differential in several variables

CHAPTER IIDIFFERENTATION IN SEVERAL VARIABLES

Department of Foundation Year,

Institute of Technology of Cambodia

2015–2016

CALCULUS II () ITC 1 / 43

Contents

1 Function of Two or More Variables

2 Limits and Continuity

3 Partial Derivatives

4 Chain Rules for Functions of Several Variables

5 Directional Derivatives and Gradient Vectors

6 Tangent Planes and Normal Lines

7 Extrema of Function of Several Variables

8 Lagrange Multipliers


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Introduction to Functions of Several Variables

Note

A real-valued function on subset D of Rn is a function whose rangeis R. That is,

f : D ⊂ Rn −→ R(x1, x2, . . . , xn) 7−→ y = f(x1, x2, . . . , xn)

Special cases for n = 2 and n = 3 will be mainly concerned since theyhelp to visualise their geometrical meaning.



Some Operations on Rn: Letx = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn) ∈ Rn and α ∈ R. We define

Addition:x+ y = (x1 + y1, x2 + y2, . . . , xn + yn)

Scalar multiplication:

αx = (αx1, αx2, . . . , αxn)

Inner product:

x · y = 〈x, y〉 = xy = x1y1 + x2y2 + · · ·+ xnyn



In this study we use only Euclidean Norm, that is ifx = (x1, x2, . . . , xn) ∈ Rn, then norm of x is defined by

‖x‖ =

(n∑

i=1

x2i

)1/2

=√x2

1 + x22 + · · ·+ x2

n.

If x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn) ∈ Rn, then norm of thedifference x and y (or Euclidean distance between x and y)is defined by

‖x− y‖ =

[n∑

i=1

(xi − yi)2

]1/2

.


Level curve, level surface and level hypersurfce

Definition 1 (Level curve, level surface and level hypersurfce)

The set of points (x1, x2, . . . , xn) in Rn where a function of nindependent variables has a constant value f(x1, x2, . . . , xn) = c iscalled a level hypersurface of f .In particular,

the set of points in the plane where a function f(x, y) has aconstant value f(x, y) = c is called a level curve of f .

the set of points in the space where a function f(x, y, z) has aconstant value f(x, y, z) = c is called a level surface of f .

Note that if n ≥ 4, the set of points satisfying the equationf(x1, x2, . . . , xn) = c is called level hypersurface.


Example of level curves

Figure: Level curves show the lines ofequal pressure (isobars) measured inmillibars

Figure: Level curves show the lines ofequal temperature (isotherms)measured in degree Fahrenheit.


Level Curves


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Limit of a Function of Several Variables

Definition 2

Let f : D ⊂ Rn → Rm and a ∈−D. Then we say that the limit of f(x)

equals L as x approaches a, written as

limx→a

f(x) = L,

if given any ε > 0, there exists a δ > 0 such that

‖f(x)− L‖Rm < ε whenever ‖x− a‖Rn < δ.

Theorem 1

If a limit exists, it is unique.



Theorem 2

Suppose that limx→a

f(x) and limx→a

g(x) both exist and that k is a scalar.

Then

limx→a

[f(x)± g(x)] = [ limx→a

f(x)]± [ limx→a

g(x)]

limx→a

[kf(x)] = k[ limx→a

f(x)]

limx→a

[f(x)g(x)] = [ limx→a

f(x)][ limx→a

g(x)]

limx→a

[f(x)/g(x)] = [ limx→a

f(x)]/[ limx→a

g(x)] provided limx→a

g(x) 6= 0 and

both f and g are real-valued functions.

If f(x) ≤ g(x) for all x, then limx→a

f(x) ≤ limx→a

g(x), where f and g

are real-valued functions.

If ‖f(x)− L‖ ≤ g(x) for all x and if limx→a

g(x) = 0, then

limx→a

f(x) = L.



Note

To show that the limit does not exist, we need to show that thefunction approaches different values as x approaches a along differentpaths in Rn.

Definition 3

Technique for Showing that lim(x,y)→(a,b)

f(x, y) Doesn’t Exist

If f(x, y) approaches two different numbers as (x,y) approaches (a, b)along two different paths, then lim

(x,y)→(a,b)f(x, y) = L does not exist.


