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Preview Multivariate functions Multivariate differential calculus Integral theory E 600 Chapter 3: Multivariate Calculus Simona Helmsmueller August 21, 2017

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Page 1: E 600 Chapter 3: Multivariate Calculus - …...E 600 Chapter 3: Multivariate Calculus Simona Helmsmueller August 21, 2017 Preview Multivariate functions Multivariate di erential calculus

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E 600

Chapter 3: Multivariate Calculus

Simona Helmsmueller

August 21, 2017

Page 2: E 600 Chapter 3: Multivariate Calculus - …...E 600 Chapter 3: Multivariate Calculus Simona Helmsmueller August 21, 2017 Preview Multivariate functions Multivariate di erential calculus

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Goals of this lecture:

• Know when an inverse to a function exists, be able to

graphically and analytically determine whether a function is

convex, concave or quasiconcave.

• Have an intuition about the meaning of derivatives as

measures of small change, be able to calculate the Gradient,

Jacobian and Hessian of a function

• Know all the calculation rules for derivatives (sum, quotient,

product, chain rule)

• Know the difference between partial and total derivatives

• Be able to write down the Taylor approximation of a function

• Know what a function’s differentials tell us about its

continuity and convexity

• Know and be able to apply the calculation rules of integral

theory; know and be able to apply Fubini’s theorem

Following lectures (both in this class and other

courses) will assume these goals have been reached!

Page 3: E 600 Chapter 3: Multivariate Calculus - …...E 600 Chapter 3: Multivariate Calculus Simona Helmsmueller August 21, 2017 Preview Multivariate functions Multivariate di erential calculus

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Contents

Multivariate functions

Invertibility

Convexity, Concavity, and Multivariate Real-Valued Functions

Multivariate differential calculus

Introduction

Revision: single-variable differential calculus

Partial Derivatives and the Gradient

Differentiability of real-valued functions

Differentiability of vector-valued functions

Higher Order Partial Derivatives and the Taylor Approximation Theorems

Characterization of convex functions

Integral theory

Univariate integral calculus

The definite integral and the fundamental theorem of calculus

Multivariate extension

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Definition(Surjective functions)

A function f : X → Y is said to be surjective or onto, if for every

y ∈ Y there exists at least one x ∈ X with f (x) = y .

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Definition(Injective functions)

A function f : X → Y is said to be injective or one-to-one, if the

following property holds for all x1, x2 ∈ X :

f (x1) = f (x2)⇒ x1 = x2.

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Theorem( Existence and definition of the inverse function)

Let f : X → Y be onto and one-to-one (i.e. bijective). Then there

exists a unique bijective function f −1 : Y → X with

f −1(f (x)) = x . Conversely, if such a function exists, then f is

bijective.

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Definition(Convex Real Valued Function)

Let X ⊆ Rn. A function f : X → R is convex if and only if X is a

convex set and for any two x , y ∈ X and λ ∈ [0, 1] we have

f (λx + (1− λ)y) ≤ λf (x) + (1− λ)f (y)

Moreover, if this statement holds strictly whenever y 6= x and

λ ∈ (0, 1), we say that f is strictly convex.

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Definition(Concave Real-Valued Function)

Let X ⊆ Rn. A function f : X → R is concave if and only if −f is

convex. Similarly f is strictly concave if and only if −f is strictly

convex.

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Page 10: E 600 Chapter 3: Multivariate Calculus - …...E 600 Chapter 3: Multivariate Calculus Simona Helmsmueller August 21, 2017 Preview Multivariate functions Multivariate di erential calculus

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Definition(Multivariate Convexity)

Let X be a convex subset of Rn. A real-valued function

f : X → Rn is (strictly) convex if and only if, for every x ∈ X and

every z ∈ {z ∈ Rn − {0} | ‖z‖= 1}, the function g(t) = f (x + tz)

is (strictly) convex on {t ∈ R | x + tz ∈ X}.

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Definition(Quasiconvexity, Quasiconcavity)

Let X be a convex subset of Rn. A real-valued function f : X → Ris quasiconvex if and only if for any c ∈ R the set

L−c := {x |x ∈ X , f (x) ≤ c},

also referred to as f lower-level set, is convex. Iff f is such that

L+c := {x |x ∈ X , f (x) ≥ c},

also referred to as f upper-level set, is convex for any c ∈ R, then

f is said to be quasiconcave.

