Holdings: Applied mathematics: body and soul

Applied mathematics: body and soul: 1 Derivatives and geometry in R 3

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Bibliographic Details
Main Authors:	Eriksson, Kenneth (Author), Estep, Donald (Author), Johnson, Claes (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2004
Online Access:	Inhaltsverzeichnis
Physical Description:	XLIII, 425 S. Ill., graph. Darst.
ISBN:	354000890X

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100	1		\|a Eriksson, Kenneth \|e Verfasser \|4 aut
245	1	0	\|a Applied mathematics: body and soul \|n 1 \|p Derivatives and geometry in R 3 \|c K. Eriksson ; D. Estep ; C. Johnson
264		1	\|a Berlin [u.a.] \|b Springer \|c 2004
300			\|a XLIII, 425 S. \|b Ill., graph. Darst.
336			\|b txt \|2 rdacontent
337			\|b n \|2 rdamedia
338			\|b nc \|2 rdacarrier
700	1		\|a Estep, Donald \|e Verfasser \|4 aut
700	1		\|a Johnson, Claes \|e Verfasser \|4 aut
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adam_text	IMAGE 1 CONTENTS VOLUME 2 INTEGRALS AND GEOMETRY IN R N 427 27 THE INTEGRAL 429 27.1 PRIMITIVE FUNCTIONS AND INTEGRALS . . . . . . . . . . . . 429 27.2 PRIMITIVE FUNCTION OF F ( X ) = X M FOR M = 0 , 1 , 2 , . . . . . 433 27.3 PRIMITIVE FUNCTION OF F ( X ) = X M FOR M = * 2 , * 3 , . . . . 434 27.4 PRIMITIVE FUNCTION OF F ( X ) = X R FOR R * = * 1 . . . . . . 434 27.5 A QUICK OVERVIEW OF THE PROGRESS SO FAR . . . . . . . 435 27.6 A "VERY QUICK PROOF" OF THE FUNDAMENTAL THEOREM . 435 27.7 A "QUICK PROOF" OF THE FUNDAMENTAL THEOREM . . . . 437 27.8 A PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS . . . 438 27.9 COMMENTS ON THE NOTATION . . . . . . . . . . . . . . . 444 27.10 ALTERNATIVE COMPUTATIONAL METHODS . . . . . . . . . . 445 27.11 THE CYCLIST'S SPEEDOMETER . . . . . . . . . . . . . . . . 445 27.12 GEOMETRICAL INTERPRETATION OF THE INTEGRAL . . . . . . . 446 27.13 THE INTEGRAL AS A LIMIT OF RIEMANN SUMS . . . . . . . 448 27.14 AN ANALOG INTEGRATOR . . . . . . . . . . . . . . . . . . . 449 28 PROPERTIES OF THE INTEGRAL 453 28.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 453 28.2 REVERSING THE ORDER OF UPPER AND LOWER LIMITS . . . . 454 28.3 THE WHOLE IS EQUAL TO THE SUM OF THE PARTS . . . . . . 454 IMAGE 2 XVIII CONTENTS VOLUME 2 28.4 INTEGRATING PIECEWISE LIPSCHITZ CONTINUOUS FUNCTIONS . . . . . . . . . . . . . . . . . . 455 28.5 LINEARITY . . . . . . . . . . . . . . . . . . . . . . . . . . 456 28.6 MONOTONICITY . . . . . . . . . . . . . . . . . . . . . . . 457 28.7 THE TRIANGLE INEQUALITY FOR INTEGRALS . . . . . . . . . . 457 28.8 DIFFERENTIATION AND INTEGRATION ARE INVERSE OPERATIONS . . . . . . . . . . . . . . . . . . 458 28.9 CHANGE OF VARIABLES OR SUBSTITUTION . . . . . . . . . . . 459 28.10 INTEGRATION BY PARTS . . . . . . . . . . . . . . . . . . . 