Definition of Limits

Example 4

Show that

a). lim(x,y)→(0,0)

x2 − y2

x2 + y2b). lim

(x,y)→(0,0)

xy

x2 + y2

doesn’t exist.(Hint: Consider different parts C1 : y = 0, C2 : x = 0, C3 : y = x)

Theorem 3

Let f : D ⊂ Rn → Rm be a vector-value functions,f = (f1, f2, . . . , fm)and L = (L1, L2, . . . , Lm). Then lim

x→af(x) = L if and only if

limx→a

fi(x) = Li for i = 1, 2, . . . ,m.


Continuity

Definition 5

Let f : D ⊂ Rn 7−→ Rm. We say that the function f is continuous ata point a in D if

limx→a

f(x) = f(a).

We say that f is a continuous function on D if it is continuous atevery point in its domain D.

Theorem 4 (Algebraic properties)

Let f, g : D ⊂ Rn → Rm be continuous vector-value function and letα ∈ R be a scalar. Then

f + αg and fg are continuous.

If both f and g are real-valued functions and if g(x) 6= 0, then f/gis continuous.


Continuity

Theorem 5

Let f : D ⊂ Rn 7−→ Rm, where f = (f1, . . . , fm). Then f is continuousat a ∈ D (respectively f is continuous on D) if and only if itscomponent functions fi : D → R are all continuous at a (respectively fiare continuous on D).


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Partial derivatives of a function of several variables

Definition 6

Let f : D ⊂ Rn 7−→ R. If y = f(x) = f(x1, x2, . . . , xn), then the firstpartial derivative of f with respect to xi, i ∈ {1, 2, . . . , n}, is definedby fxi(x) or ∂f

∂xi(x) and

fxi(x) =∂

∂xif(x) = lim

∆xi→0

f(x1, . . . , xi + ∆xi, . . . , xn)− f(x1, . . . , xn)

∆xi

provided the limit exists.

Note that if the function∂f

∂xihas a partial derivative with respect to

xj , we denote the partial derivative by

∂

∂xj

(∂f

∂xi

)=

∂2f

∂xj∂xi= fxixj .


Partial Derivatives of Functions of Two Variables

Definition 7 (Partial Derivatives of a Function of Two Variables)

Let z = f(x, y). Then the partial derivative of with respect to xis

fx =∂f

∂x= lim

h→0

f(x+ h, y)− f(x, y)

h

and the partial derivative of with respect to y is

fy =∂f

∂y= lim

h→0

f(x, y + h)− f(x, y)

h


Partial Derivatives

Definition 8

A function f : D ⊂ Rn → R is said to be of class Ck if all itspartial derivatives of order ≤ k are continuous.

A function f : D ⊂ Rn → R is said to be of class C∞ if f hascontinuous partial derivatives of all orders.

A function f : D ⊂ Rn → Rm is said to be of class Ck if each ofcomponent functions is of class Ck.

A function f : D ⊂ Rn → Rm is said to be of class C∞ if each ofcomponent functions is of class C∞.


Partial Derivatives

The function obtained by differentiating f successively with respect toxi1 , xi2 , . . . xir at x is denoted by

∂kf

∂xir∂xir−1 . . . ∂xi1= fxi1

xi2...xir

where i1 + · · ·+ ir = k.

It is called a kth-order partial derivative of f .

Theorem 6

Let f : D ⊂ Rn → R be a Ck function. Then

fxi1xi2

...xir(x) = fxj1

xj2...xjr

(x)

where i1 + i2 + · · ·+ ir = j1 + j2 + · · ·+ jr = k.


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Chain rules for functions of several variables

Theorem 7 (Chain rules for function of several variables)

Let y = f(x1, x2, . . . , xn), where f is differentiable function ofxi, i = 1, 2, . . . , n. If each xi, i = 1, 2, . . . , n is a differentiable function ofm variables t1, t2, . . . , tm, then y is a differentiable function oft1, t2, . . . , tm and

∂y

∂tj=

n∑i=1

∂y

∂xi

∂xi∂tj

.

for j = 1, 2, . . . ,m.In particular, if xi, i = 1, 2, . . . , n is a function of a single variable t,then we have

dy

dt=

n∑i=1

∂y

∂xi

dxidt.


