Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
Figure 1.22.
Motion on a circle.
1.11. (a) Show that the acceleration a of a particle which travels along a space
curve with velocity v is given by
a ˆ dv
dt ^T ‡ v2
^N;
where ^T, ^N, and are as de®ned in the text.
(b) Consider a particle P moving on a circular path of radius r with constant
angular speed ! ˆ d=dt (Fig. 1.22). Show that the acceleration a of the
particle is given by
a ˆ! 2 r:
1.12. A particle moves along the curve x 1 ˆ 2t 2 ; x 2 ˆ t 2 4t; x 3 ˆ 3t 5, where t
is the time. Find the components of the particle's velocity and acceleration
at time t ˆ 1 in the direction ^e 1 3^e 2 ‡ 2^e 3 .
1.13. (a) Find a unit vector normal to the surface x 2 1 ‡ x 2 2 x 3 ˆ 1 at the point
P(1,1,1).
(b) Find the directional derivative of ˆ x 2 1x 2 x 3 ‡ 4x 1 x 2 3 at (1, 2; 1) in
the direction 2^e 1 ^e 2 2^e 3 .
1.14. Consider the ellipse given by r 1 ‡ r 2 ˆ const: (Fig. 1.23). Show that r 1 and r 2
make equal angles with the tangent to the ellipse.
1.15. Find the angle between the surfaces x 2 1 ‡ x 2 2 ‡ x 2 3 ˆ 9andx 3 ˆ x 2 1 ‡ x 2 2 3.
at the point (2, 1, 2).
1.16. (a) Iff and g are di€erentiable scalar functions, show that
r… fg† ˆf rg ‡ grf :
(b) Find rr if r ˆ…x 2 1 ‡ x 2 2 ‡ x 2 3† 1=2 .
(c) Show that rr n ˆ nr n2 r.
1.17. Show that:
(a) r…r=r 3 †ˆ0. Thus the divergence of an inverse-square force is zero.
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