461 28.11 THE MEAN VALUE THEOREM . . . . . . . . . . . . . . . . 462 28.12 MONOTONE FUNCTIONS AND THE SIGN OF THE DERIVATIVE . . 464 28.13 A FUNCTION WITH ZERO DERIVATIVE IS CONSTANT . . . . . . 464 28.14 A BOUNDED DERIVATIVE IMPLIES LIPSCHITZ CONTINUITY . . 465 28.15 TAYLOR'S THEOREM . . . . . . . . . . . . . . . . . . . . . 465 28.16 OCTOBER 29, 1675 . . . . . . . . . . . . . . . . . . . . . 468 28.17 THE HODOMETER . . . . . . . . . . . . . . . . . . . . . . 469 29 THE LOGARITHM LOG( X ) 473 29.1 THE DEFINITION OF LOG( X ) . . . . . . . . . . . . . . . . . 473 29.2 THE IMPORTANCE OF THE LOGARITHM . . . . . . . . . . . . 474 29.3 IMPORTANT PROPERTIES OF LOG( X ) . . . . . . . . . . . . . 475 30 NUMERICAL QUADRATURE 479 30.1 COMPUTING INTEGRALS . . . . . . . . . . . . . . . . . . . 479 30.2 THE INTEGRAL AS A LIMIT OF RIEMANN SUMS . . . . . . . 483 30.3 THE MIDPOINT RULE . . . . . . . . . . . . . . . . . . . . 484 30.4 ADAPTIVE QUADRATURE . . . . . . . . . . . . . . . . . . . 485 31 THE EXPONENTIAL FUNCTION EXP( X ) = E X 491 31.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 491 31.2 CONSTRUCTION OF THE EXPONENTIAL EXP( X ) FOR X * 0 . . . 493 31.3 EXTENSION OF THE EXPONENTIAL EXP( X ) TO X 0 . . . . . 498 31.4 THE EXPONENTIAL FUNCTION EXP( X ) FOR X * R . . . . . . 498 31.5 AN IMPORTANT PROPERTY OF EXP( X ) . . . . . . . . . . . . 499 31.6 THE INVERSE OF THE EXPONENTIAL IS THE LOGARITHM . . . 500 31.7 THE FUNCTION A X WITH A 0 AND X * R . . . . . . . . . 501 32 TRIGONOMETRIC FUNCTIONS 505 32.1 THE DEFINING DIFFERENTIAL EQUATION . . . . . . . . . . . 505 32.2 TRIGONOMETRIC IDENTITIES . . . . . . . . . . . . . . . . . 509 32.3 THE FUNCTIONS TAN( X ) AND COT( X ) AND THEIR DERIVATIVES 510 32.4 INVERSES OF TRIGONOMETRIC FUNCTIONS . . . . . . . . . . . 511 32.5 THE FUNCTIONS SINH( X ) AND COSH( X ) . . . . . . . . . . . 513 32.6 THE HANGING CHAIN . . . . . . . . . . . . . . . . . . . . 514 32.7 COMPARING U + K 2 U ( X ) = 0 AND U * K 2 U ( X ) = 0 . . . 515 IMAGE 3 CONTENTS VOLUME 2 XIX 33 THE FUNCTIONS EXP( Z ) , LOG( Z ) , SIN( Z ) AND COS( Z ) FOR Z * C 517 33.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 517 33.2 DEFINITION OF EXP( Z ) . . . . . . . . . . . . . . . . . . . . 517 33.3 DEFINITION OF SIN( Z ) AND COS( Z ) . . . . . . . . . . . . . . 518 33.4 DE MOIVRES FORMULA . . . . . . . . . . . . . . . . . . . . 518 33.5 DEFINITION OF LOG( Z ) . . . . . . . . . . . . . . . . . . . . 519 34 TECHNIQUES OF INTEGRATION 521 34.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 521 34.2 RATIONAL FUNCTIONS: THE SIMPLE CASES . . . . . . . . . 522 34.3 RATIONAL FUNCTIONS: PARTIAL FRACTIONS . . . . . . . . . . 523 34.4 PRODUCTS OF POLYNOMIAL AND TRIGONOMETRIC OR EXPONENTIAL FUNCTIONS . . . . . . . . . . . . . . . . 528 34.5 COMBINATIONS OF TRIGONOMETRIC AND ROOT FUNCTIONS . . 