REFERENCES. 1. Strang, G., Linear Algebra and its Applications, 2nd ed., Academic Press Inc., New York, 1980.

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1 REFERENCES 1. Strang, G., Linear Algebra and its Applications, 2nd ed., Academic Press Inc., New York, Kolman, B., Elementary Linear Algebra, The Macmillan Company, New York, Davis, H. F., Introduction to Vector Analysis, 4th ed., Allyn & Bacon, Boston, Pipes, L.A., Matrix Methods for Engineering, Prentice Hall, Englewood Cliffs, New Jersey, Langhaar, H. L., Energy Methods in Applied Mechanics, Wiley, New York, Przemieniecki, J. S., Theory of Matrix Structural Analysis, McGraw-Hill, New York, Kreyszig, E., Advanced Engineering Mathematics, 5th ed., Wiley, New York, Taylor, A. E. and Mann, W. R., Advanced Calculus, 3rd ed., Ginn and Company, Boston, Hildebrand, F. B., Advanced Calculus for Applications, 2nd ed., Prentice Hall, Englewood Cliffs, New Jersey, Hochstadt, H., The Functions of Mathematical Physics, Wiley-Interscience, New York, Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. I, Wiley Interscience, New York, Hildebrand, F. B., Methods of Applied Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey, Lanczos, C., Linear Differential Operators, Van Nostrand, New York, Royden, H. L., Real Analysis, Macmillan, New York,

2 57 4 References 15. Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, New York, Synge, J. L. and Schild, A., Tensor Calculus, University of Toronto Press, Toronto, Tychonov, A. N. and Samarski, A. A., Partial Differential Equations of Mathematical Physics, Vol. I, Holden-Day, San Francisco, Mikhlin, S. G., Mathematical Physics, An Advanced Course, North-Holland Publishing Co., Amsterdam, Garabedian, P.R., Partial Differential Equations, Wiley, New York, Coddington, E. A. and Levington, N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, Mikhlin, S. G., Variational Methods in Mathematical Physics, Pergamon Press, Oxford, U.K., Aubin, J.P., Applied Functional Analysis, Wiley, New York, Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill, New York, Gould, P. L., Analysis of Shells and Plates, Springer-Verlag, New York, Shames, I. H. and Dym, C. L., Energy and Finite Element Methods in Structural Mechanics, McGraw-Hill, New York, Timoshenko, S. and Goodier, J. N., Theory of Elasticity, McGraw-Hill, New York, Strang, G. and Fix, G. J., An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, New Jersey, Zienkiewicz, 0. C., The Finite Element Method, 3rd ed., McGraw-Hill, New York, 1977.

3 INDEX A A-scalar product 104 Abel's Test 137 absolutely convergent 131 addition of functions 169 addition of power series 146 addition of vectors 31, 186 aerodynamics 271 algebraic representation of a geometric vector36 algebraic vector 36 alternating series 130 analogies 278 arbitrary function 282 associative law of addition 10 associative law of multiplication 10 augmented matrix 53 B B-orthogonal 110 B-orthogonal modal matrix 111 B-orthonormal eigenvectors 111 back substitution 6, 19 bars 90 basis for L 2 ( a, b) 196 basis of a vector space 43 beam bending equation 92 beam element 92 beam element stiffness matrix 93 beam equation 556 beam on an elastic foundation 503 beams92 bending matrix 555 bending moments 562 bending of a plate 537, 564 bending of a thin elastic plate 520 bending of beams 92 Bernoulli's equation 273 Bernoulli's Law 273 Bessel function of the first kind of order-v 166 Bessel function of the first kind of orderv 165 Bessel's equation 164 Bessel's inequality 195, 206 biharmonic operator 523 bilinear rectangular elements 563 Bocher's formulas 79 boundary conditions 104, 253, 259, 264,266,290 boundary-value problem 173, 261, 264, 274,290 bounded below operator 472 bounded functional 383 bounded linear operator 390 Brachistochrone Problem 453,457 buckling modes 77 c calculus of variations 452 candidate solutions 173 canonical form 298,299, 313 canonical form of a hyperbolic 575

