VECTOR CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL FORMS A Unified Approach STH EDITION JOHN H HUBBARD BARBARA BURKE HUBBARD sgn( a-) Span (J" sup Supp(!) T TxX tr v·w vxw lvl (v) J_ Varf [x]k signature of a permutation (Theorem and Definition 4.8.11) span (Definition 2.4.3) standard deviation (Definition 3.8.6) Also denotes permutation sum (Section 0.1) supremum; least upper bound (Definitions 0.5.1 , 1.6.5) support of a function f (Definition 4.1.2) (tau) torsion (Definition 3.9.14) tangent space to manifold (Definition 3.2.1) trace of a matrix (Definition 1.4.13) dot product of two vectors, (Definition 1.4.1) cross product of two vectors (Definition 1.4.17) length of vector v (Definition 1.4.2) orthogonal complement to subspace spanned by v (proof of Theorem 3.7.15) variance (Definitions 3.8.6) k-truncation (Definition Al.2) Notation particular to this book [OJ matrix with all entries (equation 7.48) equal in the sense of Lebesgue (Definition 4.11 6) L ~_ (e.g , v i) A [Alb] Br(x) /Jn D N(Ilr) [Df(a)] IIJ) D1f Wxl 8'Af X r(J) [h]R 370 llfll~S,P)' 370 abstract vector space, 207-218 basis, 212 linear transformation between, 209 active variable, 170, 267 binormal (in Frenet frame), 392 adapted coordinates, 379, 384, 392 plane curve, 379 space curve, 390 surface, 384, 751-752 asymptotic development, 326 adjacency matrix, 51, 228 augmented Hessian matrix, 360, 360, 747 adjoint matrix, 645 augmented matrix, 167, 176 affine function, 64 axiom of choice, 792 axis of symmetry, 491 affine subspace, 64, 306, 616 algebraic number, 22, 23 binomial formula, 327, 744 Axler, Sheldon, 221 algebraic topology and orientation, 619 Biot, Jean-Baptiste, 677 Biot-Savart law, 677 Borel, Emile, 714 Born, Max, 43, 119, 219 boundary, 86, 86 boundary orientation, 619-624 nonsmooth, 611, 614 of manifold, 611 of oriented k-parallelogram, 623 of piece-with-boundary, 620 of subset of manifold, 612, 612 orientation of smooth, 620 smooth, 611, 612, 614 volume of, 438, 492 almost all, almost everywhere, 431 sn alphabet, Greek, alternating (antisymmetric), 461 B, see magnetic field back substitution, 174, 174, 233 Ampere, Andre Marie, 673, 677 ball, 84 ampere (amp), 670 Banach, Stefan, 785 Ampere's law, 671 Banach space, 236 Bowditch, Nathaniel, Banach-Tarski paradox, 793 Brahe, Tycho, 99 anchored k-parallelograms, 527 (n-dimensional unit ball), 547 807 bounded bounded above, 108 bounded set, 105 bounded support, 404 808 branch of function, 12 Brouwer, Luitzen, 64, 640 Brouwer fixed point theorem, 21 Buffon, Georges Leclerc Comte de, 419 Buffon's needle, 419-420 bump function, 652, 652 Bunyakovsky, Viktor, 71 IC, 6, see also complex number C function, 149 GP function 210 GP manifold, 285 C 00 , 231 C(O, 1), C[O, 1], 208, 210 C2 (space of C functions), 210 calculus (history), 83, 96, 112, 527 Cantor, Georg, 6, 10, 22, 24, 25, 104, 787 Cardano, Girolamo, 26, 29, 112 Cardano's formula, 113, 709-711 cardinality, 22, 22, 23, 25 Cartan, Elie, 640, 644 Cartesian plane, 40 catenoid, 396 Cauchy, Augustin Louis, 87, 234, 236 Cauchy-Bunyakovsky-Schwarz, 71 Cauchy-Riemann equation, 332 Cayley, Arthur, 33, 46, 100, 249, 477 Cayley-Hamilton theorem, 46, 476, 477 center of gravity, 417, 418 centered random variable, 370, 371 central limit theorem, 421, 421, 756, 757 Gaussian integral, 511 Monte Carlo algorithm, 455 proof, 757-759 cgs units, 608, 670 chain rule, 141, 141, 275, 715-716 for Taylor polynomials, 328 map defined on manifold, 312 change of basis change of basis formula, 216 change of basis matrix, 214, 214, 215 direct basis, indirect basis, 583 change of parametrization, 533-535 change of variables cylindrical, 491, 491 finding, 495 how domains correspond, 487 justification, 767 linear, 485 Index: Page numbers in italics indicate theorems, propositions, etc change of variables, cont nonlinear, 486-496 polar, 