Mathematics of Physics and Engineering
26 Curves in Space Given a fixed frame O, the formulas of differential calculus for vector functions in this frame are easily ob ...
The Tangent Vector and Arc Length 27 EXERCISE 1.3.1/^4 (a) Show that if r is differentiable at t 0 , then r is con- tinuous at t ...
28 Curves in Space The equation of the tangent line at point P is R(s) = r(to) + su{t 0 ). (1.3.9) EXERCISE 1.3.2.c Let C be a p ...
The Tangent Vector and Arc Length^29 Lc{c,d) between the points r(c) and r(d) along a rectifiable curve C is Lc(c,d) = lim Ln, T ...
30 Curves in Space If the curve is smooth, then ds/dt > 0 and s is a monotone function of t so that t is a well-defined funct ...
Prenet's Formulas 31 the principal unit normal vector at P is p =-«'(«); (1.3.15) the unit binomial vector at P is b(s) = u(s) x ...
32 Curves in Space At every point P of the curve, the vector triple (v., p, b) is a right- handed coordinate system with origin ...
Velocity and Acceleration 33 it is clear that the tangent vector 2 points along the fuselage from the tail to the nose, and the ...
34 Curves in Space or ., d^2 s _.. Ids\^2 du(s) .„ „„ , °<«>=d* "<'>+(*) -11. (1.3.19) Equation (1.3.19) shows that ...
Velocity and Acceleration 35 and v • r — 0. So v is tangent to the circle. The acceleration a is a(t) =v'(t) = - R(9"(t) sin 9(t ...
(^36) Curves in Space The velocity v is a sum of the radial velocity component fr and the angular velocity component rujd. We ca ...
Velocity and Acceleration 37 EXERCISE 1.3.17.A Let (r(t),9(t)) be the polar coordinates of a 2-D motion of a point mass m in a f ...
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Chapter 2 Vector Analysis and Classical and Relativistic Mechanics 2.1 Kinematics and Dynamics of a Point Mass Kinematics is the ...
40 Kinematics and Dynamics of a Point Mass University. Newton's First Law: Unless acted upon by a force, a point mass is either ...
Newton's Laws of Motion and Gravitation 41 Recall that a force F acting on the point mass m has a torque, or moment, about O equ ...
42 Kinematics and Dynamics of a Point Mass L — mr x r = m(£r x 196) = m£^29 A, and ^=m£r6k. (2.1.7) at The forces acting on the ...
Newton's Laws of Motion and Gravitation 43 EXERCISE 2.1.2r Let point mass m move in a planar path C given by r(t), where r is th ...
(^44) Kinematics and Dynamics of a Point Mass The basic ideas of modern astronomy go back to the Polish astronomer NICOLAUS COPE ...
Newton's Laws of Motion and Gravitation 45 recover Newton's argument by combining equation (1.3.29), page 36, with his second la ...
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