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Theorem( Quasiconvexity, Quasiconcavity)

Let X be a convex subset of Rn. A real-valued function f : X → Ris quasiconvex if and only if

∀x , y ∈ X ∀λ ∈ [0, 1] f (λx + (1− λ)y) ≤ max{f (x), f (y)}

Iff f is such that

∀x , y ∈ X ∀λ ∈ [0, 1] f (λx + (1− λ)y) ≥ min{f (x), f (y)}

then f is said to be quasiconcave.

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Contents

Multivariate functions

Invertibility

Convexity, Concavity, and Multivariate Real-Valued Functions

Multivariate differential calculus

Introduction

Revision: single-variable differential calculus

Partial Derivatives and the Gradient

Differentiability of real-valued functions

Differentiability of vector-valued functions

Higher Order Partial Derivatives and the Taylor Approximation Theorems

Characterization of convex functions

Integral theory

Univariate integral calculus

The definite integral and the fundamental theorem of calculus

Multivariate extension

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“Calculus is the mathematical study of change, in the

same way that geometry is the study of shape and

algebra is the study of operations and their application to

solving equations. It has two major branches, differential

calculus (concerning rates of change and slopes of

curves), and integral calculus (concerning accumulation

of quantities and the areas under and between curves);

these two branches are related to each other by the

fundamental theorem of calculus, [...] [which] states that

differentiation and integration are inverse operations.”

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TheoremLet f , g : R→ R differentiable at x0 with derivatives f ′(x0), g ′(x0),

and λ, µ ∈ R. Then

1. λf + µg is differentiable in x0 with

(λf + µg)′(x0) = λf ′(x0) + µg ′(x0),

2. fg is differentiable in x0 with (fg)′(x0) = (f ′g + fg ′)(x0)

3. if g(x) 6= 0 in a neighborhood of x0, then f /g is differentiable

with (f

g

)′(x0) =

f ′g − fg ′

g2(x0),

4. if all the following expressions are well-defined, then g ◦ f is

differentiable in x0 with

(g ◦ f )′(x0) = g ′(f (x0))f ′(x0).

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• Let f be differentiable at x̄, then f is continuous at x̄.

• Let f be differentiable at x̄, then there exists a good

“linear approximation” of f in a neighborhood

• Let a and b be real numbers such that a < b. If f is

continuous and differentiable on (a,b), then:

(i) f ′(x) = 0 for all x ∈ (a,b) iff f is constant on (a,b).

(ii) f ′(x) < 0 for all x ∈ (a,b) iff f is decreasing on (a,b).

(iii) f ′(x) > 0 for all x ∈ (a,b) iff f is increasing on (a,b).

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Convince yourself of these geometric properties by looking at the

following four functions:

1. f1(x) = x2

2. f2(x) = |x |3. f3(x) = max{k ∈ Z : k ≤ x}4. f4(x) = 1 if x ∈ Q and f4(x) = 0 if x ∈ R−Q

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Definition(Partial Derivative)

Let X ⊆ Rn and suppose f : X → R. If x̄ is an interior point of X ,

then, the partial derivative of f with respect to xi at x̄ is defined as

∂f (x̄)

∂xi:= lim

h→0

f (x̄1, ..., x̄i−1, x̄i + h, x̄i+1, ..., x̄n)− f (x̄1, ..., x̄i , ..., x̄n)

h

with h ∈ R, whenever the limit exists. Another common notation

for the partial derivative of f with respect to xi at x̄ is fi (x̄).

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Definition(Gradient)

Let f : Rn → R a function which is partially differentiable with

respect ot all xi , i = 1, ..., n. Then the row vector

∇f (x̄) :=(f1(x̄) f2(x̄) · · · fn(x̄)

)is called the gradient of f at x̄ .

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https://www.khanacademy.org/math/multivariable-

calculus/multivariable-derivatives/partial-derivatives/v/partial-

derivatives-and-graphs

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Definition(Jacobian matrix)

Let f : Rn → Rm with partially differentiable component functions

f1, ..., fm : Rn → R, and let x̄ ∈ Rn. Then the Jacobian matrix of f

at x̄ is defined as

Jf (x̄) =

∂f1∂x1

(x̄)∂f1∂x2

(x̄) · · · ∂f1∂xn

(x̄)

∂f2∂x1

(x̄)∂f2∂x2

(x̄) · · · ∂f2∂xn

(x̄)

......

