By Jerry B. Marion (Auth.)
This best-selling classical mechanics textual content, written for the complicated undergraduate one- or two-semester direction, presents an entire account of the classical mechanics of debris, platforms of debris, and inflexible our bodies. Vector calculus is used generally to discover topics.The Lagrangian formula of mechanics is brought early to teach its strong challenge fixing ability.. smooth notation and terminology are used all through in aid of the text's goal: to facilitate scholars' transition to complicated physics and the mathematical formalism wanted for the quantum conception of physics. CLASSICAL DYNAMICS OF debris AND structures can simply be used for a one- or two-semester path, reckoning on the instructor's number of subject matters
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Additional info for Classical Dynamics of Particles and Systems
For this purpose, we introduce the unit vectors along the rectangular axes*: e l5 e 2 , e 3 . 71) We have seen [Eq. 72) Therefore, the scalar product of any two unit vectors (because they are * Many variants of the symbols for the unit vectors are in use. Perhaps the most common set is i, j, k ori, j, k; others include i,, i2, i3 and lx, lv, 1,. 75) *ijifiiAjBk By direct expansion and comparison with Eq. 80) PROBLEMS 29 Suggested References Texts on vector methods abound. A standard, often-quoted work is that of Phillips (Ph33).
Grad(p 2 - φ2^ταάφι) = φχΨφ2 - φ1Ψφι. 2-9. Show that div(A x B) = B · curl A - A · curl B. 2-10. Show that (A · grad)A = \ grad A2 — A x curl A. 2-11. If φί and φ2 describe two different scalar fields, show that (gradc^) x (grad φ2) is solenoidal. 2-12. Use r = ^ / £ x 2 to prove the following relations : i (n 2) (a) gradr" = «r ~ r (b) div(rnr) = (n + 3)r", (c) (d) W curlr = 0 = n(n + l)r (n " 2) , n # -3 n Φ -1 2-13. A particle rotates with constant angular velocity ω about an axis which passes through the origin.
Let A be a vector from the origin to a point P fixed in space. Let r be a vector from the origin to a variable point Q(xu x2> ^3). Show that A-r = A2 is the equation of a plane perpendicular to A and passing through the point P. 1-8. Show that the triple scalar product (A x B)· C may be written as A1 (A x B ) C = \Bl A2 A31 B2 B3 Show also that the product is unaffected by an interchange of the scalar and vector product operations or by a change in the order of A, B, C, as long as they are in cyclic order ; that is, (A x B) · C = A · (B x C) = B · (C x A) = (C x A) · B, etc.