1. foundations of numerics from advanced mathematics ...€¦tu munc¨ hen exercise functions which...

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1. Foundations of Numerics from Advanced Mathematics

Calculus

Miriam Mehl: 1. Foundations of Numerics from Advanced Mathematics

Calculus, October 25, 2012 1

http://www5.in.tum.de/

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1.1. Calculus

Functions Revisited

• notions of a function, its range, and its image

• graph of a function

• isolines and isosurfaces

• sums and products of functions

• composition of functions

• inverse of a function: when existing?

• simple properties: (strictly) monotonous

• explicit and implicit definition

• parametrized representations (curves, ...)




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Exercise Functions

Which of the following curves is a graph of a function f (x)?

x x x

Graphically determine the image of the function graph(s).




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Exercise Functions – Solution

Which of the following curves is a graph of a function f (x)?

x x x

Graphically determine the image of the function graph(s).




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Exercise Functions

Sketch the isolines of the function f : R2 → R, (x , y) 7→ x2 + y2.




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Exercise Functions

Sketch the isolines of the functionf : R2 → R2, (x , y) 7→ x2 + y2.

x

y




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Exercise Functions

Assume, we have the functions f ,g : R→ R with

f (x) = sin(x), g(x) = cos(x).

(f 2 + g2)(x) =?

(f ◦ g)(x) = ?




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Exercise Functions – Solution

Assume, ew have the functions f ,g : R→ R with

f (x) = sin(x), g(x) = cos(x).

(f 2 + g2)(x) = sin2(x) + cos2(x) = 1.

(f ◦ g)(x) = sin(cos(x)).




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Examples for Functions

Explicit function:

f : R→ R, x 7→ (1− x)2 + ex

Implicit function:

f : R+0 → R+

0 , x 7→ y with x2 + y2 = 1

Parametrized function:f : R→ R, x 7→ gy(gx−1(x)) with g : R→ R2,g(t) =( gx(t),gy(t) ).




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Continuity

• remember the ε and the δ!

• definition of local (“in x0”) and global continuity (“∀x”)

• what about sums, products, quotients, ... of continuous functions?

• what about compositions of continuous functions?

• what about continuity of the inverse?

• intermediate value theorem

• continuous functions on compact sets – maximum and minimum value

• uniform continuity




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Exercise Continuity

We have two continuous functions f : R→ R and g : R→ R. fis continuous iff . . .




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Exercise Continuity – Solution

We have two continuous functions f : R→ R and g : R→ R. fis continuous iff∀ε > 0, x ∈ R ∃δ > 0 : |f (x)− f (y)| < ε ∀y : |x − y | < δ.




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Exercise Continuity

We have two continuous functions f : R→ R and g : R→ R.

Are f + g, f − g, f · g, fg , and f ◦ g continuous?




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Exercise Continuity – Solution

We have two continuous functions f : R→ R and g : R→ R.

Are f + g, f − g, f · g, fg , and f ◦ g continuous?

The continuity of all these functions can be shown easily using

|f (x)± g(x)− (f (y)± g(y))| ≤ |f (x)− f (y)|+ |g(x)− g(y)|

|f (x) · g(x)− f (y) · g(y)| = |(f (x)− f (y))g(x) + f (y)(g(x)− g(y))| ≤|g(x)||f (x)− f (y)|+ |f (y)||f (x)− f (y)|∣∣∣ 1

g(x)− 1

g(y)

∣∣∣ =∣∣∣ g(y)−g(x)

g(x)·g(y)

∣∣∣Proof the continuity of f ◦ g on your own! It’s easy, but a few lines to write.




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Limits

• meaning of ε→ 0 and N →∞ and x →∞

• accumulation point of a set

• limit (value) of a set

• limits from the left or from the right, respectively: f (x+), f (x−)

• limits at infinity: limx→∞ f (x)

• infinite limits: f (x)→∞

• how can discontinuities look like?

– jumps: f (x+) 6= f (x−)

– holes: f (x+) = f (x−) 6= f (x)

– second kind: f (x) = 0 in x = 0 and f (x) = sin(

1x

)elsewhere




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Exercise Limits

Determine the accumulation point of S ={1

n ;n ∈ N}

.




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Exercise Limits – Solution

Determine the accumulation point of{1

n ;n ∈ N}

.

The accumulation point is 0 since for all ε > 0 there is a e ∈ S with |e − 0| < ε.




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Limits – Visualization

N →∞:

x →∞:

ε→ 0:

0




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Examples for limx→∞ f (x):




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Two examples for f (x)→∞:




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Sequences

• definition of a sequence: a function f defined on N

• if f (n) = an, write(an) or a1, a2, a3, ...

• bounded / monotonously increasing / monotonously decreasing sequences

• notion of convergence of a sequence: existence of a limit for n→∞

• Cauchy sequence

• subsequences




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Sequences – Visualization

∀ε > 0 ∃ N ∈ N :|aN − aM | < ε ∀ N,M > N.

