Table Of ContentChristian Karpfinger
Calculus and
Linear Algebra
in Recipes
Terms, phrases and numerous examples in
short learning units
Calculus and Linear Algebra in Recipes
Christian Karpfinger
Calculus and Linear Algebra
in Recipes
Terms, phrases and numerous examples
in short learning units
ChristianKarpfinger
TechnischeUniversitätMünchen
ZentrumMathematik
München,Germany
ISBN978-3-662-65457-6 ISBN978-3-662-65458-3 (eBook)
https://doi.org/10.1007/978-3-662-65458-3
Thetranslationwasdonewiththehelpofartificialintelligence(machinetranslationbytheserviceDeepL.com).
Asubsequenthumanrevisionwasdoneprimarilyintermsofcontent.
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Foreword to the Third Edition
In the present third edition, we have revised and extended the recipe book: all known
errors have been corrected, new tasks have been added, and further topics have been
added,namelyresidualelementestimationinTaylorexpansion,thenumericalsolutionof
boundaryvalueproblems,and the solution of first-orderpartialdifferentialequationsby
meansofthecharacteristicmethod.Thus,wehavepresentedfurthertopicsonnumerical
mathematics or differentialequations, which are importantfor the engineer,in a proven
andunderstandablemanner.
On the website of the book you will find, besides the previous add-ons, video
animationsandappsthatillustratesomeofthemathematicalcontentsoftherecipebook.
Thewebsiteforthisbookcanbefoundvia
http://www.springer-spektrum.de/
Munich,Germany ChristianKarpfinger
April2017
v
Preface to the Second Edition
The main new features in this second edition are the chapter on the solution of partial
differential equations using integral transformations and a section on the numerical
solution of the wave equation. This is intended to introduce further important methods
forfindingsolutionsofthepartialdifferentialequationswhichareso fundamentalinthe
naturalsciencesandtechnology.
Many small improvementsor additionsto explanationsor exampleshave foundtheir
wayintothesecondedition,andofcourseallknownerroneouspassagesinthetexthave
been corrected. We have also increased the number of exercises, especially in the later
chapters, in order to promote the practice of the formulas and the understanding of the
laterandmoredifficulttopics.
On the website for this book, as a special extra, you will find video recordings of
many lectures that follow the present text. In addition, we have supplemented the script
“Introductionto MATLAB”, which can be found on the website for this book, with the
part“MATLAB—agreatcalculator”.Mr.BenjaminRüthcontributedsignificantlytothe
creationofthisscript,andIwouldliketothankhimverymuch.AlsotheMATLABcodes,
whichareusedintherecipebookaswellasintheaccompanyingworkbook,andalsothe
solutionstomostoftheexercisescanbefoundonthementionedwebsite.Thelinktothe
websitecanbefoundvia
http://www.springer-spektrum.de/
Furthercommentsfromthereadershiparealwayswelcome.
Munich,Germany ChristianKarpfinger
April2014
vii
Preface to the First Edition
Joiningthemanybooksonhighermathematicsisanother,thepresentbookCalculusand
Linear Algebra in Recipes. In writing the book, the author had the following aspects in
mind:
• Many typical problems in higher mathematics can be solved in recipes. The book
providesacollectionofthemostimportantrecipes.
• AclearpresentationofthetopicsthatcanbecoveredinfoursemestersofCalculusand
LinearAlgebra.
• Aninexpensivebookforundergraduatesthatcoversallthemajorcontent.
• Numerousexamplesthatreinforcethetheoriesandpracticeusingtherecipes.
• A division of the materialinto many roughlyequally short teaching or learning units
(eachchaptercanbecoveredinabouta90-minutelecture).
• Omitting content that is usually only actually understood by about 10 percent of the
studentbodyandthatisoflesserimportanceforpractice.
• NumericalmathematicsandalsotheuseofMATLABareanintegralpartofthecontent.
It is customary, but perhaps not entirely correct, to teach higher mathematics as proof-
completely as possible. The disadvantages are obvious: desperate students who then
quickly realize that the exams can largely be passed without the proofs. It might make
more sense to use the time gained by omitting proofs to cover the topics that are so
importantforpractice,suchasnumericsandMATLAB.Wecoverafewtopicsinnumerical
mathematics, punctuating them with numerous examples, and always showing how to
use MATLAB as a great calculator in the engineering mathematics problems covered.
Occasionally,especiallyin the assignments,we also solve programmingproblemsusing
MATLAB.TherebyhardlyanypreviousknowledgeforMATLABarenecessary.Weputon
theInternetpagetothisbookunder
http://www.springer-spektrum.de/
ashortintroductioncoursetoMATLABonafewpages.
ix
x PrefacetotheFirstEdition
The inputs to MATLAB we always formulate with an excellent font type. And
instead of a comma we do like MATLAB we also put a dot, so we write 1.25 for 5/4.
