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! colspan="3" | 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY | ! colspan="3" | 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY | ||
|- | |- | ||
| 1 || Introduction || 471 | | 13.1 || Introduction || 471 | ||
|- | |- | ||
| 2 || Lines in n-space || 472 | | 13.2 || Lines in n-space || 472 | ||
|- | |- | ||
| 13.3 || Some simple properties of straight lines || 473 | |||
|- | |- | ||
| | | 13.4 || Lines and vector-valued functions || 474 | ||
|- | |- | ||
| | | 13.5 || Exercises || 477 | ||
|- | |- | ||
| | | 13.6 || Planes in Euclidean n-space || 478 | ||
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| 13.7 || Planes and vector-valued functions || 481 | |||
|- | |- | ||
| | | 13.8 || Exercises || 482 | ||
|- | |- | ||
| | | 13.9 || The cross product || 483 | ||
|- | |- | ||
| | | 13.10 || The cross product expressed as a determinant || 486 | ||
|- | |- | ||
| | | 13.11 || Exercises || 487 | ||
|- | |- | ||
| | | 13.12 || The scalar triple product || 488 | ||
|- | |- | ||
| | | 13.13 || Cramer's rule for solving a system of three linear equations || 490 | ||
|- | |- | ||
! colspan="2" | Index || | | 13.14 || Exercises || 491 | ||
|- | |||
| 13.15 || Normal vectors to planes || 493 | |||
|- | |||
| 13.16 || Linear Cartesian equations for planes || 494 | |||
|- | |||
| 13.17 || Exercises || 496 | |||
|- | |||
| 13.18 || The conic sections || 497 | |||
|- | |||
| 13.19 || Eccentricity of conic sections || 500 | |||
|- | |||
| 13.20 || Polar equations for conic sections || 501 | |||
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| 13.21 || Exercises || 503 | |||
|- | |||
| 13.22 || Conic sections symmetric about the origin || 504 | |||
|- | |||
| 13.23 || Cartesian equations for the conic sections || 505 | |||
|- | |||
| 13.24 || Exercises || 508 | |||
|- | |||
| 13.25 || Miscellaneous exercises on conic sections || 509 | |||
|- | |||
! colspan="3" | 14. CALCULUS OF VECTOR-VALUED FUNCTIONS | |||
|- | |||
| 14.1 || Vector-valued functions of a real variable || 512 | |||
|- | |||
| 14.2 || Algebraic operations. Components || 512 | |||
|- | |||
| 14.3 || Limits, derivatives, and integrals || 513 | |||
|- | |||
| 14.4 || Exercises || 516 | |||
|- | |||
| 14.5 || Applications to curves. Tangency || 517 | |||
|- | |||
| 14.6 || Applications to curvilinear motion. Velocity, speed, and acceleration || 520 | |||
|- | |||
| 14.7 || Exercises || 524 | |||
|- | |||
| 14.8 || The unit tangent, the principal normal, and the osculating plane of a curve || 525 | |||
|- | |||
| 14.9 || Exercises || 528 | |||
|- | |||
| 14.10 || The definition of arc length || 529 | |||
|- | |||
| 14.11 || Additivity of arc length || 532 | |||
|- | |||
| 14.12 || The arc-length function || 533 | |||
|- | |||
| 14.13 || Exercises || 535 | |||
|- | |||
| 14.14 || Curvature of a curve || 536 | |||
|- | |||
| 14.15 || Exercises || 538 | |||
|- | |||
| 14.16 || Velocity and acceleration in polar coordinates || 540 | |||
|- | |||
| 14.17 || Plane motion with radial acceleration || 542 | |||
|- | |||
| 14.18 || Cylindrical coordinates || 543 | |||
|- | |||
| 14.19 || Exercises || 543 | |||
|- | |||
| 14.20 || Applications to planetary motion || 545 | |||
|- | |||
| 14.21 || Miscellaneous review exercises || 549 | |||
|- | |||
! colspan="3" | 15. LINEAR SPACES | |||
|- | |||
| 15.1 || Introduction || 551 | |||
|- | |||
| 15.2 || The definition of a linear space || 551 | |||
|- | |||
| 15.3 || Examples of linear spaces || 552 | |||
|- | |||
| 15.4 || Elementary consequences of the axioms || 554 | |||
|- | |||
| 15.5 || Exercises || 555 | |||
|- | |||
| 15.6 || Subspaces of a linear space || 556 | |||
|- | |||
| 15.7 || Dependent and independent sets in a linear space || 557 | |||
|- | |||
| 15.8 || Bases and dimension || 559 | |||
|- | |||
| 15.9 || Exercises || 560 | |||
|- | |||
| 15.10 || Inner products, Euclidean spaces, norms || 561 | |||
|- | |||
| 15.11 || Orthogonality in a Euclidean space || 564 | |||
|- | |||
| 15.12 || Exercises || 566 | |||
|- | |||
| 15.13 || Construction of orthogonal sets. The Gram-Schmidt process || 568 | |||
|- | |||
| 15.14 || Orthogonal complements. Projections || 572 | |||
|- | |||
| 15.15 || Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace || 574 | |||
|- | |||
| 15.16 || Exercises || 576 | |||
|- | |||
! colspan="3" | 16. LINEAR TRANSFORMATIONS AND MATRICES | |||
|- | |||
| 16.1 || Linear transformations || 578 | |||
|- | |||
| 16.2 || Null space and range || 579 | |||
|- | |||
| 16.3 || Nullity and rank || 581 | |||
|- | |||
| 16.4 || Exercises || 582 | |||
|- | |||
| 16.5 || Algebraic operations on linear transformations || 583 | |||
|- | |||
| 16.6 || Inverses || 585 | |||
|- | |||
| 16.7 || One-to-one linear transformations || 587 | |||
|- | |||
| 16.8 || Exercises || 589 | |||
|- | |||
| 16.9 || Linear transformations with prescribed values || 590 | |||
|- | |||
| 16.10 || Matrix representations of linear transformations || 591 | |||
|- | |||
| 16.11 || Construction of a matrix representation in diagonal form || 594 | |||
|- | |||
| 16.12 || Exercises || 596 | |||
|- | |||
| 16.13 || Linear spaces of matrices || 597 | |||
|- | |||
| 16.14 || Isomorphism between linear transformations and matrices || 599 | |||
|- | |||
| 16.15 || Multiplication of matrices || 600 | |||
|- | |||
| 16.16 || Exercises || 603 | |||
|- | |||
| 16.17 || Systems of linear equations || 605 | |||
|- | |||
| 16.18 || Computation techniques || 607 | |||
|- | |||
| 16.19 || Inverses of square matrices || 611 | |||
|- | |||
| 16.20 || Exercises || 613 | |||
|- | |||
| 16.21 || Miscellaneous exercises on matrices || 614 | |||
|- | |||
! colspan="2" | Answers to exercises || 617 | |||
|- | |||
! colspan="2" | Index || 657 | |||
|- | |- | ||
|} | |} | ||
[[Category:Mathematics]] | [[Category:Mathematics]] |