528 34.6 PRODUCTS OF EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS 529 34.7 PRODUCTS OF POLYNOMIALS AND LOGARITHM FUNCTIONS . . 529 35 SOLVING DIFFERENTIAL EQUATIONS USING THE EXPONENTIAL 531 35.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 531 35.2 GENERALIZATION TO U * ( X ) = * ( X ) U ( X ) + F ( X ) . . . . . . . 532 35.3 THE DIFFERENTIAL EQUATION U ** ( X ) * U ( X ) = 0 . . . . . . 536 35.4 THE DIFFERENTIAL EQUATION * N K =0 A K D K U ( X ) = 0 . . . . 537 35.5 THE DIFFERENTIAL EQUATION * N K =0 A K D K U ( X ) = F ( X ) . . . 538 35.6 EULER'S DIF FERENTIAL EQUATION . . . . . . . . . . . . . . . 539 36 IMPROPER INTEGRALS 541 36.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 541 36.2 INTEGRALS OVER UNBOUNDED INTERVALS . . . . . . . . . . . 541 36.3 INTEGRALS OF UNBOUNDED FUNCTIONS . . . . . . . . . . . . 543 37 SERIES 547 37.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 547 37.2 DEFINITION OF CONVERGENT INFINITE SERIES . . . . . . . . . 548 37.3 POSITIVE SERIES . . . . . . . . . . . . . . . . . . . . . . . 549 37.4 ABSOLUTELY CONVERGENT SERIES . . . . . . . . . . . . . . 552 37.5 ALTERNATING SERIES . . . . . . . . . . . . . . . . . . . . . 552 37.6 THE SERIES * * I =1 1 I THEORETICALLY DIVERGES! . . . . . . . 553 37.7 ABEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 37.8 GALOIS . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 38 SCALAR AUTONOMOUS INITIAL VALUE PROBLEMS 559 38.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 559 38.2 AN ANALYTICAL SOLUTION FORMULA . . . . . . . . . . . . . 560 38.3 CONSTRUCTION OF THE SOLUTION . . . . . . . . . . . . . . . 563 IMAGE 4 XX CONTENTS VOLUME 2 39 SEPARABLE SCALAR INITIAL VALUE PROBLEMS 567 39.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 567 39.2 AN ANALYTICAL SOLUTION FORMULA . . . . . . . . . . . . . 568 39.3 VOLTERRA-LOTKA'S PREDATOR-PREY MODEL . . . . . . . . . 570 39.4 A GENERALIZATION . . . . . . . . . . . . . . . . . . . . . 571 40 THE GENERAL INITIAL VALUE PROBLEM 575 40.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 575 40.2 DETERMINISM AND MATERIALISM . . . . . . . . . . . . . . 577 40.3 PREDICTABILITY AND COMPUTABILITY . . . . . . . . . . . . 577 40.4 CONSTRUCTION OF THE SOLUTION . . . . . . . . . . . . . . . 579 40.5 COMPUTATIONAL WORK . . . . . . . . . . . . . . . . . . . 580 40.6 EXTENSION TO SECOND ORDER INITIAL VALUE PROBLEMS . . 581 40.7 NUMERICAL METHODS . . . . . . . . . . . . . . . . . . . . 582 41 CALCULUS TOOL BAG I 585 41.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 585 41.2 RATIONAL NUMBERS . . . . . . . . . . . . . . . . . . . . 585 41.3 REAL NUMBERS. SEQUENCES AND LIMITS . . . . . . . . . . 586 41.4 POLYNOMIALS AND RATIONAL FUNCTIONS . . . . . . . . . . 586 41.5 LIPSCHITZ CONTINUITY . . . . . . . . . . . . . . . . . . . 587 41.6 DERIVATIVES . . . . . . . . . . . . . . . . . . . . . . . . 