. . ....

∂fm∂x1

(x̄)∂fm∂x2

(x̄) · · · ∂fm∂xn

(x̄)

.

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https://www.khanacademy.org/math/multivariable-

calculus/multivariable-derivatives/partial-derivatives-of-vector-

valued-functions/v/computing-the-partial-derivative-of-a-vector-

valued-function

https://www.khanacademy.org/math/multivariable-

calculus/multivariable-derivatives/partial-derivatives-of-vector-

valued-functions/v/partial-derivative-of-a-parametric-surface-part-1

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Definition(Multivariate Derivative)

Let X ⊆ Rn and suppose f : X → R. If x̄ is an interior point of X ,

then f is differentiable at x̄ if and only if there exists a row vector

Df (x̄) such that

lim‖h‖→0

‖f (x̄ + h)− f (x̄)− Df (x̄) · h‖‖h‖

= 0

where h is a vector in Rn. If such a vector Df (x̄) exists, we

interpret it as the derivative of f at x̄ .

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Theorem( Total Differential, Partial Derivative, and Gradient)

Let X ⊆ Rn and suppose f : X → R. If x̄ is an interior point of X ,

and if f is differentiable at x̄ , then:

(i) all partial derivatives1 of f exist at x̄ and,

(ii) ∀ z in X , ‖z‖= 1: df (x̄ , z) := Df (x̄) · z = ∇f (x̄) · z

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Theorem( Partial Differentiablility and Differentiability)

Let X ⊆ Rn, suppose f : X → R, and let x̄ be an interior point of

X . If all the partial derivatives of f at x̄ exist and are continuous,

then f is differentiable.

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Definition(Multivariate Derivative of Vector-Valued Functions)

Let X ⊆ Rn and suppose f : X → Rm. If x̄ is an interior point of

X , then f is differentiable at x̄ if and only if there exists a matrix

Df (x̄) such that

lim‖h‖→0

‖f (x̄ + h)− f (x̄)− Df (x̄) · h‖‖h‖

= 0

where h is a vector in Rn. If such a matrix Df (x̄) exists, we

interpret it as the derivative of f at x̄ .

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Theorem( Multivariate Derivative and Jacobian Matrix)

Let X ⊆ Rn, suppose f : X → Rm, and let x̄ be an interior point

of X . Then, f is differentiable at x̄ if and only if each of its

component functions is differentiable at x̄ . Moreover, if f is

differentiable at x̄ , then:

(i) all partial derivatives of the component functions exist at x̄ ,

and

(ii) the derivative of f at x̄ is the matrix of partial derivatives of

the component functions at x̄ :

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Jf (x̄) := Df (x̄) =

∇f 1(x̄)...

∇f m(x̄)

=

∂f 1

∂x1(x̄) · · · ∂f 1

∂xn(x̄)

... · · ·...

∂f m

∂x1(x̄) · · · ∂f m

∂xn(x̄)

∈Rm×n

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Theorem( Multivariate Chain Rule)

Let X ⊆ Rn suppose g : X → Y , where Y ⊆ Rm. Further,

suppose f : Y → Z , where Z ⊆ Rp. If x̄ is an interior point of X ,

g(x̄) an interior point of Y , and g and f are differentiable at x̄ and

g(x̄), respectively, then f ◦ g is differentiable at x̄ and:

D[f ◦ g ](x) = Df (g(x̄))Dg(x̄)

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Definition(Hessian matrix)

Let X ⊆ Rn be open, suppose f : X → R, and let x̄ be an element

of X . If all second order partial derivatives of f are defined at x̄ ,

then, in the same way as the first order partial derivatives could be

gathered in a vector – the gradient –, all the second order partial

derivatives can be gathered in a matrix. Such a matrix, denoted

Hf(x̄), is called the Hessian of f at x̄ , is square, and should be

thought of as a generalized second order derivative for multivariate

real valued functions.

Hf(x̄) =

∇f1(x̄)

∇f2(x̄)...

∇fn(x̄)

=

f1,1(x̄) f1,2(x̄) · · · f1,n(x̄)

f2,1(x̄) f2,2(x̄) · · · f2,n(x̄)...

.... . .

