## A Course in Mathematics for Students of Physics:This textbook, available in two volumes, has been developed from a course taught at Harvard over the last decade. The course covers principally the theory and physical applications of linear algebra and of the calculus of several variables, particularly the exterior calculus. The authors adopt the 'spiral method' of teaching, covering the same topic several times at increasing levels of sophistication and range of application. Thus the reader develops a deep, intuitive understanding of the subject as a whole, and an appreciation of the natural progression of ideas. Topics covered include many items previously dealt with at a much more advanced level, such as algebraic topology (introduced via the analysis of electrical networks), exterior calculus, Lie derivatives, and star operators (which are applied to Maxwell's equations and optics). This then is a text which breaks new ground in presenting and applying sophisticated mathematics in an elementary setting. Any student, interpreted in the widest sense, with an interest in physics and mathematics, will gain from its study. |

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### Contents

viii | |

II | xi |

III | 407 |

IV | 411 |

V | 419 |

VI | 429 |

VII | 431 |

VIII | 433 |

LIX | 641 |

LX | 646 |

LXI | 651 |

LXII | 653 |

LXIII | 660 |

LXIV | 662 |

LXV | 664 |

LXVI | 674 |

IX | 436 |

X | 444 |

XI | 446 |

XII | 451 |

XIV | 458 |

XV | 461 |

XVI | 466 |

XVII | 469 |

XVIII | 474 |

XIX | 477 |

XX | 482 |

XXI | 485 |

XXII | 487 |

XXIII | 492 |

XXV | 502 |

XXVI | 520 |

XXVII | 526 |

XXIX | 532 |

XXX | 535 |

XXXI | 539 |

XXXII | 541 |

XXXIII | 553 |

XXXIV | 564 |

XXXV | 574 |

XXXVII | 583 |

XXXVIII | 585 |

XXXIX | 586 |

XL | 588 |

XLI | 596 |

XLII | 597 |

XLIII | 600 |

XLIV | 602 |

XLV | 604 |

XLVI | 606 |

XLVII | 608 |

XLVIII | 612 |

L | 615 |

LI | 616 |

LII | 621 |

LIII | 626 |

LIV | 628 |

LV | 633 |

LVI | 636 |

LVIII | 638 |

LXVIII | 686 |

LXIX | 689 |

LXX | 692 |

LXXI | 695 |

LXXII | 697 |

LXXIII | 700 |

LXXIV | 704 |

LXXVI | 706 |

LXXVII | 707 |

LXXVIII | 709 |

LXXIX | 711 |

LXXX | 715 |

LXXXI | 724 |

LXXXII | 729 |

LXXXIII | 735 |

LXXXIV | 740 |

LXXXV | 744 |

LXXXVII | 750 |

LXXXVIII | 755 |

LXXXIX | 758 |

XC | 761 |

XCI | 762 |

XCII | 764 |

XCIII | 766 |

XCV | 768 |

XCVI | 775 |

XCVII | 780 |

XCVIII | 785 |

XCIX | 796 |

C | 800 |

CI | 805 |

CII | 808 |

CIII | 814 |

CIV | 816 |

CV | 823 |

CVI | 826 |

CVII | 831 |

CVIII | 835 |

CIX | 836 |

CX | 838 |

CXI | 845 |

848 | |

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### Common terms and phrases

assume basis element boundary nodes branch voltages Calculate called capacitive capacitors Chapter charge Clifford algebra coboundary components compute conductors consider constant construct converges coordinates corresponding defined definition denote derivatives determine differential form dim C0 dimension Dirichlet problem dual dual space dx A dy dy A dz electrical networks electrostatics energy entropy equilibrium equivalence class evaluate example expression exterior fc-form finite formula given Green's hence holomorphic function integral interior nodes kernel Kirchhoff's Kirchhoff's voltage law linear differential form linear function linear map matrix maximal tree Maxwell's mesh currents method multiplication null curve one-cochain one-dimensional one-form orientation orthogonal particles Poisson Poisson's equation potential proof prove quotient space region resistors scalar product shown in figure shows Similarly solution solve sphere star operator Stokes subspace Suppose surface temperature tetrahedron theorem theory two-form vanishes vector field vector space voltage law write zero