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Analytic Geometry and Calculus, with Vectors by Ralph Palmer Agnew

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Analytic Geometry and Calculus, with Vectors written by Ralph Palmer Agnew , Professor of Mathematics, Department of Mathematics, Cornell University. There is an element of truth in the old saying that the Euler textbook Introductioin Analysin Infinitorum (Lausannae, 1748) was the first great calculus textbook, and that all elementary calculus textbooks published since that time have been copied from Euler or have been copied from books that were copied from Euler. Euler, the greatest mathematician of his day and in many respects the greatest mathematician of all time, held sway when, except where the geometry of Euclid was involved, it was not the fashion to try to base mathematical work upon accurately formulated basic concepts. Problems were the important things, and meaningful formulations of axioms, postulates, definitions, hypotheses, conclusions, and theorems either were not written or played minor roles. Through most of the first half of the twentieth century, elementary textbooks in our subject taught unexplained but "well motivated" intuitive ideas along with their problems. Enthusiasm for this approach to calculus waned when it was realized that students were not nourished by stews in which problems, motivations, fuzzy definitions, and fuzzy theorems all boiled together while something approached something else without ever quite getting there. About the middle of the twentieth century, precise formulations of basic concepts began to occupy minor but increasingly important roles. So far as calculus is concerned, this book attaches primary importance to basic concepts. These concepts comprise the solid foundation upon which advanced as well as elementary applications of calculus are based. Applications, including those that have great historical interest, occupy secondary roles. With this shift in our emphasis, we can remove the mystery from old mathematics and learn modern mathematics when we sometimes spend a day or two studying basic concepts and attaining mastery of ideas, language, and notation that are used. The mathematical counterparts of hydrogen and electrons are important, and we study them before trying to construct the mathematical counterparts of carbohydrates and television receivers.

Analytic Geometry and Calculus, with Vectors written by Ralph Palmer Agnew cover the following topics.

  • 1. Analytic geometry in two dimensions
    1.1 Real numbers
    1.2 Slopes and equations of lines
    1.3 Lines and linear equations; parallelism and perpendicularity
    1.4 Distances, circles, and parabolas
    1.5 Equations, statements, and graphs
    1.6 Introduction to velocity and acceleration

  • 2. Vectors and geometry in three dimensions
    2.1 Vectors in E3
    2.2 Coordinate systems and vectors in E3
    2.3 Scalar products, direction cosines, and lines in E3
    2.4 Planes and lines in E3
    2.5 Determinants and applications
    2.6 Vector products and changes of coordinates in E3

  • 3. Functions, limits, derivatives
    3.1 Functional notation
    3.2 Limits
    3.3 Unilateral limits and asymptotes
    3.4 Continuity
    3.5 Difference quotients and derivatives
    3.6 The chain rule and differentiation of elementary functions
    3.7 Rates, velocities
    3.8 Related rates
    3.9 Increments and differentials

  • 4. Integrals
    4.1 Indefinite integrals
    4.2 Riemann sums and integrals
    4.3 Properties of integrals
    4.4 Areas and integrals
    4.5 Volumes and integrals
    4.6 Riemann-Cauchy integrals and work
    4.7 Mass, linear density, and moments
    4.8 Moments and centroids in B2 and E3
    4.9 Simpson and other approximations to integrals

  • 5. Functions, graphs, and numbers
    5.1 Graphs, slopes, and tangents
    5.2 Trends, maxima, and minima
    5.3 Second derivatives, convexity, and flexpoints
    5.4 Theorems about continuous and differentiable functions
    5.5 The Rolle theorem and the mean-value theorem
    5.6 Sequences, series, and decimals
    5.7 Darboux sums and Riemann integrals

  • 6. Cones and conics
    6.1 Parabolas
    6.2 Geometry of cones and conics
    6.3 Ellipses
    6.4 Hyperbolas
    6.5 Translation and rotation of axes
    6.6 Quadric surfaces

  • 7. Curves, lengths, and curvatures
    7.1 Curves and lengths
    7.2 Lengths and integrals
    7.3 Center and radius of curvature

  • 8. Trigonometric functions
    8.1 Trigonometric functions and their derivatives
    8.2 Trigonometric integrands
    8.3 Inverse trigonometric functions
    8.4 Integration by trigonometric and other substitutions
    8.5 Integration by substituting z = tan e/2

  • 9. Exponential and logarithmic functions
    9.1 Exponentials and logarithms
    9.2 Derivatives and integrals of exponentials and logarithms
    9.3 Hyperbolic functions
    9.4 Partial fractions
    9.5 Integration by parts

  • 10. Polar, cylindrical, and spherical coordinates
    10.1 Geometry of coordinate systems
    10.2 Polar curves, tangents, and lengths
    10.3 Areas and integrals involving polar coordinates

  • 11. Partial derivatives
    11.1 Elementary partial derivatives
    11.2 Increments, chain rule, and gradients
    11.3 Formulas involving partial derivatives

  • 12. Series
    12.1 Definitions and basic theorems
    12.2 Ratio test and integral test
    12.3 Alternating series and Fourier series
    12.4 Power series
    12.5 Taylor formulas with remainders
    12.6 Euler-Maclaurin summation formulas

  • 13. Iterated and multiple integrals
    13.1 Iterated integrals
    13.2 Iterated integrals and volumes
    13.3 Double integrals
    13.4 Rectangular coordinate applications of double and iterated integrals
    13.5 Integrals in polar coordinates
    13.6 Triple integrals; rectangular coordinates
    13.7 Triple integrals; cylindrical coordinates
    13.8 Triple integrals; spherical coordinates

    1. Proofs of basic theorems on limits
    2. Volumes


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