Occasionally we calculate with MATLAB symbolically, which is possible thanks to the
SYMBOLIC MATH TOOLBOX this is also possible, note that you also have this toolbox
installed.
Wesummarizethespecialfeaturesofthisbookonceagain:
• Wedonotattempttoerecttheabstractedificeofmathematics,createdovermillennia,
inafew100pagesascomprehensivelyandproof-completelyaspossible.We address
the topics of mathematics that are important to engineers, make concepts and rules
plausibleifonlythatisfeasible,anduseexamplesandmanyproblemstolearnhowto
solveproblems.
• We dividethetopicsofhighermathematicsintonearly100chaptersofroughlyequal
length and formulate numerous problem-solving strategies in a recipe-like fashion.
Each chapter covers about the material of a 90-minute lecture. This provides an
overviewandtheopportunitytoplan,bothforstudentsandlecturers.
• We use the computer, in particular MATLAB as a powerful calculator to deal with
realisticexamplesinsteadoftheusualacademicexamples.
Attheendofthechapterssomeexercisesaregiven,whicharerecommendedtowork
on.These exercisescanbe usedto checktheunderstandingofthe presentedrecipesand
methods.OntheInternetpagetothebookunder
http://www.springer-spektrum.de/
wehaveprovideddetailedsolutionsuggestionsforalotoftheexercises.Theexercisesand
solutionsarealsoprintedintheaccompanyingworkbook.
The creation of this comprehensive book was not possible without the help of
many colleagues and collaborators. For proof-reading,for numerous hints, suggestions,
proposals for improvement, tasks, examples, sketches and MATLAB programs, I would
like to thankDr. L. Barnerßoi,Prof.Dr. D. Castrigiano, S. Dorn,F. Ellensohn,Dr. H.-J.
Flad, P. Gerhard, S. Held, Dr. F. Himstedt, Dr. F. Hofmaier, Prof. Dr. O. Junge, Dr. S.
Kaniber,B.Kleinherne,Y.Kochnova,A.Köhler,Dr.M.Kohls,Dr.P.Koltai,A.Kreisel,
Prof. Dr. C. Lasser, Dr. D. Meidner, N. Michels, S. Otten, M. Perner, P. Piprek, Dr. M.
Prähofer, F. Reimers, Dr. K.-.D. Reinsch, Prof. Dr. P. Rentrop, B. Rüth, M. Ritter, Th.
Simon, A. Schreiber,Dr. Th. Stolte, Prof. Dr. B. Vexler, Dr. H. Vogel, J. Wenzel and E.
Zindan.
PrefacetotheFirstEdition xi
Special thanks go to Dr. Ch. Ludwig, who not only always had an open ear for my
questions,whetherduringthedayoratnight,healsoalwayshadasolutionready.Finally,
mythanksalsogotoTh.Epp,whocreatedmostofthepictures,andtoB.AltonandDr.A.
RüdingerofSpringerSpektrum,whoaccompaniedthecreationofthebookwithnumerous
piecesofadvice.
Munich,Germany ChristianKarpfinger
August2013
Contents
1 Speech,SymbolsandSets....................................................... 1
1.1 SpeechPatternsandSymbolsinMathematics............................ 1
1.2 SummationandProductSymbol .......................................... 4
1.3 PowersandRoots .......................................................... 5
1.4 SymbolsofSetTheory..................................................... 6
1.5 Exercises.................................................................... 8
2 TheNaturalNumbers,IntegersandRationalNumbers.................... 11
2.1 TheNaturalNumbers...................................................... 11
2.2 TheIntegers ................................................................ 15
2.3 TheRationalNumbers..................................................... 15
2.4 Exercises.................................................................... 17
3 TheRealNumbers............................................................... 19
3.1 Basics....................................................................... 19
3.2 RealIntervals............................................................... 20
3.3 TheAbsoluteValueofaRealNumber.................................... 21
3.4 n-thRoots................................................................... 22
3.5 SolvingEquationsandInequalities........................................ 23
3.6 Maximum,Minimum,SupremumandInfimum.......................... 25
3.7 Exercises.................................................................... 26
4 MachineNumbers............................................................... 29
4.1 b-adicRepresentationofRealNumbers................................... 29
4.2 FloatingPointNumbers.................................................... 31
4.3 Exercises.................................................................... 35
5 Polynomials....................................................................... 37
5.1 Polynomials:MultiplicationandDivision ................................ 37
5.2 FactorizationofPolynomials.............................................. 41
xiii
