Mathematical Tools for Physics - Department of Physics - University

12—Tensors 323

basis, in its reciprocal basis, and mixed (A~in one basis andB~ in the reciprocal basis). A fourth way

uses thexˆ-ˆybasis.

A

B

C

D

O

~v

12.13 Show that if the direct basis vectors have unit length along the directions
of

−−→

OAand

−−→

OBthen the components of~vin the direct basis are the lengthsOA

andOB. What are the components in the reciprocal basis?

12.14 What happens to the components of the alternating tensor when a change
of basis is made? Show that the only thing that can happen is that all the
components are multiplied by the same number (which is the determinant of the
transformation). Compute this explicitly in two dimensions.

12.15 A given tensor (viewed as a function of two vector variables) has the property that it equals

zero whenever the two arguments are the same,T(~v,~v) = 0. Show that it is antisymmetric. This is

also true if it is a function of more than two variables and the above equation holds on some pair of

arguments. Consider~v=α~u+β~w

12.16 If the components of the alternating tensor are (in three dimensions)eijk wheree 123 = 1,

Computeeijk. Compute

eijkemk, eijkejk, eijkeijk

12.17 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 problem12.15.) 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.

12.18 Find the direct coordinate basis in spherical coordinates, also the reciprocal basis.

Ans: One set isˆr,θ/rˆ ,φ/rˆ sinθ(now which ones are these?)

12.19 Draw examples of the direct and reciprocal bases (to scale) for the example in Eq. (12.45). Do
this for a wide range of angles between the axes.

12.20 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).

12.21 Compute the transformation laws for the other components of the electromagnetic field tensor.

12.22 The divergence of a tensor field is defined as above for a vector field.Tis a tensor viewed as a

vector valued function of a vector variable

divT= lim

V→ 0

1

V

∮

T

(

dA~

)

It is a vector. Note: As defined here it requires the addition of vectors at two different points of the
manifold, so it must be assumed that you can move a vector over parallel to itself and add it to a vector
at another point. Had I not made the point in the text about not doing this, perhaps you wouldn’t

have noticed the assumption involved, but now do it anyway. ComputedivTin rectangular coordinates