4 576 equation 302 canonical form of a parabolic equation 303 canonical form of an elliptic equation 305 canonical form of the quadratic form 94 Cartesian components of a vector 33 Cartesian reference frame 32 Cauchy data 286, 291 Cauchy problem 288, 361 Cauchy product 146 Cauchy Root Test 128 Cauchy sequence 190 Cauchy-Kowalewski Theorem 289 characteristic curves 292, 300, 306, 375 characteristic equation 76, 108 characteristic initial-value problem 352 characteristic polynomial76 characteristic surface 315 characteristic values 75 characteristic variables 308, 361 circulation 272 clamped beam 534 clamped plate 541 closure under addition 40 closure under multiplication by a scalar40 coefficient of convectivity 267 coefficient of porosity 266 coefficient of viscosity 262 cofactor24 column 8 column index 8 column matrices 9 column rank 49 column space 49 column vector 39 columns of a matrix 49 commutative law of addition 10 Comparison Test 127 complete in energy 436 complete set of functions 196 complete vector space 191 completeness of eigenfunctions 415 complex conjugate 80, 303 components of velocity 268 compression 27 6 conditionally convergent 131 conduction 2 conservation of mass 270 conservative systems 250 consistent 53 constant entropy 270 continuity 145 continuity equation 247, 270 continuous dependence on data 296 continuous functional 383 continuous linear operator 391 continuum168,230,243 convection 2 convection heat transfer coefficient 3 converge 125, 189 converge in energy 434 converge in the mean 189, 206 converge pointwise 133 convergent series 126 coordinate functions 478,481, 487 Cramer's rule 27 cross-sectional moment of inertia 92 cubic element 553 cylindrical coordinates 338 cylindrically symmetric solution 340 cycloids 458 D D'Alembert's formula 323, 362 determinant of a matrix 20 density 263 diagonal matrix 11 difference quotient 233 differentiable functional 449 differential arc length 234 differential equation of motion 253, 258 differential form 247 differential surface area 234 differentiation and integration 146 diffusion coefficient 265 diffusion equation 266

5 diffusion of gas 265 dimension of a vector space 45 dimension of matrix 8 direct methods 478 directional derivatives 238 Dirichlet formula 213 Dirichlet problem 338, 511, 514 discriminant62,298 distance between two points 63 distributive laws 10 divergence form 270 divergence of a vector valued function 237 Divergence Theorem 237 divergent series 126 domain 169, 170, 383, 390 domain of dependence 363 domain of influence 363 double Fourier series 227 double node 553 E eigenfunction expansion technique 379 eigenfunction expansions 407 eigenfunctions 260, 353, 400, 403 eigenvalue equation 260 eigenvalue problem 75, 260 eigenvalues 75, 260, 353, 400, 402 eigenvector expansion 83 eigenvectors 75 elastic spring 89 electrostatics 348 element of matrix 8 element mass matrix 550 element stiffness matrix 550 elementary matrices 18, 50 elementary operations 4 elementary row operation 50 elements 549 ellipse 299 elliptic differential equations 337 elliptic equation 313 elliptic partial differential equation embedding theorem 445 energy 89 energy functional 438 energy methods 439 energy norm 106, 434 energy of operator A 492 energy of operator B 492 energy orthogonal 436 energy scalar product 106, 402, 433 equality of vector functions 171 equality, addition, and multiplication of matrices 8 equation of acoustics 277 equation of state 271 equations of fluid dynamics 270 equations of fluid motion 270 Equations of Mathematical Physics 278 equilibrium equations 2 equilibrium of a membrane 261 error function 494 Euler- Lagrange equations 451 even extension 208 even function 202 even permutation 21 expansion by cofactors 24 exact differential form 247 exterior problem 351 external force 269 F family of solutions 53 Fick's law 265 finite dimensional vector space 45 finite element equation 552 finite element subspace 548 finite elements 549 first boundary-value problem 338 first order partial differential equations 280 first variation 449 flow field 268 fluid density 269 fluid dynamics 268

6 578 forced vibration 410 formal operations 151 fonnally differentiated 138 forward elimination 6, 19 forward elimination algorithm 17 Fourier coefficients 206 Fourierseries 198,205 Fourier Series Differentiation Theorem220 Fourier Series Integration Theorem 222 Fourier's law 3, 262 free vibration of a beam 500 Friedrichs' inequality 396 Frobenius method 155 Frobenius norm 397 function space 169, 171 functional 383 functions of finite energy 441 fundamental solution of Laplace's equation in three dimensions 340 fundamental solution of Laplace's equation in two dimensions 340 G Galerkin's method 86, 496 Gamma function 165 gas 170, 265, 276 gas flow 265 Gauss~ Jordan reduction 53 Gaussian elimination 6, 17 general integral 339 general solution 151, 282, 294 general solution of the wave equation 317 generalized coordinates 387 generalized eigenvalue problem 108, 492 generalized force 387 generalized functions 441 generalized solution 448 geometric sequence 126 geometric series 127 geometric vector 30 Gibbs phenomenon 218 global stiffness matrix 550, 551 gradient operator 237 Gram determinant 68, 483 Gram matrix 68 Gram~Schmidtorthononnalization process 70, 182 Gram-Schmidtrecurrrence relation 70, 182 Gramian68 greatest lower bound 477 Green's Formulas 242 Green's function 347, 419, 420, 425 Green's Identities 239 Green's Theorem 235 H harmonic 343 harmonic excitation 117 harmonic function 340 harmonic oscillation 259 heat equation 263, 303, 327, 351, 569 heat flow 262 heat transfer rate 3 Hermite bicubic coordinate functions 563 Hermite bicubic space 563 Hermite cubics 554 Hilbert space 193 homogeneous equation 7, 298 homogeneous first order partial differential equation 280 homogeneous ordinary differential equation 149 homogeneous wave equation 253,316 Householder transformation 73 hydrodynamics271 hydrostatics 277 hyperbola 299 hyperbolic differential equation 361 hyperbolic equation 314, 571 hyperbolic partial differential equation 301

