then the limitof f (x, y) as (x, y) approaches (x0, y0) is L, written as

limP→(x0,y0)

f (x, y) = L


if 0 <√

(x− x0)2 + (y− y0)2 < δ then |f (x, y)− L| < �.



ExampleUse the definition of the limit to show that

lim(x,y)→(2,1)

3x + y = 7



ExampleEvaluate the following limits:1. lim

(x,y)→(2,3)3x2 + xy− 2y2

2. lim(x,y)→(0,1)

x4 − (y− 1)4

x2 + (y− 1)2

3. lim(x,y)→(−2,4)

y 3√

x3 + 2y



TheoremIf g is a function of two variables and lim

(x,y)→(x0,y0)g(x, y) = b, and f is a

function of a single variable continuous at b, then

lim(x,y)→(x0,y0)

f (g(x, y)) = f(

lim(x,y)→(x0,y0)

g(x, y)).



DefinitionA point P0 is an accumulation point of a set S of points in Rn if everyopen ball B(P0; r) contains infinitely many points of S.



DefinitionLet f be a function of two variables defined on a set of points S in R2,and let (x0, y0) be an accumulation point of S. Then the limit of f (x, y)as (x, y) approaches (x0, y0) in S is L, written as

limP→(x0,y0)

f (x, y) = L


if 0 < ||(x, y)− (x0, y0)|| < δ then |f (x, y)− L| < �,

where (x, y) is in S



TheoremSuppose that the function f is defined for all points on an open diskhaving its center at (x0, y0), except possibly at (x0, y0) itself, and

lim(x,y)→(x0,y0)

f (x, y) = L.

If S is any set of points in R2 having (x0, y0) as an accumulation point,then

lim(x,y)→(x0,y0)

f (x, y)

exists and always has the value L.



TheoremIf the function f has different limits as (x, y) approaches (x0, y0)through two distinct sets of points having (x0, y0) as an accumulationpoint, then lim

(x,y)→(x0,y0)f (x, y) does not exist.



ExampleShow that the following limits do not exist.

1. lim(x,y)→(0,0)

x2 − y2

x2 + y2

2. lim(x,y)→(0,0)

xyx2 + y2



DefinitionSuppose that f is a function of n variables and A is a point in Rn. Thenf is said to be continuous at the point A if and only if the followingthree conditions are satisfied:

i) f (A) exists.ii) lim

P→Af (P) exists.

iii) limP→A

f (P) = f (A).

If one or more of these three conditions fails to hold at the point A,then f is said to be discontinuous at A.



DefinitionThe function f of two variables x and y is said to be continuous at thepoint (x0, y0) if and only if the following three conditions are satisfied:

i) f (x0, y0) exists.ii) lim

(x,y)→(x0,y0)f (x, y) exists.

iii) lim(x,y)→(x0,y0)

f (x, y) = f (x0, y0).



DefinitionIf a function f of two variables is discontinuous at the point (x0, y0) but

lim(x,y)→(x0,y0)

f (x, y) exists, then f is said to have a removable

discontinuity. Otherwise, it is called an essential discontinuity.



TheoremIf f and g are two functions continuous at the point (x0, y0), then

i) f + g is continuous at (x0, y0).ii) f − g is continuous at (x0, y0).

iii) fg is continuous at (x0, y0).iv) f/g is continuous at (x0, y0), provided that g(x0, y0) 6= 0.



TheoremA polynomial function of two variables is continuous at every point inR2.



TheoremA rational function of two variables is continuous at every point in itsdomain.



DefinitionThe function f of n variables is continuous on an open ball if it iscontinuous at every point in the open ball.



TheoremSuppose that f is a function of a single variable and g is a function oftwo variables such that g is continuous at (x0, y0) and f is continuous atg(x0, y0). Then the composite function f ◦ g is continuous at (x0, y0).


Partial Derivatives

DefinitionIf f is a function of two variables, its partial derivatives are thefunctions fx and fy defined by

fx(x, y) = limh→0

f (x + h, y)− f (x, y)h

,

and

fy(x, y) = limh→0

f (x, y + h)− f (x, y)h

,


Partial Derivatives

Example1. Use the definition of the partial derivative to obtain fx(x, y) and

fy(x, y) if f (x, y) = 3x2 − 2xy + y2.

2. Find the slope of the tangent line to the curve of intersection of thesurface

z =12

√24− x2 − 2y2

with the plane y = 2 at the point (2, 2,√

3).3. Let f (x, y) = ex sin y + ln xy. Determine fxx, fxy, fxyy.