Chain rules for functions of several variables

Theorem 9 (Chain rule: Implicit Differentiation)

If the equation F (x1, x2, . . . , xn, y) = 0 defines y implicitly as adifferentiable function of xi, i = 1, 2, . . . , n, then

∂y

∂xi= −Fxi(x1, x2, . . . , xn, y)

Fy(x1, x2, . . . , xn, y), Fy(x1, x2, . . . , xn, y) 6= 0

for i = 1, 2, . . . , n.

Example 10

Given xy3 + zy − 5z3 + 6 = 0. Find ∂z/∂x and ∂z/∂y.


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Gradient

Definition 11

For a real-valued function f(x1, x2, . . . , xn), the gradient of f at apoint a, denoted by ∇f(a), is the vector

∇f(a) =

(∂f

∂x1(a),

∂f

∂x2(a), . . . ,

∂f

∂xn(a)

).

Theorem 8

Let D ⊂ Rn be open, and suppose f : D → R is differentiable at a ∈ D.Then the directional derivative of f at a exists for all directions (unitvectors) u and, moreover, we have

Duf(a) = ∇f(a) · u.


The Directional Derivative

Example 12

Find the directional derivative of f(x, y) = 4− 2x2 − y2 at the point

(1, 1) in along unit vector(

12 ,√

32

).


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Tangent Planes and Normal Lines

So far, you have represented surfaces in space primarily by equations ofthe form

z = f(x, y)

In the development to follow, however, it is convenient to use the moregeneral representation F (x, y, z) = 0. For a surface S given byz = f(x, y), you can convert to the general form by defining F as

F (x, y, z) = f(x, y)− z.

Because f(x, y)− z = 0, you can consider S to be the level surface of Fgiven by

F (x, y, z) = 0.



Let S be a surface given byF (x, y, z) = 0 and let P (x0, y0, z0)be a point on S. Let C be a curve Son through P that is defined by thevector-valued function

r(t) = x(t)i + y(t)j + z(t)k

Then, for all t

F (x(t), y(t), z(t)) = 0.

If F is differentiable and x′(t), y′(t)and z′(t) all exist, then we have

F ′(t) = Fx(x, y, z)x′(t)+Fy(x, y, z)y′(t)+Fz(x, y, z)z′(t) = 0



At point P (x0, y0, z0), the equivalent vector form is

∇F (x0, y0, z0).r′(t0) = 0.

Meaning that the gradient at P is orthogonal to the tangent vector ofevery curve on S through P . So, all tangent lines on S lie in a planethat is normal to ∇F (x0, y0, z0) and contains P , as shown in theFigure.

Definition 13

Let F be differentiable at the point P (x0, y0, z0) on the surface givenby F (x, y, z) = 0 such that ∇F (x0, y0, z0) 6= 0.

The plane through P that is normal to ∇F (x0, y0, z0) is called thetangent plane to S at P .

The line through P having the direction of ∇F (x0, y0, z0) is calledthe normal line to S at P .



Theorem 9 (Equation of Tangent plane)

If F is differentiable at (x0, y0, z0) then an equation of the tangentplane to the surface given by F (x, y, z) = 0 at (x0, y0, z0) is

Fx(x0, y0, z0)(x− x0) + Fy(x0, y0, z0)(y − y0) + Fz(x0, y0, z0)(z − z0) = 0

Theorem 10

If the surface given by equation z = f(x, y), then an equation oftangent line to the the surface at the point (x0, y0, z0) is

fx(x0, y0)(x− x0) + fy(x0, y0)(y − y0)− (z − z0) = 0.


Tangent planes and normal lines

Example 14

Show that the tangent plane to the Ellipsoidx2

a2+y2

b2+z2

c2= 1 at the

point (x0, y0, z0) is given byx0x

a2+y0y

b2+z0z

c2= 1.