7 579 l ideal fluid 269 identity matrix 10 Implicit Function Theorem 240 improper integral 342 impulse 252 incompressibility 271 incompressible flow 273 indicia! equation 156 infinite dimensional 183 infinite series 126 influence function 420 initial conditions 253, 259, 264, 266, 286,291 initial curve 291 initial temperature distribution 264 initial-boundary-value problem 253, 259,264,365 initial-value problem 288, 361 integral equation of motion 252, 258 integral form of the remainder 141 integral operator 390 Integral Test 129 integrating factor 249 integration by parts 239 interval of convergence 145 inverse 14 irrotational flow 272 isentropic flow 271 isentropic flow field 272 isotropic body 262 iterated integral 235 Lagrange multipliers 487 Lagrangian form of the remainder 142 Laplace operator 238, 351 Laplace's equation 273, 278, 306, 313, 333, 337, 344 law of conservation of mass 243, 246,247 least square 193 least square approximation 194 least square approximate solution 495 least square method 494 left inverse 29 left sided derivative 201 Legendre equation 163 Legendre polynomials 164 Leibniz Test 130 Leibniz's Rule of Differentiation 233 length of a vector 60 line integrals 234 linear combination 42 linear coordinate function 558 linear equations 1, 7, 48 linear finite elements 549 linear functional 385 linear operator 255, 259, 390 linearly dependent 41, 180 linearly independent 41, 180 liquids 271 local basis 549 lower triangular 19 LU factorization 19 J Jacobian matrix 240 K kemel420 Kronecker delta 11 M Maclaurin expansion 142 Maclaurin series 142 magnitude of a vector 30 mass density of gas 265 mass matrix 554 material derivative 245, 246 material particles 243 mathematical physics 253

8 580 matrix 8 matrix equation 9 matrix of the quadratic form 88 Maximum Principle 343, 356 maximum principle for parabolic equations 355 mean value 232 Mean Value Theorem 232 member loads 1 membrane 255 metric 64 method of even and odd extensions 368 metric space 64 minimizing sequence 477 Minimum Functional Theorem 119, 438 mixed problem 338, 511, 514 modal matrix 94 modal transformation 312 momentum 252 momentum of fluid 269 monotonic decreasing function 129 multiple integral theorems 230 multiple integrals 230 multiple root 82 multiplication by a scalar 187 multiplication of a function by a scalar 169 multiplication of a matrix by a scalar 12 multiplication of a vector by a scalar 31 multiplication of a vector function by a scalar 171 multiplication of power series 146 multiplicity 82 N n-dimensional real space 36, 39 n-vector 36 natural boundary conditions 463 natural frequencies 260, 409 necessary condition for convergence 126 negative of a matrix 12 negative of a vector 31 negative of a vector function 171 Neumann problem 338, 512, 514 nodal coordinate 559 nodal values 549 nodes 549 non-homogeneous boundary conditions 466 nonsingular matrix 14 nontrivial solution 7, 56 norm 188 norm of a vector 61 norm of a vector function 176 norm of a functional383 norm of a linear operator 391 normal derivative 238 normal hyperbolic 314 normal velocity of the boundary 271 normalized 64 normed vector space odd extension 208 odd function 202 odd permutation 21 one-dimensional isentropic flow 275 one-sided limits 200 open set 170 operator 390 operator equation 255,261 operator norm 397 ordinary point 149 ordinary solution 448 orthogonal 34, 64 orthogonal matrix 66 orthogonal modal matrix 94 orthogonal reference frame 31 orthogonal with respect to B 110 orthonormal 64 orthonormal functions 181 oscillating series 126 p parabola 299 parabolic differential equation 351 parabolic equation 314