587 41.7 DIF FERENTIATION RULES . . . . . . . . . . . . . . . . . . . 587 41.8 SOLVING F ( X ) = 0 WITH F : R * R . . . . . . . . . . . . 588 41.9 INTEGRALS . . . . . . . . . . . . . . . . . . . . . . . . . . 589 41.10 THE LOGARITHM . . . . . . . . . . . . . . . . . . . . . . 590 41.11 THE EXPONENTIAL . . . . . . . . . . . . . . . . . . . . . 591 41.12 THE TRIGONOMETRIC FUNCTIONS . . . . . . . . . . . . . . 591 41.13 LIST OF PRIMITIVE FUNCTIONS . . . . . . . . . . . . . . . . 594 41.14 SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 41.15 THE DIFFERENTIAL EQUATION * U + * ( X ) U ( X ) = F ( X ) . . . . 595 41.16 SEPARABLE SCALAR INITIAL VALUE PROBLEMS . . . . . . . . 595 42 ANALYTIC GEOMETRY IN R N 597 42.1 INTRODUCTION AND SURVEY OF BASIC OBJECTIVES . . . . . . 597 42.2 BODY/SOUL AND ARTIFICIAL INTELLIGENCE . . . . . . . . . . 600 42.3 THE VECTOR SPACE STRUCTURE OF R N . . . . . . . . . . . . 600 42.4 THE SCALAR PRODUCT AND ORTHOGONALITY . . . . . . . . . 601 42.5 CAUCHY'S INEQUALITY . . . . . . . . . . . . . . . . . . . . 602 42.6 THE LINEAR COMBINATIONS OF A SET OF VECTORS . . . . . 603 42.7 THE STANDARD BASIS . . . . . . . . . . . . . . . . . . . . 604 42.8 LINEAR INDEPENDENCE . . . . . . . . . . . . . . . . . . . 605 42.9 REDUCING A SET OF VECTORS TO GET A BASIS . . . . . . . . 606 42.10 USING COLUMN ECHELON FORM TO OBTAIN A BASIS . . . . 607 42.11 USING COLUMN ECHELON FORM TO OBTAIN R ( A ) . . . . . 608 IMAGE 5 CONTENTS VOLUME 2 XXI 42.12 USING ROW ECHELON FORM TO OBTAIN N ( A ) . . . . . . . 610 42.13 GAUSSIAN ELIMINATION . . . . . . . . . . . . . . . . . . . 612 42.14 A BASIS FOR R N CONTAINS N VECTORS . . . . . . . . . . . 612 42.15 COORDINATES IN DIFFERENT BASES . . . . . . . . . . . . . . 614 42.16 LINEAR FUNCTIONS F : R N * R . . . . . . . . . . . . . . 615 42.17 LINEAR TRANSFORMATIONS F : R N * R M . . . . . . . . . . 615 42.18 MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . 616 42.19 MATRIX CALCULUS . . . . . . . . . . . . . . . . . . . . . . 617 42.20 THE TRANSPOSE OF A LINEAR TRANSFORMATION . . . . . . . 619 42.21 MATRIX NORMS . . . . . . . . . . . . . . . . . . . . . . . 620 42.22 THE LIPSCHITZ CONSTANT OF A LINEAR TRANSFORMATION . . 621 42.23 VOLUME IN R N : DETERMINANTS AND PERMUTATIONS . . . . 621 42.24 DEFINITION OF THE VOLUME V ( A 1 , . . . , A N ) . . . . . . . . . 623 42.25 THE VOLUME V ( A 1 , A 2 ) IN R 2 . . . . . . . . . . . . . . . 624 42.26 THE VOLUME V ( A 1 , A 2 , A 3 ) IN R 3 . . . . . . . . . . . . . 624 42.27 THE VOLUME V ( A 1 , A 2 , A 3 , A 4 ) IN R 4 . . . . . . . . . . . 625 42.28 THE VOLUME V ( A 1 , . . . , A N ) IN R N . . . . . . . . . . . . 625 42.29 THE DETERMINANT OF A TRIANGULAR MATRIX . . . . . . . . 625 42.30 USING THE COLUMN ECHELON FORM TO COMPUTE DET A . . 625 42.31 THE MAGIC FORMULA DET AB = DET A DET B . . . . . . . 