488, 488, 489, 492 spherical, 489, 490, 490, 492 substitution method, 487 symmetry, 487 change of variables formula, 492, 492-496 Lebesgue integral, 510, 790-791 rigorous proof, 765-772 characteristic function, 403 see also indicator function characteristic polynomial, 74, hard to compute, 475 roots eigenvalues, 475 composition, cont of onto functions is onto, 16 pullback of, 643 computer, 199, 236 computer graphics, 314 definition of function, 11 round-off errors, 164 row reduction, 198 concrete to abstract function, 211-214 cone operator, 691, 693 475 Charles V (on language), 213 Cicero (tomb of Archimedes), 549 closed path, 688 closed set, 85, 84-86 closed under limits, 90 cone over k-parallelogram, 691, 692 configuration space, 290, 291 conjugate matrix, 220 connectedness connected manifold, 587, 587, 587 connected set, 290 definition in topology different, 290 unconnected examples, 287, 587, 591 conservation of charge, 671 closure, 86, 86, 91 conservative force fields, 636 codomain, 9, 12, 56 conservative vector fields, 636, 689 Cohen, Paul, 24, 792 constant form, 575 column operation, 161, 463 equivalent to row operation, 468 constrained critical point, 350, 350, 349-359 checking boundary, 356-358 classifying, 359 finding using parametrization, 352 finding with Lagrange multipliers, column space, 197 compact set, 105, 105, 108, 109 convergent subsequence, 105 decreasing intersection, 714 existence of minima and maxima, 109 Heine-Borel theorem, 714 in general topology, 714 352 signature of, 360, 361, 747 constraint function, 352 complement, constructivists, 108 completing squares, 336, 745-746 contented set (pavable set), 411 complex conjugate, 27 complex exponential, see exponential continuity, 5, 6, 84, 96, 98 criterion for, 96 see also continuous function complex number addition, 26 absolute value, 27 argument, 28 imaginary part, 26 modulus, 27 multiplication, , 26, 28 real part, 26 roots, 28 complex vector space, 208 basis of, 589 composition, 15, 15, 16 associative, 15 matrix multiplication, 61 little o and big 0, 737 of 1-1 functions is 1-1, 16 of continuous functions, 97 continuous function, 96 combining, 97 composition, 97 differentiable implies continuous, 127 on compact set, 11 O when graph volume 0, 427 when integrable, 427, 428 continuously differentiable function 147 149 150 I l continuum hypothesis, 24, 792 convergence, 20, 88, 88, 90, 99 absolute, 99, 99 almost everywhere, 501, 782 elegance not required, 89 Riemann integral, 500 series of matrices, 100 Index: Page numbers in bold indicate definitions, possibly informal convergence, cont uniform, 499 convergent sequence, 19, 88 convergent series, 19 convergent subsequence, 91, 105, 105 convex domain, 690, 691 coordinate, 34 coordinates (independence of) piece-with-boundary, 617 manifold, 297 signature of quadratic form, 338 curvature, cont Theorema Egregium, 550-554 total, 542, 548 see also Gaussian curvature see also mean curvature curve, 104, 284, 287, 542 parametrized by arc length, 383 see also plane curve, space curve cycle notation (for permutations), 470 cycloid, 396 cylindrical coordinates, 491, 491, 491 corner point, 613 correlation, 371, 372, 373-374, 420 ][) (finite decimals), 705 cosine law, 69 d'Alembert, Jean, 1, 29, 112-114, 201, 683 Cotes, Roger, 99 de la Vallee-Poussin, Charles, 326 Coulomb, Charles Augustin, 672 de Moivre, Abraham, 28, 756 coulomb, 608, 670 de Moivre's formula, 28 Coulomb's law, 608, 673 de Rham cohomology, 690 countable additivity, 369, 787, 791 decoupling, 223 countably infinite set, 22, 787 Dedekind, Richard, 25 counterclockwise orientation, 585, 590 Dedekind cuts, 18 covariance, 370, 371, 420 degenerate critical point, 345, 347 covariance matrix, 374, 376 principal component of, 375 symmetric, 374 degenerate quadratic form, 339, 346, 347 degree of form, 566 of freedom (footnote), 