This is a Cauchy sequence!

This is NOT!




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Series

• notion of an (infinite) series– elements of a series– partial sums of a series– convergence defined by convergence of the sequence of the partial sums– convergence and absolute convergence

• examples:– geometric series:

∑∞k=1 xk = 1

1−x

– harmonic series:∑∞

k=11k = ∞

– alternating harmonic series:∑∞

k=1(−1)k−1 1k = ln(2)

• criteria for convergence: quotient and root criterion

• power series:∑∞

k=0 ak (z − a)k

– coefficients ak and centre point a– radius of convergence R: absolute convergence for |z − a| < R– identity theorem for power series

• re-arrangement

• sums of series, nested series, products of series (Cauchy product)




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Series – Convergence Criteria

quotient criterion: lim supn→∞|an+1||an| = q < 1 (convergence)

(Cauchy’s) root criterion:

lim supn→∞n√|an| = C

{< 1 absolute conv.> 1 divergence= 1 (abs.) con-/divvergence




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Series – Identitity Theorem for Power Series

If the radii of convergence of the power series∑∞

n=0 an(z − z0)n

and∑∞

n=0 bn(z − z0)n are positive and the sums of the series

are equal in infinitely many points which have z0 as anaccumulation point, then the both series are identical, i.e.an = bn for each n = 0,1,2, . . ..




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Differentiation

• first step: functions f of one real variable, complex values allowed

• derivative or differential quotient of f :

– defined via limit process of difference quotients– write f ′ or f or df

dx– geometric meaning?– local and global differentiability– derivative from the left / from the right

• rules for the daily work:

– derivative of f + g, fg, and f/g?– derivative of f (g) (chain rule)?– derivative of the inverse function?

• higher derivatives f (k)(x); meaning

• notion of continuous differentiability

• smoothness of a function

• space of k -times continuously differentiable functions: Ck




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Visualization Differention

f(x+h)−f(x)

hf(x)−f(x−h)

h

x

h 0




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Exercise Differentiation Rules

(f + g)′ = ?

(f · g)′ = ?

(f ◦ g)′ = ?

(f−1)′ = ?




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Exercise Differentiation Rules – Solution

(f + g)′ = f ′ + g′.

(f · g)′ = f ′ · g + f · g′. ?

(f ◦ g)′ = (f ′ ◦ g) · g′.

(f−1)′ = (f ′)−1.




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Differential Calculus of one Real Variable

• notion of a global/local minimum/maximum

• local extrema and the first derivative

• mean value theorem:

∃ξ ∈ (a, b) : f ′(ξ) =f (b)− f (a)

b − a

• monotonous behaviour and the first derivative

• local extrema and the second derivative

• rule of de l’Hospital

• notions of convexity and concavity

• convexity/concavity and the second derivative

• notion of a turning point

• turning points and the second derivative




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Visualization Mean Value Theorem

a b

f(b)

f(a)

ξ a b

f(b)

f(a)

ξξ 12




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Meaning of Derivatives

first derivative second derivativeincreasing > 0 –decreasing < 0 –maximum = 0 < 0minimum = 0 > 0convex – > 0concave – < 0turning point – = 0 con ex conCAVE




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Function Classes (1)

• polynomials– definition, degree, sums and products, division with rest, identity theorem,

roots and their multiplicity• rational functions

– poles and their multiplicity, partial fraction decomposition• exponential function and logarithm

– characterising law of the exponential function:

exp(s + t) = exp(s) · exp(t) or y ′ = y

(functional equation of natural growth)– series expansion of the exponential function, speed of growth– natural logarithm as exp’s inverse:

y = exp(x) = ex , x = ln(y)

– functional equation: ln(xy) = ln(x) + ln(y)

– exponential function and logarithm for general basis a:

ax := ex ln a, loga(y) :=ln yln a




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Function Classes (2)

• hyperbolic functions

– cosh(z), sinh(z), ...

• trigonometric functions

– sin(x), cos(x): solutions of y (2) + y = 0– geometric meaning?– Euler’s formula: eix = cos(x) + i · sin(x)

– derivatives, addition theorem– periodicity– series expansion




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Integral Calculus of one Variable

• Riemann integral, upper and lower sums• approximation by staircase functions• properties:

– linearity– monotonicity

• mean value theorem:

ξ ∈ (a, b) :

∫ b

af (x)dx = (b − a) · f (ξ)

• main theorem of differential and integral calculus:– define F (x) :=

∫ xa f (t)dt

– then∫ b

a f (t)dt = F (b)− F (a)

• rules for everyday work:– partial integration: ∫

uv ′dx = uv −∫

vu′dx

– substitution: ∫ b

af (t(x))t ′(x)dx =

∫ t(b)

t(a)f (t)dt




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Visualization Riemann-Integral




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Local Approximation: Taylor Polynomials and Series

• local approximation of functions with polynomials• generalization of the tangent approximation used for the definition of the derivative• Taylor polynomials:

– let f be n-times differentiable in a– we look for a polynomial T with T (k) = f (k) for k = 0, 1, ..., n– obviously:

T (x) :=n∑

k=0

1k!

f (k)(a)(x − a)k

– unique, degree n, write Tnf (x ; a)

– remainder Rn+1(x) := f (x)− Tnf (x : a)

Rn+1(x) =f (n+1)(ξ)

(n + 1)!(x − a)n+1

• Taylor series:– for infinitely differentiable functions (exp, sin, cos, ...)– sum up to∞ instead of n only– examples




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Visualization Taylor Polynomials

T(1)

T(2)

aa




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Global Approximation: Uniform Convergence

• convergence of sequences of functions fn defined on D:

• pointwise: for each x ∈ D; then

f (x) := limn→∞

fn(x)

defines a function

– problems: are properties such as continuity or differentiability inherited fromthe fn to f , and how to calculate derivatives or integrals of f?

– i.e., can the order of limit processes be changed?

• therefore the notion of uniform convergence:

– definition: ‖fn − f‖D → 0 for n→∞– with that, the inheritance and change-order problems from above are solved!– criteria: Cauchy, ...

• approximation theorem of Weierstrass: each continuous function f on acompact set can be arbitrarily well approximated with some polynomial




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Simple Differential Equations

• notion of a differential equation

– ordinary: one variable– partial: more than one variable (several spatial dimensions or space and

time)

• examples:

– growth: y = k · y or y = k(t , y) · y– oscillation: y + y = 0 or similar

• example of an analytic solution strategy: separation of variables

y ′ = g(x) · h(y), y(x) = y0

– formal separation:dy

h(y)= g(x)dx

– integration of the left and right side– some requirements for applicability




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Periodic Functions

• target now: periodic functions, period typically 2π

• trigonometric polynomials

• definition:

T (x) :=n∑

k=−n

ck eikx =a0

2+

n∑k=1

(ak cos(kx) + bk sin(kx))

(coefficients ck , ak , and bk are unique)

• formula for the coefficients:

ck =1

2π

∫ 2π

0T (x)e−ikx dx

• T is real iff all ak , bk are real iff ck = c−k

• Weierstrass: 2π-periodic continuous functions can be arbitrarily wellapproximated by trigonometric polynomials




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Fourier Series

• consider vector space of 2π-periodic complex functions f on R

• Fourier coefficients: f (k) :=1

2π

∫ 2π

0f (x)e−ikx dx

• Fourier polynomial: Snf (x) :=n∑

k=−n

f (k)eikx

• Fourier series:∞∑−∞

f (k)eikx

• sine-cosine representation of Snf :

Snf (x) =a0

2+

n∑k=1

(ak cos(kx) + bk sin(kx))

• coefficients:ak = f (k) + f (−k) =

1π

∫ π

−πf (x) cos(kx)dx

bk = i(f (k)− f (−k)) =1π

∫ π

−πf (x) sin(kx)dx

• all ak vanish for odd f , all bk vanish for even f




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Functions of Several Variables

• f now defined on Rn or a subset of it

• notion of differentiability: now via existence of a linear map, the differential

• directional derivatives

• partial derivatives

• prominent differentiability criterion: existence and continuity of all partialderivatives

• the gradient of a scalar function f and its interpretation

• the Jacobian of a vector-valued function f

• mean value theorem

• higher partial derivatives, Taylor approximation, Hessian

• local minima and maxima, criteria

• saddle points




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Visualization Saddle Point




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Integration over Domains

• a huge field, from which we only mention a few results• theorem of Fubini:

– shows that, in many cases, a multi-dimensional integration domain can betackled dimension by dimension

– statement (we neglect the requirements, for which more integration theory isneeded):∫

X×Yf (x , y)d(x , y) =

∫Y

(∫X

f (x , y)dx)

dy =

∫X

(∫Y

f (x , y)dy)

dx

– related to Cavalieri’s principle– will also be of relevance for numerical quadrature

• transformation theorem:– a generalisation of integration by substitution– statement, again without requirements:∫

Uf (T (x)) ·

∥∥det T ′(x)∥∥ dx =

∫V

f (y)dy

– allows for a change of the coordinate system (polar coordinates), e.g.




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Gauss Theorem

• we further generalise integration, now allowing for integration over hyper-surfaces(a sphere, e.g.)

• this is important for the physical modelling in many scenarios (heat flux through apot’s surface, ...)

• the famous Gauss theorem allows to combine integrals over volumes andsurfaces, which occurs in the derivation of many physical models (conservationlaws) and, hence, is of special relevance for CSE

• prerequisites:– a vector field: a vector-valued function on Rn (example: the velocity field in

fluid mechanics)– the divergence of a vector field F :

div F (x) =n∑

i=1

∂i Fi (x)

• finally the Gauss theorem:– several regularity assumptions needed∫

Gdiv Fdx =

∫∂G

F ~dS




1. foundations of numerics from advanced mathematics ...€¦tu munc¨ hen exercise functions which...

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