03 Handout Calculus

7/28/2019 03 Handout Calculus

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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/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) x2 + y2.




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Exercise Functions

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

x




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Exercise Functions

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

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

(f2 + 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).

(f2 + 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 (1 x)2 + ex

Implicit function:

f : R+0 R+0 , x y with x

2 + y2 = 1

Parametrized function:

f : R R, x gy(gx1(x)) with g : R R2, g(t) =( gx(t),gy(t) ).




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Continuity

remember the and the !

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

> 0, x R > 0 : |f(x) f(y)| < y : |x y| < .




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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 gcontinuous?




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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 gcontinuous?

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)|

1g(x)

1

g(y) = g(y)g(x)

g(x)g(y)

Proof the continuity of f gon your own! Its easy, but a few lines to write.




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Limits

meaning of 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+) = f(x)

holes: f(x+) = f(x) = 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=

1n;n N

.



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Exercise Limits Solution

Determine the accumulation point of

1n;n N

.

The accumulation point is 0 since for all > 0 there is a e Swith |e 0| < .



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Limits Visualization

N :

x :

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

> 0 N N :|aN aM| < 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 x

k

=

1

1x

harmonic series:

k=11k

=

alternating harmonic series:

k=1(1)k1 1

k= 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)

(Cauchys) root criterion:

lim supnn

|an| = C

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

Miriam Mehl: 1. Foundations of Numerics from Advanced MathematicsCalculus, October 25, 2012 25

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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 seriesare 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 dfdx

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(xh)

h

x

h 0


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Exercise Differentiation Rules

(f+ g) = ?

(f g) = ?

(f g) = ?

f1

= ?


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Exercise Differentiation Rules Solution

(f+ g) = f + g.

(f g) = f g+ f g. ?

(f g) = (f g) g.

f1

= (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:

(a, b) : f() =f(b) f(a)

b a

monotonous behaviour and the first derivative local extrema and the second derivative

rule of de lHospital

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)

a b

f(b)

f(a)

12


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Meaning of Derivatives

first derivative second derivative

increasing > 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 exps 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 y

ln 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?

Eulers 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:

(a, b) : b

a

f(x)dx = (b a) f()

main theorem of differential and integral calculus:

define F(x) :=xaf(t)dt

thenbaf(t)dt = F(b) F(a)

rules for everyday work:

partial integration: uvdx = uv

vudx

substitution: ba

f(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

1

k! 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)()

(n+ 1)!(x a)n+1

Taylor series:

for infinitely differentiable functions (exp, sin, cos,...)

sum up to instead of nonly

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 from

the 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 fD 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 2

trigonometric polynomials

definition:

T(x) :=n

k=n

ckeikx =

a0

2+

nk=1

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

(coefficients ck, ak, and bk are unique)

formula for the coefficients:

ck =1

2

20

T(x)eikxdx

T is real iff all ak, bk are real iff ck = ck Weierstrass: 2-periodic continuous functions can be arbitrarily well

approximated by trigonometric polynomials



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F i S i


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Fourier Series

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

Fourier coefficients: f(k) :=1

22

0f(x)eikxdx

Fourier polynomial: Snf(x) :=n

k=n

f(k)eikx

Fourier series:

f(k)eikx

sine-cosine representation of Snf:

Snf(x) =a0

2+

nk=1

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

coefficients:

ak = f(k) + f(k) = 1

f(x) cos(kx)dx

bk = i(f(k) f(k)) =1

f(x) sin(kx)dx

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



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F ti f S l V i bl


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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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Vi li ti S ddl P i t


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Visualization Saddle Point



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Integration over Domains


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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 be

tackled dimension by dimension

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

XY f(x, y)d(x, y) =

Y

X f(x, y)dxdy =

X

Y f(x, y)dydx

related to Cavalieris principle

will also be of relevance for numerical quadrature

transformation theorem:

a generalisation of integration by substitution

statement, again without requirements:U

f(T(x)) detT(x) dx =

V

f(y)dy

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



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Gauss Theorem


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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 apots 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 influid mechanics)

the divergence of a vector field F:

divF(x) =n

i=1

iFi(x)

finally the Gauss theorem:

several regularity assumptions neededG

divFdx =

G

F dS



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03 Handout Calculus