Appel W. Mathematics for Physics and Physicists

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Change of coordinates 457

How are they deﬁned? Suppose that we have two nearby points M and

′

in space. Then we can write

M M

′

= δx

where δ x

is the variation of the µ-th coordinate x

when going from M

to M

′

. This same vector, which is an intrinsic quantity, may also be written

M M

′

= δu

′

where δu

is the variation of the µ-th curvilinear coordinate u

when going

from M to M

′

. But, to ﬁrst order, we have the relation

δu

∂ u

∂ x

δx

which, after identifying, gives

δx

∂ u

∂ x

δx

′

∂ u

∂ x

δx

′

(since µ and ν are mute indices), and hence, since the δ x

are independent

and arbitrary

∂ u

∂ x

′

or conversely e

′

∂ x

∂ u

or ag ain, expanding the notation,



















∂ u

∂ x

···

∂ u

∂ x

∂ u

∂ x

···

∂ u

∂ x













′







and







′













∂ x

∂ u

···

∂ x

∂ u

∂ x

∂ u

···

∂ x

∂ u



















Note that the matrices

∂ u

∂ x

and L

∂ x

∂ u

represent, respectively, the jacobian matrices of Φ and Φ

−1

. In particular, they

are invertible (the differential of Φ is bijective and therefore invertible), and

each is in fact the inverse of the other. This may be w ritten as

458 Ten sors

∂ u

∂ x

∂ u

= δ

We recover the notation used in an afﬁne space

; the only difference is that,

now, the matrices L and Λ depend on the point M where t he transformation

is performed. To summarize:

THEOREM 16.63 (Transformation of basis vectors) During a change of coordi-

nates Φ : (x

)

7→ (u

)

, the basis vectors are transformed acc ording to the following

relations:

′

∂ x

∂ u

and e

∂ u

∂ x

′

Note that these formulas are “balanced” from the point of view of indices,

obeying the following rule: a superscripted index of a term in the denominator of a

fraction is equivalent to a subscripted index.

16.5.c Transformation of physical quantities

Now, assume we have a vector which corresponds to some physical quantity

(and hence is an intrinsic object, also called a vector ﬁeld). It can be expressed

v = v

= v

′µ

′

in each system of coordinates. Applying the rule for the transformation of

basis vectors, we derive the rule for t he, transformation of the contravariant

coordinates of a vector:

THEOREM 16.64 (Transformation of contravariant c oordinates)

During a change of coordinates Φ : (x

)

7→ (u

)

, the contravariant coordinates v

of a vector ﬁeld are transformed according to the relation

′µ

∂ u

∂ x

and v

∂ x

∂ u

′ν

that is, following the opposite rule as that used for basis vectors.

Consider in this manner the formula of Proposition 16.13. We may write αα

αα

′µ

= Λ

αα

the form dx

′µ

= Λ

, which gives the well-known formula

′µ

∂ x

′µ

∂ x

One should of c ou rse indicate e very time at which p o int the vectors are tangent, which is

also where the partial derivatives are evaluated. In this case, the preceding formulas should be

Change of coordinates 459

Remark 16.65 Here is a way to memorize those formulas. Let x

′µ

be the new coordinates. We

recover the relation expressing e

′

as a functio n of e

, namely,

′

∂ x

′µ

by applying the following rules:

• since the index of the vector on the left-hand side of the equation (which is e

′

) is

subscripted, we write e

′

∂ x

and put the “prime” sign at the bottom:

′

∂ x

′

;

• since this index µ is subscripted, it can appear as “x

” only in the denominator, so we

write

′

∂ x

′µ

;

• ﬁnally, we complete the indices with a “mute” ν:

′

∂ x

′µ

(and then o ne should check that the index is balanced as it should).

Similarly, to recover the formula giving v

′µ

in terms of the v

’s,

• we write the formula with a “prime” si gn in the numerator, since the index is super-

scripted:

′µ

∂ x

′

∂ x

;

• then we complete the indices in order that the result be balanced:

′µ

∂ x

′µ

∂ x

The reader is invited to che ck that the converse formulas (giving v

and e

in terms of the

′ν

’s and e

′

’s) may be obtained with similar techniques.

16.5.d Transformation of linear forms

Since linear forms a re dual to vectors, the reader will not ﬁnd the following

result very surprising, except for the terminology: what might be called a ﬁeld

of linear forms is called a differential form:

written more rigorously:

′µ

′

) =

∂ u

∂ x

(Y) v

(Y) and v

(Y) =

∂ x

∂ u

′

) v

′

which complicates the notation quite a bit.

460 Tensors

THEOREM 16.66 (Transformation of linear forms) Let ωω

ωω be a differential form.

During a change of coordinates Φ : (x

)

7→ (u

)

, its covariant coordinates are

transformed according to the relations

′

∂ x

∂ u

and ω

∂ u

∂ x

′

The basic linear forms are transformed according to

αα

′µ

∂ u

∂ x

αα

and αα

αα

∂ x

∂ u

αα

′ν

The rule sketched above to memorize th ose relations remains valid.