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Local Extrema

Definition 15

Let D ⊂ Rn, f : D → R be a function and a ∈ D.f(a) is called a local minimum of f if there is an r > 0 such thatf(a) ≤ f(x) for all x ∈ Br(a) ∩D.f(a) is called a local maximum of f if there is an r > 0 suchthat f(a) ≥ f(x) for all x ∈ Br(a) ∩D.f(a) is called a local extremum of f if there f(a) is a localminimum or a local maximum of f .


Differential fo Vector-Valued Functions

Definition 16

Let f : D ⊂ Rn → Rm be a vector-valued function of n variables. Letx = (x1, . . . , xn) denote a point of Rn and f = (f1, . . . , fm). We definethe matrix of partial derivatives of f , denoted Df , to be the

m× n matrix whose (i, j) entry is∂fi∂xj

. That is, Df =

(∂fi∂xj

).

The matrix

Df(a) =

(∂fi∂xj

(a)

)is also called Jacobian matrix of f at a.


Hessian Matrix

Definition 17 (Principal Minor)

Let A = (aij)n×n be a square matrix. The determinant

Ak = det((aij)k×k

), 1 ≤ k ≤ n, is called the kth principal minor of

matrix A.

Definition 18 (The Hessian of a Function)

Let D ⊂ Rn and f : D → R be a function of class C2. The Hessian off , denoted by Hf , is the matrix whose (i, j) entry is fxixj .. That is,

Hf (a) =

(∂2f

∂xj∂xi(a)

)n×n

.

We denote Hk = det(

∂2f∂xj∂xi

)k×k

the kth principal minor of Hf for

k = 1, 2, . . . , n.


Local Extrema

Theorem 19

Let f have continuous second partial derivatives on an open region

containing a = (x0, y0) for which

{fx(a) = 0

fy(a) = 0To test for relative extrema of f , consider the quality

H1 = fxx(a) , H2 = fxx(a)fyy(a)− (fxy(a))2.

1 If H2 > 0 and H1 > 0, then f has a local minimum at a = (x0, y0).

2 If H2 > 0 and H1 < 0, then f has a local maximum at a = (x0, y0).

3 If H2 < 0, then a = (x0, y0) is a saddle point.

4 If H2 = 0, the test is inconclusive.


Local Extrema

Example 20

Determine the local extrema/saddle points of the function

f(x, y) = 3x2 + 2y2 − 6x− 4y + 16.


Local Extrema

Theorem 21

Let f have continuous second partial derivatives on an open region

containing a = (x0, y0, z0) for which

fx(a) = 0

fy(a) = 0

fz(a) = 0To test for relative extrema of f , consider the quality

H1 = fxx(a, b), H2 =

∣∣∣∣ fxx(a) fxy(a)fyx(a) fyy(a)

∣∣∣∣ , H3 =

∣∣∣∣∣∣fxx(a) fxy(a) fxz(a)fyx(a) fyy(a) fyz(a)fyz(a) fzy(a) fzz(a)

∣∣∣∣∣∣1 If H1 > 0, H2 > 0 and H3 > 0, then f has a local minimum at a.

2 If H1 < 0, H2 > 0 and H3 < 0, then f has a local maximum at a.

3 If neither case 1 nor 2 holds, then a is a saddle point.

4 If H3 = 0, the test is inconclusive.CALCULUS II () ITC 39 / 43

Local Extrema

Example 22

Determine the local extrema/saddle points of the function

f(x, y, z) = x2 − xy + z2 − 2xz + 6z.


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Lagrange Multipliers

Theorem 23

Let f, g1, . . . , gk : D → R be C1 functions where D ⊂ Rn is open andk < n. Suppose there is an a ∈ D such that

det

(∂gi∂xj

(a)

)6= 0.

If f(a) is local extremum of f subject to the constraints gi(x) = ci fori = 1, 2, . . . , k, then there exist scalars λ1, . . . , λk (called LagrangeMultipliers) such that{

∇f(a) +∑k

i=1 λi∇gi(a) = 0

g(x) = ci i = 1, 2, ..., k.


Local Extrema

Example 24

Use Lagrange Multipliers, determine the local extrema of functionf(x, y) = 5x+ 2y subject to constraint 5x2 + 2y2 = 14.


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