Mathematical Tools for Physics

12—Tensors 394

12.19 In three dimensions three non-collinear vectors from a point define a volume, that of the parallelepiped
included between them. This defines a number as a function of three vectors. Show that if the volume is allowed
to be negative when one of the vectors reverses that this defines a trilinear functional and that it is completely
antisymmetric, an alternating tensor. (Note problem 17 .) If the units of volume are chosen to correspond to the
units of length of the basis vectors so that three one inch long perpendicular vectors enclose one cubic inch as
opposed to 16. 387 cm^3 then the functional is called ”the” alternating tensor. Find all its components.

12.20 Find the direct coordinate basis in spherical coordinates, also the reciprocal basis. Ans:ˆr,θ/rˆ ,φ/rˆ sinθ
(now which is it?)

12.21 Draw examples of the direct and reciprocal bases (to scale) for the example in Eq. ( 39 ). Do this for a
wide range of angles between the axes.

12.22 Show that the angle between two surfacesφ=constant andψ=constant is

cosθ=gij∂iφ∂jψ

/

[gmn∂mφ∂nφgpq∂pψ∂qψ

] 1 / 2

where∂i=∂/∂xi. In particular, what is the angle between two coordinate surfacesx^1 =constant andx^2 =
constant?

12.23 Show that the area in two dimensions enclosed by the infinitesimal parallelogram betweenx^1 andx^1 +dx^1
andx^2 andx^2 +dx^2 is

√

gdx^1 dx^2 wheregis the determinant of(gij).

12.24 Same as the preceding problem, but in three dimensions.

12.25 The divergence of a vector field is the limit of a surface integral divided by the enclosed volume, Eqs. (9.5)
and (9.9), and the result is that when you use a coordinate basis,

divF~= lim

V→ 0

1

V

∮

F~.dA~=√^1

g

∂

∂xk

(√

gFk

)

Verify this result explicitly in cylindrical coordinates.