16.5.e Transformation of an arbitrary tensor ﬁeld

Of course, the previous results extend easily to tensors of arbitrary type.

THEOREM 16.67 (Transformation of a tensor) Let T be a tensor ﬁeld with coor-

dinates

T(Y) = T

...,µ

,...,ν

(Y ) ⊗···⊗e

(Y ) ⊗αα

αα

(Y) ⊗···⊗αα

αα

(Y).

During a change of coordinates Φ : (x

)

7→ (u

)

, these coordinates are transformed

according to the rule

′

...,µ

,...,ν

∂ u

∂ x

···

∂ u

∂ x

∂ u

···

∂ x

∂ u

,...,ρ

,...,σ

Finally, we note that it is customary, both in relativity theory and in ﬁeld

theory, to introduce the following notation:

∂

def

∂

∂ x

and T

,ν

def

= ∂

T ,

so that, for instance, the divergence of a vector T = T

is written

div T = T

,µ

Change of coordinates 461

16.5.f Conclusion

Le Calcul Tensoriel sait mieux la physique que le Physicien lui-même.

Tensor calculus knows physics better than the physic ist himself does.

Paul Langevin (quoted in [10]).

Physical applications are too numerous to be studied individually. Among

others, we may mention:

Special relativity: the natural setting of special relativity was deﬁned by Min-

kowski. It is a space isomorphic to R

, with a pseudo-metric with

signature (1, 3). Generalizing the scalar product is done very simply.

The notion of covariance, in other words, of the invariance of a quantity

or a the form of an equation, is ex pressed mathematically in the invari-

ance under change of basis of quantities which are “balanced” in terms

of their expressions using indices.

General relativity: Riemannian geometry is unfortunately outside the scope

of this book. We simply mention that the notion of Minkowski space

is extended to non-euclidean geometry. Tensor calculus (in the context

of manifolds) is then, for intrinsic reasons, an indispensable ingredient

of the theory.

Fluid mechanics: using tensors, the notion of current ca n be generalized.

The current of a vector is a scalar. The current of a vector may be

expressed in matrix form (one index for each component of the vector,

one index for the direction of the current). It is then convenient to

go farther and to denote as a tensor with three indices the current of a

tensor with two indices, and so on.

Electricity, signal theory, optics: tensor calculus may be used as a mathemat-

ical trick to express general linear transformations. Then the underlying

notions such as linear forms, metric, duality,... do not carry special physi-

cal signiﬁcance.

Remark 16.68 A reader interested in general tensor calculus is invited to read the books of

Delachet [27]; the excellent treatise on modern geometry [30] in three volumes; the bible

concerning th e mathematics of general relativity (including all non-euclidean geometry) by

Wheeler

et al. [67]; or ﬁnally, one of t h e best physics books ever, that of Steven Weinberg [92].

On voit [...] se substituer à l’homo faber l’homo mathematicus.

Par exemple l’outil tensoriel est un merveilleux opérateur de généralité ;

à le manier, l’esprit acquiert des capacités nouvelles de généralisation.

We see [...] homo mathematicus substituting hims elf to homo faber.

For instance, tensor calculus is a wo nde rful operator in generalities; handling it,

the mind develops new capacities for generalization.

Gaston Bachelard, Le nouvel esprit scientiﬁque [10]

Who invented th e words “black hole.”

462 Tensors

SOLUTIONS

 Solution of exercise 16.1 on page 439. Using the formulas for change of basis, we get

′µ

αα

′ν

(x) e

′





αα

(x)











αα

(x) e

= δ

αα

(x) e

= Φ

αα

(x) e

and this is the required formula.

 Solution of exercise 16.2 on page 440. The ﬁrst questions are only a matter of writing

things down. In the case n = p = 2 , we see, for instance, that







0 1 0 −1

2 3 −2 −3

0 0 0 2

0 0 4 6









1 −1

0 2



⊗



0 1

2 3



but







0 0 0 0

0 1 0 0

0 0 2 3

0 0 0 7







cannot be a tensor p roduct of two matrices.

Chapter

Diﬀerential forms

In all this chapter, E denotes a real vector space of ﬁnite dimension n (hence

we have E ≃ R

, but E does not, a priori, have a euclidean structure, for

instance).

17.1

Exterior algebra

17.1.a 1-forms

DEFINITION 17.1 A linea r form or exterior 1-form on E is a linear map

from E to R:

ω : E −→ R.

The dual of E is the vector space E

∗

of linear forms on E.

Let B = ( b

, . . . , b

) be a basis of E. Any vector x ∈ E can be expressed

in a unique way as combination of vectors of the basis B:

x = x

+ ···+ x

We often denote simply x = (x

, . . . , x

) (identifying E and R

, the basis

of E being assumed given). For any i ∈ [[1, n]], we denote either b

∗i

or dx

the linear form on E which is deﬁned by

∗i

( x) =



∗i

, x



= x



∗i

, b



= δ

∀i, j ∈ [[1, n]].