626 42.32 TEST OF LINEAR INDEPENDENCE . . . . . . . . . . . . . . . 626 42.33 CRAMER'S SOLUTION FOR NON-SINGULAR SYSTEMS . . . . . . 628 42.34 THE INVERSE MATRIX . . . . . . . . . . . . . . . . . . . . 629 42.35 PROJECTION ONTO A SUBSPACE . . . . . . . . . . . . . . . 630 42.36 AN EQUIVALENT CHARACTERIZATION OF THE PROJECTION . . . 631 42.37 ORTHOGONAL DECOMPOSITION: PYTHAGORAS THEOREM . . . 632 42.38 PROPERTIES OF PROJECTIONS . . . . . . . . . . . . . . . . . 633 42.39 ORTHOGONALIZATION: THE GRAM-SCHMIDT PROCEDURE . . . 633 42.40 ORTHOGONAL MATRICES . . . . . . . . . . . . . . . . . . . 634 42.41 INVARIANCE OF THE SCALAR PRODUCT UNDER ORTHOGONAL TRANSFORMATIONS . . . . . . . . . . . 634 42.42 THE QR-DECOMPOSITION . . . . . . . . . . . . . . . . . 635 42.43 THE FUNDAMENTAL THEOREM OF LINEAR ALGEBRA . . . . . 635 42.44 CHANGE OF BASIS: COORDINATES AND MATRICES . . . . . . 637 42.45 LEAST SQUARES METHODS . . . . . . . . . . . . . . . . . . 638 43 THE SPECTRAL THEOREM 641 43.1 EIGENVALUES AND EIGENVECTORS . . . . . . . . . . . . . . 641 43.2 BASIS OF EIGENVECTORS . . . . . . . . . . . . . . . . . . . 643 43.3 AN EASY SPECTRAL THEOREM FOR SYMMETRIC MATRICES . . 644 43.4 APPLYING THE SPECTRAL THEOREM TO AN IVP . . . . . . . 645 43.5 THE GENERAL SPECTRAL THEOREM FOR SYMMETRIC MATRICES . . . . . . . . . . . . . . . . . 646 43.6 THE NORM OF A SYMMETRIC MATRIX . . . . . . . . . . . . 648 43.7 EXTENSION TO NON-SYMMETRIC REAL MATRICES . . . . . . 649 IMAGE 6 XXII CONTENTS VOLUME 2 44 SOLVING LINEAR ALGEBRAIC SYSTEMS 651 44.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 651 44.2 DIRECT METHODS . . . . . . . . . . . . . . . . . . . . . . 651 44.3 DIRECT METHODS FOR SPECIAL SYSTEMS . . . . . . . . . . . 658 44.4 ITERATIVE METHODS . . . . . . . . . . . . . . . . . . . . . 661 44.5 ESTIMATING THE ERROR OF THE SOLUTION . . . . . . . . . . 671 44.6 THE CONJUGATE GRADIENT METHOD . . . . . . . . . . . . 674 44.7 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . 676 45 LINEAR ALGEBRA TOOL BAG 685 45.1 LINEAR ALGEBRA IN R 2 . . . . . . . . . . . . . . . . . . . 685 45.2 LINEAR ALGEBRA IN R 3 . . . . . . . . . . . . . . . . . . . 686 45.3 LINEAR ALGEBRA IN R N . . . . . . . . . . . . . . . . . . . 686 45.4 LINEAR TRANSFORMATIONS AND MATRICES . . . . . . . . . . 687 45.5 THE DETERMINANT AND VOLUME . . . . . . . . . . . . . . 688 45.6 CRAMER'S FORMULA . . . . . . . . . . . . . . . . . . . . . 688 45.7 INVERSE . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 45.8 PROJECTIONS . . . . . . . . . . . . . . . . . . . . . . . . 689 45.9 THE FUNDAMENTAL THEOREM OF LINEAR ALGEBRA . . . . . 689 45.10 THE QR-DECOMPOSITION . . . . . . . . . . . . . . . . . 689 45.11 CHANGE OF BASIS . . . . . . . . . . . . . . . . . . . . . . 690 45.12 THE LEAST SQUARES METHOD . . . . . . . . . . . . . . . 