Partial Derivatives


fy(x, y) if f (x, y) = 3x2 − 2xy + y2.2. Find the slope of the tangent line to the curve of intersection of the

surfacez =

12

√24− x2 − 2y2


3).

3. Let f (x, y) = ex sin y + ln xy. Determine fxx, fxy, fxyy.


Partial Derivatives


fy(x, y) if f (x, y) = 3x2 − 2xy + y2.2. Find the slope of the tangent line to the curve of intersection of the

surfacez =

12

√24− x2 − 2y2


3).3. Let f (x, y) = ex sin y + ln xy. Determine fxx, fxy, fxyy.


Partial Derivatives

Theorem (Clairaut’s Theorem)Suppose f is defined on a disk D that contains the point (x0, y0). If thefunctions fxy and fyx are both continuous on D, then

fxy(x0, y0) = fyx(x0, y0).


Differentiability and the Total Differential

DefinitionIf f is a function of two variables x and y, then the increment of f at thepoint (x0, y0), denoted by ∆f (x0, y0), is given by

∆f (x0, y0) = f (x0 + ∆x, y0 + ∆y)− f (x0, y0).



DefinitionIf f is a function of two variables x and y, and the increment of f at thepoint (x0, y0) can be written as

∆f (x0, y0) = D1f (x0, y0)∆x + D2f (x0, y0)∆y + �1∆x + �2∆y

where �1 and �2 are functions of ∆x and ∆y such that �1 → 0 and�2 → 0 as (∆x,∆y)→ (0, 0), then f is differentiable at (x0, y0).



TheoremIf a function f of two variables is differentiable a a point, it iscontinuous at that point.

TheoremLet f be a function of x and y such that D1f and D2f exist on an opendisk B(P0; r), where P0 is the point (x0, y0). Then if D1f and D2f arecontinuous at P0, f is differentiable at P0.



ExampleShow that f (x, y) = xey − y ln x is differentiable at all points in itsdomain.



DefinitionIf f is a function of two variables x and y, and f is differentiable at(x, y), then the total differential of f is the function df having functionvalues given by

df (x, y,∆x,∆y) = D1f (x, y)∆x + D2f (x, y)∆y,

or (if z = f (x, y))

dz =∂z∂x

dz +∂z∂y

dy.

Remark: The same concepts can be extended to functions of more thantwo variables.



ExampleA closed metal container in the shape of a right-circular cylinder hasan inside height of 6in, an inside radius of 2in, and a thickness of 0.1in.Approximate the amount of metal used in the container.

ExampleA closed container in the shape of a rectangular solid is to have aninside length of 8m, an inside width of 5m, an inside height of 4m, anda thickness of 4cm. Use differentials to approximate the amount ofmaterial needed to construct the container.


The Chain Rule and Implicit Differentiation

DefinitionIf u is a differentiable function of x and y, defined by u = f (x, y), where

x = F(r, s), y = G(r, s), and∂x∂r

,∂x∂s

,∂y∂r

, and∂y∂s

all exist, then u is afunction of r and s, and

∂u∂r

=∂u∂x

∂x∂r

+∂u∂y

∂y∂r

∂u∂s

=∂u∂x

∂x∂s

+∂u∂y

∂y∂s.



Example

Apply the chain rule to obtain∂u∂r

and∂u∂s

, where u = x2 + y2, x = rex,

and y = re−s.

Remark: The chain rule can be extended to functions involving morethan two variables.



Example

Apply the chain rule to obtain∂u∂r

and∂u∂t

, where u = x2 + yz,

x = r sin t, y = r cos t, and z = r sin2 t.



TheoremIf f is a differentiable function of the single variable x such thaty = f (x), and f is defined implicitly by the equation F(x, y) = 0, then ifF is differentiable and Fy(x, y) 6= 0,

dydx

= −Fx(x, y)Fy(x, y)

.



TheoremIf f is a differentiable function of x and y such that z = f (x, y), and f isdefined implicitly by the equation F(x, y, z) = 0, then if F isdifferentiable and Fz(x, y, z) 6= 0,

∂z∂x

= −Fx(x, y, z)Fz(x, y, z)

and∂z∂y

= −Fy(x, y, z)Fz(x, y, z)



Example

Obtain∂z∂x

and∂z∂x

if 4z3 + 3xz2 − xyz− 2xy2 + 7 = 0.


Functions of Several VariablesLimits and ContinuityPartial DerivativesDifferentiability and the Total DifferentialThe Chain Rule and Implicit Differentiation

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