290 of Taylor polynomial, 315 total (of multi-exponent), 316 Cramer, Gabriel, 198 Cramer's rule, 479 critical point, 334, 343, 348 degenerate, nondegenerate, 345 signature of, 345, 345 see also constrained critical point critical value, 343 cross product, 47, 77, 77, 78, 78 crossed partials equal, 317, 318, 448, 732-733 cube, unit n-dimensional, 416, 480 cubic equation, 708-711 Cardano's formula, 709 discriminant, 710, 710 cubic form, 333 cubic splines, 451 curl, 633, 670, 635-637, 638, 691 curl probe, 636 current density J, 671 curvature, 378-393 of closed curve, 542 of parametrized curve, 393 of plane curve, 379, 380, 384 of space curve, 391, 393, 393, 752 of surfaces, 385, 387 de!, 633, see also nabla derivative, 120 chain rule, 141, 312, 328 computing from definition, 131-134 continuously differentiable map, 149 differentiable implies continuous, 127 differentiating under integral sign, 512 directional, 121, 129, 129, 121-130 example not Lipschitz, 718 f(k) notation, 315 in closed set, 85 in one dimension, 120, 120-121 in several variables, 122-123 Jacobian matrix, 125, 146 of C mapping Lipschitz, 239 of composition, 141, 312 of determinant, 472, 473 of inverse function for matrices, 133 of squaring function for matrices, 131 reinterpreted, 626 rules for computing, 137, 137-143 second partial, 239 several variables, 126, 126, 126, 129 809 derivative, cont see also Lipschitz condition determinant, 461, 461, 469, 461-477, 569 derivative of, 473 development by first column, 462 existence, 462-465 geometric interpretation, 75, 79 history of linear algebra, 198 independent of basis, 468 measures volume, 479, 479-485 of x matrix, 75, 75, 75 of x matrix, 76, 76, 79 of AT A, 657 of elementary matrix, 468 of product of matrices, 467 of transpose, 468 of triangular matrix, 469 permutations, 471 product of eigenvalues, 479 right-hand rule, 79 diagonal, diagonal matrix, 50 diagonalizable matrix, 223, 476 diagonalization, 222-223, 223 diameter, 460 Dieudonne, Jean, 11, 67, 325 diffeomorphism, 209 differentiability, see also derivative criteria, 145-151, 150 pathological function, 146, 149 of polynomial, 138 of rational function, 138 smoothness, 285 differential equation, 219 differential form, 539, 575 addition of forms, 569 bounding integral of in terms of volume, 657 cone operator, see cone operator constant, 566, 566-675 degree of, 566 elementary, 569, 570, 568-571 exterior derivative see exterior derivative flux form, 601 form field, 575, 575 geometric interpretation, 567-568 integrating over oriented manifold 589, 595, 596, 595-598 ' integrating over parametrized domain, 578, 577-581 mass form, 603 multiplication by scalar, 569 Index: Page numbers in italics indicate theorems, propositions, etc 810 differential form, cont vector space A~(JR.n), 570, 570 wedge product, see wedge product work form, 600 differential operator, 633, 635, 644 differentiation under integral sign, see derivative dilation, 415 dimension, 39, 189, 189 fractional, 560 in physics, 601 of A~(JR.n), 571 of subspace, 189 of vector space, 187, 217, 217 dimension formula, 192, 196, 196, 204 applications, 199-204 eigenface, 375-377 event (in probability), 369 eigenspace, 221 event (point of spacetime), 685 existence of solutions, 13-14, 181, 183, 192, 196, 196 expectation, see expected value eigenvalue, 221, 220-228, 474-477 AT A and AAT have same, 368 Lagrange multiplier, 362 leading, 228 Perron-Frobenius theorem, 228 root of characteristic polynomial, 475 eigenvector, 221, 221-228, 474-477 finding, 221, 224-227, 363 leading, 228 linear independence, 224 of symmetric matrix, 363 Einstein, Albert, 119, 236, 672 , 685 Einstein's equation, 314 elasticity, 337, 635 expected value, 370, 370, 371, 372, 373, 420 exponential, 99, 100, 99-100 of matrix, 102 exterior derivative, 626, 626-631, 643, 794-797 computing, 627, 628 d(d