690 45.13 EIGENVALUES AND EIGENVECTORS . . . . . . . . . . . . . . 690 45.14 THE SPECTRAL THEOREM . . . . . . . . . . . . . . . . . . 690 45.15 THE CONJUGATE GRADIENT METHOD FOR AX = B . . . . . . 690 46 THE MATRIX EXPONENTIAL EXP( XA ) 691 46.1 COMPUTATION OF EXP( XA ) WHEN A IS DIAGONALIZABLE . . 692 46.2 PROPERTIES OF EXP( AX ) . . . . . . . . . . . . . . . . . . 694 46.3 DUHAMEL'S FORMULA . . . . . . . . . . . . . . . . . . . . 694 47 LAGRANGE AND THE PRINCIPLE OF LEAST ACTION* 697 47.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 697 47.2 A MASS-SPRING SYSTEM . . . . . . . . . . . . . . . . . . 699 47.3 A PENDULUM WITH FIXED SUPPORT . . . . . . . . . . . . 700 47.4 A PENDULUM WITH MOVING SUPPORT . . . . . . . . . . . 701 47.5 THE PRINCIPLE OF LEAST ACTION . . . . . . . . . . . . . . 701 47.6 CONSERVATION OF THE TOTAL ENERGY . . . . . . . . . . . . 703 47.7 THE DOUBLE PENDULUM . . . . . . . . . . . . . . . . . . 703 47.8 THE TWO-BODY PROBLEM . . . . . . . . . . . . . . . . . 704 47.9 STABILITY OF THE MOTION OF A PENDULUM . . . . . . . . . 705 IMAGE 7 CONTENTS VOLUME 2 XXIII 48 N -BODY SYSTEMS* 709 48.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 709 48.2 MASSES AND SPRINGS . . . . . . . . . . . . . . . . . . . . 710 48.3 THE N -BODY PROBLEM . . . . . . . . . . . . . . . . . . 712 48.4 MASSES, SPRINGS AND DASHPOTS: SMALL DISPLACEMENTS . . . . . . . . . . . . . . . . . . . 713 48.5 ADDING DASHPOTS . . . . . . . . . . . . . . . . . . . . . 714 48.6 A COW FALLING DOWN STAIRS . . . . . . . . . . . . . . . 715 48.7 THE LINEAR OSCILLATOR . . . . . . . . . . . . . . . . . . . 716 48.8 THE DAMPED LINEAR OSCILLATOR . . . . . . . . . . . . . 717 48.9 EXTENSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 719 49 THE CRASH MODEL* 721 49.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 721 49.2 THE SIMPLIF IED GROWTH MODEL . . . . . . . . . . . . . . 722 49.3 THE SIMPLIF IED DECAY MODEL . . . . . . . . . . . . . . . 724 49.4 THE FULL MODEL . . . . . . . . . . . . . . . . . . . . . . 725 50 ELECTRICAL CIRCUITS* 729 50.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 729 50.2 INDUCTORS, RESISTORS AND CAPACITORS . . . . . . . . . . . 730 50.3 BUILDING CIRCUITS: KIRCHHOFF'S LAWS . . . . . . . . . . . 731 50.4 MUTUAL INDUCTION . . . . . . . . . . . . . . . . . . . . . 732 51 STRING THEORY* 735 51.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 735 51.2 A LINEAR SYSTEM . . . . . . . . . . . . . . . . . . . . . 736 51.3 A SOFT SYSTEM . . . . . . . . . . . . . . . . . . . . . . . 737 51.4 A STIF F SYSTEM . . . . . . . . . . . . . . . . . . . . . . . 737 51.5 PHASE PLANE ANALYSIS . . . . . . . . . . . . . . . . . . . 738 52 PIECEWISE LINEAR APPROXIMATION 741 52.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 741 52.2 LINEAR INTERPOLATION ON [0 , 1] . . . . . . . . . . . . . . . 