Mathematical Methods in Engineering and Science …

1 Mathematical Methods in Engineering and Science1, Mathematical Methods in Engineering andScience[ dasgupta/MathCourse]Bhaskar Applied Mathematics course forgraduate and senior undergraduate studentsand also forrising Methods in Engineering and Science2,Textbook:Dasgupta B.,App. Math. Meth.(Pearson Education 2006, 2007). dasgupta/MathBookMathematical Methods in Engineering and Science3,Contents IPreliminary BackgroundMatrices and Linear TransformationsOperational Fundamentals of Linear AlgebraSystems of Linear EquationsGauss Elimination Family of MethodsSpecial Systems and Special MethodsNumerical Aspects in Linear SystemsMathematical Methods in Engineering and Science4,Contents IIEigenvalues and EigenvectorsDiagonalization and Similarity TransformationsJacobi and Givens Rotation MethodsHouseholder Transformation and Tridiagonal MatricesQR Decomposition MethodEigenvalue Problem of General MatricesSingular Value DecompositionVector Spaces: Fundamental Concepts* Mathematical Methods in Engineering and Science5,Contents IIIT opics in Multivariate CalculusVector Analysis.

2 Curves and SurfacesScalar and Vector FieldsPolynomial EquationsSolution of Nonlinear Equations and SystemsOptimization: IntroductionMultivariate OptimizationMethods of Nonlinear Optimization* Mathematical Methods in Engineering and Science6,Contents IVConstrained OptimizationLinear and Quadratic Programming Problems*Interpolation and ApproximationBasic Methods of Numerical IntegrationAdvanced Topics in Numerical Integration*Numerical Solution of Ordinary Differential EquationsODE Solutions: Advanced IssuesExistence and Uniqueness TheoryMathematical Methods in Engineering and Science7,Contents VFirst Order Ordinary Differential EquationsSecond Order Linear Homogeneous ODE sSecond Order Linear Non-Homogeneous ODE sHigher Order Linear ODE sLaplace TransformsODE SystemsStability of Dynamic SystemsSeries Solutions and Special FunctionsMathematical Methods in Engineering and Science8,Contents VISturm-Liouville TheoryFourier Series and IntegralsFourier TransformsMinimax Approximation*Partial Differential EquationsAnalytic FunctionsIntegrals in the Complex PlaneSingularities of Complex FunctionsMathematical Methods in Engineering and Science9,Contents VIIV ariational Calculus*EpilogueSelected ReferencesMathematical Methods in Engineering and SciencePreliminary Background10.

3 Theme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundOutlinePreliminary BackgroundTheme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundMathematical Methods in Engineering and SciencePreliminary Background11,Theme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundTheme of the CourseTo develop a firm Mathematical background necessary for graduatestudies and research a fast-paced recapitulation of UG mathematics extension with supplementary advanced ideas for a matureand forward orientation exposure and highlighting of interconnectionsTopre-emptneeds of thefuturechallenges trade-off betweensufficientandreasonable target mid-spectrummajorityof studentsNotable beneficiaries (at two ends) would-be researchers in analytical/computational areas students who are till now somewhatafraidof mathematicsMathematical Methods in Engineering and SciencePreliminary Background12,Theme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundCourse Contents Applied linear algebra Multivariate calculus and vector calculus Numerical Methods Differential equations + + Complex analysisMathematical Methods in Engineering and SciencePreliminary Background13,Theme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundSources for More Detailed StudyIf you have the time, need and interest, then you may consult individual bookson individual topics; another umbrella volume, like Kreyszig, McQuarrie,O Neilor Wylie and Barrett.

4 A good book of numerical analysis or scientific computing, likeActon,Heath, Hildebrand, Krishnamurthy and Sen,Press etal, Stoer and Bulirsch; friends, injoint-study Methods in Engineering and SciencePreliminary Background14,Theme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundLogistic Strategy Study in the given sequence, to the extent possible. Do not read mathematics. Use lots of pen and mathematics books anddomathematics. Exercises aremust. Use as many Methods as you can think of, certainly includingthe one which is recommended. Consult the Appendix after you work out the solution. Followthe comments, interpretations and suggested extensions. Think. Get excited. Discuss. Bore everybody in your knowncircles. Not enough time to attempt all? Want aselection? Program implementation is needed in algorithmic exercises.

5 Master a programming environment. Use Mathematical /numerical a MATLAB tutorial session? Mathematical Methods in Engineering and SciencePreliminary Background15,Theme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundLogistic StrategyTutorial PlanChapter Selection Tutorial ChapterSelectionTutorial22,33261,2,4,643 2,4,5,64,5271,2,3,43,441,2,4,5,74,5282,5 ,6651,4,54291,2,5,6661,2,4,74301,2,3,4,5 471,2,3,42311,21(d)81,2,3,4,64321,3,5,77 91,2,44331,2,3,7,88102,3,44341,3,5,65112 ,4,55351,3,43121,33361,2,44131,213711(c) 142,4,5,6,74381,2,3,4,55156,77392,3,4,54 162,3,4,88401,2,4,54171,2,3,66411,3,6,88 181,2,3,6,73421,3,66191,3,4,66432,3,4320 1,2,32441,2,4,7,9,107,10211,2,5,7,87451, 2,3,4,7,94,9221,2,3,4,5,63,4461,2,5,7723 1,2,33471,2,3,5,8,9,109,10241,2,3,4,5,61 481,2,4,55251,2,3,4,55 Mathematical Methods in Engineering and SciencePreliminary Background16,Theme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundExpected Background moderate background of undergraduate mathematics firm understanding of school mathematics and undergraduatecalculusTake the preliminary test.

