Tensors for Physics

Appendix: Exercises... 393

3.5 Velocity of a Particle Moving on a Screw Curve(p.46)
Hint: Useα=ωt for the parameter occurring in the screw curve(3.48),ωis a
frequency.
Differentiation with respect to the timetyields the velocity

v=ρω[−esin(ωt)+ucos(ωt)]+χ

ω

2 π

e×u,

where it is assumed that not only the orthogonal unit vectorseandu, but also the
radiusρand the pitch parameterχare constant.

Exercise Chapter 4

4.1 2D Dual Relation in Complex Notation(p.54)
Let the two 2D vectors(x 1 ,y 1 )and(x 2 ,y 2 )be expressed in terms of the complex
numbers z 1 =x 1 +iy 1 and z 2 =x 2 +iy 2 .Write the dual relation corresponding to
(4.25)in terms of the complex numbers z 1 and z 2 .How about the scalar product of
these 2D vectors?
Hint: the complex conjugate of z=x+iy is z∗=x−iy.
The productz∗ 1 z 2 isx 1 x 2 +y 1 y 2 +i(x 1 y 2 −x 2 y 1 ), thus the dual scalar is

x 1 y 2 −x 2 y 1 =

1

2 i

(z∗ 1 z 2 −z 1 z∗ 2 ).

Similarly, the scalar product of the two vectors is

x 1 y 1 +x 2 y 2 =

1

2

(z∗ 1 z 2 +z 1 z∗ 2 ).

In other words, the scalar product is the real part and the dual scalar is the imaginary
part ofz∗ 1 z 2.

Exercises Chapter 5

5.1 Show that the Moment of Inertia Tensors for Regular Tetrahedra and
Octahedra are Isotropic(p.62)
Hint: Use the coordinates( 1 , 1 , 1 ),(− 1 ,− 1 , 1 ),( 1 ,− 1 ,− 1 ),(− 1 , 1 ,− 1 )for the four
corners of the tetrahedron and( 1 , 0 , 0 ),(− 1 , 0 , 0 ),( 0 , 1 , 0 ),( 0 ,− 1 , 0 ),( 0 , 0 , 1 ),
( 0 , 0 ,− 1 ),for the six of the octahedron.