Skip to main content Accessibility help
×
  1. Home
  2. Browse subjects
  3. Physics and Astronomy
  4. General and Classical Physics
  5. Content listing

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

4305 results in General and Classical Physics

Page 183 of 216

  • Chapter 3 - DERIVATIVES (Program 3)

  • Richard P. Olenick, University of Dallas, Tom M. Apostol, California Institute of Technology, David L. Goodstein, California Institute of Technology
  • Book:
    The Mechanical Universe
    Published online:
    28 May 2018
    Print publication:
    30 August 1985, pp 39-68

  • Summary

    It is most useful that the true origins of memorable inventions be known, especially of those which were conceived not by accident but by an effort of meditation … One of the noblest inventions of our time has been a new kind of mathematical analysis, known as the differential calculus.

    Gottfried Wiihelm von Leibniz. Historia et Origo Calculi Differentialis (1714)

    THE DEVELOPMENT OF DIFFERENTIAL CALCULUS

    After the advent of algebra in the sixteenth century, mathematical discoveries inundated Europe. The most important were differential calculus and integral calculus, bold new methods for attacking a host of problems that had challenged the world's best minds for more than 2000 years. Differential calculus deals with ideas such as speed, rate of growth, tangent lines, and curvature, whereas integral calculus treats topics such as area, volume, arc length, and centroids.

    Work begun by Archimedes in the third century B.C. led ultimately to the birth of integral calculus in the seventeenth century. This development has a long and fascinating history to which we shall return in Chapter 7.

    Differential calculus has a relatively short history. Its principles were first formulated early in the seventeenth century when a French mathematician, Pierre de Fermat, tried to devise a way of finding the smallest and largest values of a given function. He imagined the graph of a function having, at each of its points, a direction given by a tangent line, as suggested by the points labeled in Fig. 3.1.

  • Chapter 5 - VECTORS (Program 5)

  • Richard P. Olenick, University of Dallas, Tom M. Apostol, California Institute of Technology, David L. Goodstein, California Institute of Technology
  • Book:
    The Mechanical Universe
    Published online:
    28 May 2018
    Print publication:
    30 August 1985, pp 81-110

  • Summary

    If I wished to attract the student of any of these sciences to an algebra for vectors, I should tell him that the fundamental notions of this algebra were exactly those with which he was daily conversant … In fact, I should tell him that the notions which we use in vector analysis are those which he who reads between the lines will meet on every page of the great masters of analysis, or of those who have probed the deepest secrets of nature.

    J W Gibbs, in Nature, 16 March 1893

    THE RISE OF VECTOR ANALYSIS

    Throughout their history, mathematics and physics have been intimately related; a discovery in one field led to an improvement in the other. Early natural philosophers grappling with quantities such as distance, speed, and time used geometry inherited from the Greeks to explore physical problems. But in the tumultuous years of the seventeenth century, physics underwent a transformation – a shift in emphasis from numerical quantities, such as distance and speed, to vector quantities, such as displacement and velocity, which have direction as well as magnitude.

    The transition was neither abrupt nor confined to that century. It was necessary to invent new mathematical objects – vectors – and new mathematical machinery for manipulating them – vector algebra – to embody the properties of the physical quantities they were to represent.

  • Chapter 14 - ENERGY AND STABILITY (Program 14)

  • Richard P. Olenick, University of Dallas, Tom M. Apostol, California Institute of Technology, David L. Goodstein, California Institute of Technology
  • Book:
    The Mechanical Universe
    Published online:
    28 May 2018
    Print publication:
    30 August 1985, pp 267-286

  • Summary

    I do not consider these principles to be certain mysterious qualities feigned as arising from characteristic forms of things, but as universal laws of Nature, by the influence of which these very things have been created. For the phenomena of Nature show that these principles do indeed exist, although their nature has not yet been elucidated. To assert that each and every species is endowed with a mysterious property characteristic to it, due to which it has a definite mode in action, is really equivalent to saying nothing at all. On the other hand, to derive from the phenomena of Nature two or three general principles, and then to explain how the properties and actions of all corporate things follow from those principles, this would indeed be a mighty advance in philosophy, even if the causes of those principles had not at the time been discovered.

    Roger Boscovich, A Theory of Natural Philosophy (1763)

    FORMS OF ENERGY

    The concept of energy, as we saw in Chapter 13, is subtle, elegant, and rich. It describes a dynamic property of the universe which is strictly and absolutely conserved; energy can neither be created nor destroyed. Not even in the presence of friction is energy ever lost; it is simply transformed into other forms. Nevertheless, the universe is winding down. Energy tends to be transformed from well-organized forms into more disorganized forms, until it becomes completely useless.

Page 183 of 216