6 [p 3,App. Math. Meth.]Grade yourself sincerely.[p 4,App. Math. Meth.]Prerequisite Problem Sets*[p 4 8,App. Math. Meth.] Mathematical Methods in Engineering and SciencePreliminary Background17,Theme of the CourseCourse ContentsSources for More Detailed StudyLogistic StrategyExpected BackgroundPoints to note Put in effort, keep pace. Stress concept as well as problem-solving. Follow Methods diligently. Ensure background Exercises:Prerequisite problem sets ?? Mathematical Methods in Engineering and ScienceMatrices and Linear Transformations18,MatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyOutlineMatrices and Linear TransformationsMatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyMathematical Methods in Engineering and ScienceMatrices and Linear Transformations19,MatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyMatricesQuestion:What is a matrix ?

7 Answers: a rectangular array of numbers/elements ? a mappingf:M N F, whereM={1,2,3, ,m},N={1,2,3, ,n}andFis the set of real numbers orcomplex numbers ?Question:What does a matrixdo?Explore:With anm nmatrixA,y1=a11x1+a12x2+ +a1nxny2=a21x1+a22x2+ + +am2x2+ +amnxn orAx=yMathematical Methods in Engineering and ScienceMatrices and Linear Transformations20,MatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyMatricesConsider these definitions: y=f(x) y=f(x) =f(x1,x2, ,xn) yk=fk(x) =fk(x1,x2, ,xn),k= 1,2, ,m y=f(x) y=AxFurther Answer:A matrix is thedefinitionof a linear vector function of avector deeper?Caution:Matricesdo notdefine vector functions whose components areof the formyk=ak0+ak1x1+ak2x2+ + Methods in Engineering and ScienceMatrices and Linear Transformations21,MatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyGeometry and AlgebraLet vectorx= [x1x2x3]Tdenote a point (x1,x2,x3) in3-dimensional space in frame of :Withm= 2 andn= 3,y1=a11x1+a12x2+a13x3y2=a21x1+a22x2+a23 x3.

8 Ploty1andy2in theOY1Y2plane. 32AR2: R Co domain Domain1xy 3 XYYXX12 OOFigure:Linear transformation: schematic illustrationWhat is matrixAdoing? Mathematical Methods in Engineering and ScienceMatrices and Linear Transformations22,MatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyGeometry and AlgebraOperatingon pointxinR3, matrixAtransformsit theimageof pointxunder the mapping defined R3,co-domain R2with reference to thefigureandverify thatA:R3 R2fulfils the requirements of amapping, matrix givesadefinition of alinear transformationfrom one vector space to Methods in Engineering and ScienceMatrices and Linear Transformations23,MatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyLinear TransformationsOperateAon a large number of pointsxi corresponding imagesyi linear transformation represented byAimplies the totality ofthese decide to use a differentframe of reference OX 1X 2X 3forR3.

9 [And, possiblyOY 1Y 2forR2at the same time.]Coordinateschange, tox i(and possiblyyitoy i).Now, we need a different matrix, sayA , to get back thecorrespondence asy =A x .A matrix: :How to get the new matrixA ? Mathematical Methods in Engineering and ScienceMatrices and Linear Transformations24,MatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyMatrix Terminology Matrix product Transpose Conjugate transpose Symmetric and skew-symmetric matrices Hermitian and skew-Hermitian matrices Determinant of a square matrix Inverse of a square matrix Adjoint of a square matrix Mathematical Methods in Engineering and ScienceMatrices and Linear Transformations25,MatricesGeometry and AlgebraLinear TransformationsMatrix TerminologyPoints to note A matrix defines a linear transformation from one vector spaceto another. Matrix representation of a linear transformation depends onthe selected bases (or frames of reference) of the source andtarget :Revise matrix algebra basics as necessary Exercises:2,3 Mathematical Methods in Engineering and ScienceOperational Fundamentals of Linear Algebra26,Range and Null Space: Rank and NullityBasisChange of BasisElementary TransformationsOutlineOperational Fundamentals of Linear AlgebraRange and Null Space: Rank and NullityBasisChange of BasisElementary TransformationsMathematical Methods in Engineering and ScienceOperational Fundamentals of Linear Algebra27,Range and Null Space: Rank and NullityBasisChange of BasisElementary TransformationsRange and Null Space: Rank and NullityConsiderA Rm nas a mappingA.

10 Rn Rm,Ax=y,x Rn,y Rnhas an imagey Rm, but everyy Rmneednot have a pre-image (or range space) as subset/subspace ofco-domain: containing images ofallx ofx RninRmis unique, but pre-image ofy Rmneed not may be non-existent, unique or infinitely space as subset/subspace of domain:containing pre-images ofonly0 Methods in Engineering and ScienceOperational Fundamentals of Linear Algebra28,Range and Null Space: Rank and NullityBasisChange of BasisElementary TransformationsRange and Null Space: Rank and NullityRmRnNull ( ) AORange ( ) ADomain Co domain0 AFigure:Range and null space: schematic representationQuestion:What is the dimension of a vector space?Linear dependence and independence:Vectorsx1,x2, ,xrin a vector space are called linearly independent ifk1x1+k2x2+ +krxr=0 k1=k2= =kr= (A) ={y:y=Ax,x Rn}Null(A) ={x:x Rn,Ax=0}Rank(A) = dimRange(A)Nullity(A) = dimNull(A) Mathematical Methods in Engineering and ScienceOperational Fundamentals of Linear Algebra29,Range and Null Space: Rank and NullityBasisChange of BasisElementary TransformationsBasisTake a set of vectorsv1,v2, ,vrin a vector :Given a vectorvin the vector space, can we describe itasv=k1v1+k2v2+ +krvr=Vk,whereV= [v1v2 vr] andk= [k1k2 kr]T?