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.

2 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: 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.

3 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.

4 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)

5 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; a good book of numerical analysis or scientific computing, likeActon,Heath, Hildebrand, Krishnamurthy and Sen,Press etal, Stoer and Bulirsch.

6 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.

7 Think. Get excited. Discuss. Bore everybody in your knowncircles. Not enough time to attempt all? Want aselection? Program implementation is needed in algorithmic exercises. 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)

8 142,4,5,6,74381,2,3,4,55156,77392,3,4,54 162,3,4,88401,2,4,54171,2,3,66411,3,6,88 181,2,3,6,73421,3,66191,3,4,66432,3,4320 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.

9 [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 ?

10 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?