...

fn,1(x̄) fn,2(x̄) · · · fn,n(x̄)

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Definition(Function of class C k)

Let X ⊆ Rn be an open set, Y ⊆ R, and suppose f : X → Y . f is

said to be of class C k on X , denoted f ∈ C k(X ,Y )2, if all partial

derivatives of order less or equal to k exist and are continuous on

X .

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Theorem( Schwarz’s Theorem / Young’s Theorem)

If f ∈ C k(X ), then the order in which the derivatives up to order k

are taken can be permuted.

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Theorem( nth Order Univariate Taylor Approximation)

Let X ⊆ R be an open set and consider f ∈ Cn+1(X ). Then f can

be best nth order approximated around x̄ by the nth order Taylor

expansion:

f (x̄ + h) ≈ f (x̄) +n∑

k=1

f (k)(x̄)hk

k!

where h ∈ R is such that x̄ + h ∈ X and f (k)(x) denotes f ’s

derivative of order k at x̄ . The error of approximation, also known

as the remainder of the Taylor approximation, is given by the

following formula:

Rn(h | x̄) := f (x̄ +h)− f (x̄)−n∑

k=1

f (k)(x̄)hk

k!=

f (n+1)(x + λh)

(n + 1)!hn+1

for some λ ∈ (0, 1).

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Theorem( Second Order Multivariate Taylor Approximation)

Let X ⊆ Rn be an open set, f ∈ C 3(X ). Then f can be best 2nd

order approximated around x̄ by the second order Taylor expansion:

f (x̄ + h) ≈ f (x̄) +∇f (x̄) · h +1

2h′ ·Hf(x̄)h

where h ∈ Rn is such that x̄ + h ∈ X . As ‖h‖ approaches zero, the

remainder approaches zero at a faster rate than h itself.

R2(h | x̄) := f (x̄ + h)− f (x̄)−∇f (x̄) · h +1

2h′ ·Hf(x̄)h =

f (n)(x + λh)

(n + 1)!hn+1

for some λ ∈ (0, 1).

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Theorem( Multivariate Convexity )

Let X be a convex subset of Rn. A real-valued function f : X → Rthat is also an element of C 2(X ) is convex if and only if, Hf (x) is

positive semidefinite for all x ∈ Int(X ). Further, if Hf (x) is

positive definite for all x ∈ Int(X ), then f is strictly convex.

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Contents

Multivariate functions

Invertibility

Convexity, Concavity, and Multivariate Real-Valued Functions

Multivariate differential calculus

Introduction

Revision: single-variable differential calculus

Partial Derivatives and the Gradient

Differentiability of real-valued functions

Differentiability of vector-valued functions

Higher Order Partial Derivatives and the Taylor Approximation Theorems

Characterization of convex functions

Integral theory

Univariate integral calculus

The definite integral and the fundamental theorem of calculus

Multivariate extension

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Theorem( Calculation rules for indefinite integrals)

Let f , g be two integrable functions3 and let a,C be constants,

n ∈ N. Then

•∫

(af (x) + g(x))dx = a

∫f (x)dx +

∫g(x)dx

•∫

xndx =xn+1

n + 1+ C if n 6= −1 and

∫1

xdx = lnx + C

•∫

exdx = ex + C and

∫ef (x)f ′(x)dx = ef (x) + C

•∫

(f (x))nf ′(x)dx =1

n + 1(f (x))n+1 + C if n 6=

−1 and

∫f (x)

f ′(x)dx = lnf (x) + C

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Theorem( Integration by parts)

Let u, v be two differentiable functions. Then,∫u(x)v ′(x)dx = u(x)v(x)−

∫u′(x)v(x)dx .

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Theorem

Let f : [a, b]→ R be continuous and define F (x) =

∫ x

af (t)dt for

all x ∈ [a, b]. Then, F is differentiable on (a, b) with

F ′(x) = f (x) for all x ∈ (a, b).

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Theorem( Fubini’s theorem)

Let Ix = [a, b] and Iy = [c, d ] be two intervals in Rn and define

I := Ix × Iy . Let f : I → R continuous. Then∫If (x , y)d(x , y) =

∫Ix

(∫Iy

f (x , y)dy

)dx ,

and all the integrals on the right-hand side are well-defined.

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TheoremLet A,B ∈ R two closed intervals, f : A→ R, g : B → Rcontinuous functions. Then∫

A×Bf (x)g(y)d(x , y) =

(∫Af (x)dx

)(∫Bg(y)dy

).