742 52.3 THE SPACE OF PIECEWISE LINEAR CONTINUOUS FUNCTIONS . 747 52.4 THE L 2 PROJECTION INTO V H . . . . . . . . . . . . . . . . 749 53 FEM FOR TWO-POINT BOUNDARY VALUE PROBLEMS 755 53.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 755 53.2 INITIAL BOUNDARY-VALUE PROBLEMS . . . . . . . . . . . . 758 53.3 STATIONARY BOUNDARY VALUE PROBLEMS . . . . . . . . . . 759 53.4 THE FINITE ELEMENT METHOD . . . . . . . . . . . . . . . 759 IMAGE 8 XXIV CONTENTS VOLUME 2 53.5 THE DISCRETE SYSTEM OF EQUATIONS . . . . . . . . . . . . 762 53.6 HANDLING DIFFERENT BOUNDARY CONDITIONS . . . . . . . . 765 53.7 ERROR ESTIMATES AND ADAPTIVE ERROR CONTROL . . . . . . 768 53.8 DISCRETIZATION OF TIME-DEPENDENT REACTION-DIFFUSION-CONVECTION PROBLEMS . . . . . . . . 773 53.9 NON-LINEAR REACTION-DIFFUSION-CONVECTION PROBLEMS . 773 REFERENCES 777 INDEX 779
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author	Eriksson, Kenneth Estep, Donald Johnson, Claes
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id	DE-604.BV017518571
illustrated	Illustrated
indexdate	2024-07-20T07:39:15Z
institution	BVB
isbn	354000890X
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-010550931
oclc_num	1070736194
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physical	XLIII, 425 S. Ill., graph. Darst.
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spelling	Eriksson, Kenneth Verfasser aut Applied mathematics: body and soul 1 Derivatives and geometry in R 3 K. Eriksson ; D. Estep ; C. Johnson Berlin [u.a.] Springer 2004 XLIII, 425 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Estep, Donald Verfasser aut Johnson, Claes Verfasser aut (DE-604)BV017518560 1 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010550931&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Eriksson, Kenneth Estep, Donald Johnson, Claes Applied mathematics: body and soul
title	Applied mathematics: body and soul
title_auth	Applied mathematics: body and soul
title_exact_search	Applied mathematics: body and soul
title_full	Applied mathematics: body and soul 1 Derivatives and geometry in R 3 K. Eriksson ; D. Estep ; C. Johnson
title_fullStr	Applied mathematics: body and soul 1 Derivatives and geometry in R 3 K. Eriksson ; D. Estep ; C. Johnson
title_full_unstemmed	Applied mathematics: body and soul 1 Derivatives and geometry in R 3 K. Eriksson ; D. Estep ; C. Johnson
title_short	Applied mathematics: body and soul
title_sort	applied mathematics body and soul derivatives and geometry in r 3
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010550931&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV017518560
work_keys_str_mv	AT erikssonkenneth appliedmathematicsbodyandsoul1 AT estepdonald appliedmathematicsbodyandsoul1 AT johnsonclaes appliedmathematicsbodyandsoul1

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