By Roger Blanford, Kip Thorne
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Extra info for Applications of Classical Physics (web draft april 2013)
By traveling along a sequence of such dx’s, laying them down tail-at-tip, one after another, we can map out a curve to which these dx’s are tangent (Fig. 1b). The curve is P(λ), with λ a parameter along the curve; and the infinitesimal vectors that map it out are dx = (dP/dλ)dλ. The product of a scalar with a vector is still a vector; so if we take the change of location dx of a particular element of a fluid during a (universal) time interval dt, and multiply it by 1/dt, we obtain a new vector, the fluid element’s velocity v = dx/dt, at the fluid element’s location P.
The Euclidean distance ∆σ between two points P and Q in 3-space can be measured with a ruler and so, of course, requires no coordinate system for its definition. (If one does have a Cartesian coordinate system, then ∆σ can be computed by the Pythagorean formula, a precursor to the “invariant interval” of flat spacetime, Sec. ) This distance ∆σ is also the length |∆x| of the vector ∆x that reaches from P to Q, and the square of that length is denoted |∆x|2 ≡ (∆x)2 ≡ (∆σ)2 . 2) Of particular importance is the case when P and Q are neighboring points and ∆x is a differential (infinitesimal) quantity dx.
Introduction to Electrodynamics, Upper Saddle River NJ: Prentice-Hall, third edition. Jackson, John David 1999. Classical Electrodynamics, New York: Wiley, third edition. Thornton, Stephen T. and Marion, Jerry B. 2004. Classical Dynamics of Particles and Systems, Belmont, CA, Brooks/Cole—Thomson Learning, fifth edition. 1 Overview . . . . . . . . . . . . . . . . . . . 2 Foundational Concepts . . . . . . . . . . . . . . . 1 Inertial frames, inertial coordinates, events, vectors, and spacetime diagrams .
Applications of Classical Physics (web draft april 2013) by Roger Blanford, Kip Thorne