9 parabolic partial differential equation 302 parallelogram rule 31 Parseval relation 197, 436 partial differential equation 253 partial sum 125 period204 periodic function 204 permutation 21 permutation symbol 20 physical space 170 piecewise bilinear coordinate functions 563 piecewise bilinear functions 563 piecewise continuous 200 piecewise cubic coordinate function 561 piecewise cubic polynomial 561 piecewise quadratic coordinate function 560 piecewise quintic coordinate function 562 piecewise smooth 201 plane flow 268 plucked string problem 317 point 63 point force 420 pointwise convergence of Fourier series 212 Pointwise Convergence Theorem 213 Poisson's equation 337, 511 porous material 265 positive bounded below linear operator 393 positive defmite 100 positive definite linear operator 393 positive semidefmite 102 positive semidefmite linear operator 393 potential340, 348 potential flow 272 Potential Theory 340 power of a matrix 79 power series 143, 145 power series solution 285 pressure 269 principal boundary conditions 463 principal minor 101 principle of minimum total potential energy 121, 439, 511 principle of superposition 298 propagation of a wave 319 propagation of sound 277 proper.subspace 46 pyramid functions 559 Q quadratic coordinate function 560 quadratic elements 552 quadratic form 88 R radius of convergence 145 range 390 rank of a matrix 51 rate of heat flow 262 rate of mass flow 265 ratio of specific heats 271 Ratio Test 128 Ratio Test for Absolute Convergence 131 Rayleigh quotient 122, 472 real line 170 real-valued function 168 reciprocity 423 rectangular elements 563 recurrence formula 151, 156 reflected 321 regular singular point 154 relative minimum 450 repeated Latin index 7 residual 86, 494 Riemann Zeta function 130 Riemann's lemma 220 Riesz Representation Theorem 387 right inverse 28 right sided derivative 201 right-hand x-y-z orthogonal reference 581

10 582 frames 32 Ritz approximate solution 483 Ritz equations 482 Ritz method 121, 503 Ritz method for the eigenvalue problem 124 Ritz method for the generalized eigenvalue problem 124 Rodrigues' formula 164 row8 row index 8 rowrank49 row space of a matrix 49 row vector 39,49 rows of a matrix 49 s scalar product 33, 60, 174, 187 scalar product with respect to the operator B 401 Schwartz Inequality 61, 177 second boundary-value problem 338 second order differential operator 390 second order linear differential equations 114 second order partial differential equation 284, 297 separated equation 328 separation of variables 328, 334, 344, 352, 376, 407' 415 sequence 125 sequence of partial sums 125 simply connected region 232 simply supported beam 531 simply supported plate 538 singular matrix 14 singular point 154 skew symmetric 12 solid-fluid boundary 271 solution curves 281 solution surface 281 solve a linear equation 49 span a vector space 42 special functions 162, 405 specific heat 263 spectrum 398 spherical coordinates 338 spherically symmetric solution 339 spring constant 89 square integrable functions 179 stability 296, 394 stability of a column 506 stable solution 344 stable with respect to the initial conditions 296 static displacement 259 steady flow 274 steady state temperature distribution 264 steady state vibration 259 steady, isentropic, irrotational flow 275 steady-state conditions 2 steady-state heat transfer 2 stiffness matrix 90, 555 strain energy 90 stream function 274 stream lines 274 Sturm-Liouville equation 547 Sturm-Liouville operator 425 Sturm-Liouville problem 404 submatrix 56 subset 46 subspace 46 subtraction of matrices 12 subtraction of vector functions 171 sum of the series 126 sum of two vector functions 171 summation notation 7 surface integrals 234 symmetric 12 symmetric function 348 symmetric linear operator 392 symmetrized matrix 89 T Taylor expansion 141, 144 Taylor series 142

11 583 Taylor's Theorem 141 temperature 262 Theorem of the Alternative 84, 472 thermal conductivity 3, 262 thermodynamics270 third boundary-value problem 338 torsion of a rectangular rod 515 total energy 324 total potential energy 121 trace 78 transformation 240 transpose 11 transversality condition 452 traveling waves 373 Triangle Inequality 63, 178 Tricomi Equation 308 triple node 561 trivial solution 7, 56 trusses 90 u ultrahyperbolic 314 uniform and absolute convergence 145 Uniform and Absolute Convergence Theorem216 uniformly convergent 134 uniqueness 146, 343 unit coordinate vectors 33 unit normal 236 unit tangent 236 unit vector 31 upper triangular 17 vector sum 31 vector variable 170 velocity potential 273 vibrate 251 vibrating membrane 258,411 vibrating string 169, 253, 376 vibration of a cantilever beam 407 vibration of a thin elastic plate 523 vibration of beams and plates 545 virtual displacement 119, 387 virtual work 119, 387 w wave equation 253, 258, 302 wave front 320 wave propagation 373 wave speed 320 waves 375 Weierstrass M-test 135 well-posed problem 297 y Young's Modulus 90 z zero function 169 zero matrix 10 zero vector 31 zero vector function 170 v variational methods 439 vector 30, 39 vector analysis 30 vector function 170 vector product 34, 38 vector space 40 vector space postulates 40

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