464 Differential forms

THEOREM 17.2 The family (b

∗1

, . . . , b

∗n

) is a basis of E

∗

Proof. Let ϕ ∈ E

∗

. Then, denoting α

= ϕ(b

), we have obviously (by linearity)

ϕ =

∗i

, whi ch shows that the family (b

∗1

, . . . , b

∗n

) generates E

∗

. It is also a

free family (indeed, if

∗i

= 0, applying this linear form to the vector b

yields

= 0), and hence is a basis.

Any linear form may therefore be writ ten

ω = ω

+ ···+ ω

with ω

= ω( b

17.1.b Exterior 2-forms

DEFINITION 17.3 An exterior form of degree 2 or a 2-form is an antisym-

metric bilinear form on E × E, that is, a bilinear form ω

: E × E → R such

that we have

( x, y) = −ω

( y, x)

for all x, y ∈ E.

Example 17.4 The determinant map in a basis B = ( b

, b

) of R

, namely,

det : E × E −→ R,

( x, y) 7−→ det

( x, y) =



is an exterior 2-form on R

DEFINITION 17.5 The vector space of exterior forms of degree 2 on E is

denoted Λ

∗2

(E).

We now look for a basis of Λ

∗2

(E). Clearly, the bilinear form

⊗dx

−dx

⊗dx

: E × E −→ R,

( x, y) 7−→ x

− x

is an element of Λ

∗2

(E). Similarly, the bilinear forms dx

⊗dx

−dx

⊗dx

with i, j ∈ [[1, n]] are also in this space. A moment of thought s ufﬁces to

see that any exterior 2-form may be expressed as linear combination of those

elements. So we have found a generating family. But clearly, this family is not

free, since

• dx

⊗dx

−dx

⊗dx

= 0 for all i ∈ [[1, n]];

• dx

⊗dx

−dx

⊗dx

= −



⊗dx

−dx

⊗dx



for all i, j ∈ [[1, n]].

It is sufﬁcient to keep the elements of this type with i < j; t here are n(n−1)/2

of them.

THEOREM 17.6 A basis of Λ

∗2

(E) is given by the family of the exterior 2-forms

∧dx

def

= dx

⊗dx

−dx

⊗dx

where 1 ¶ i < j ¶ n. Hence we have dim Λ

∗2

(E) = n(n −1)/2.

Exterior algebra 465

Notice that th e space E

∗⊗2

def

= E

∗

⊗ E

∗

admits for basis the family of

elements



⊗dx

; i, j ∈ [[1, n]]



from which it is natural to create two fur ther families:

• the family we have seen of elements of the type dx

∧dx

(with i < j);

• the family of elements of the type

(dx

⊗dx

+ dx

⊗dx

Those two families are bases, respectively, for the space Λ

∗2

(E), which is the

space of antisymmetric





-tensors, and for the space of symmetric





-tensors.

PROPOSITION 17.7 Any





-tensor is the sum of a symmetric and an antisymmetric

tensor.

This may b e generalized to





-tensors.

17.1.c Exterior k-forms

We ﬁrst recall the following deﬁnition:

DEFINITION 17.8 A permuta tion of [[1, n]] is any bijection σ of the set [[1, n]]

into its elf; in other words, σ permutes the integers from 1 to n. A transposi-

tion is any permutation which exchanges two elements exactly and leaves the

others invariants. The set of permutations of [[1, n]] is denoted S

The composition of applications deﬁnes the product of permutations.

Any permutation may be written as a product of ﬁnitely many transposi-

tions. The signature of a permutation σ ∈ S

is the integer ǫ(σ) given by:

ǫ(σ) = 1 if σ can be written a s a product a an even number of transpositions,

and ǫ(σ) = −1 if σ can be written as the product of an odd number of

transpositions.

For any permutations σ and τ, we have

ǫ(στ) = ǫ(σ) ǫ(τ).

DEFINITION 17.9 An exterior k-form, or more simply a k-form, is a ny an-

tisymmetric k-multilinear form ω

on E

, that is, the map ω

: E

→ R

satisﬁes the relation

( x

, . . . , x

) = (−1)

ǫ(σ)

( x

σ(1)

, . . . , x

σ(k)

)

for any permutation σ ∈ S

Implicit in this deﬁnition is the fact, which we admit, that the parity of the number

of transpositions is independent of the choice of a formula expressing σ as a product o f

transpositions.

466 Differential forms

Example 17.10 The most famous example is that of an exterior k-form on a space of dimension

k, namely, the determinant map ex pressed using the canonical basis: if ( e

, . . . , e

) is the

canonical basis of R

, and if ξξ

ξξ

∈ R

for i = 1, . . . ,k, then denoting ξξ

ξξ

= ξ

+ ···+ ξ

this is a map

det : (R

)

−→ R,

(ξξ

ξξ

, . . . ,ξξ

ξξ

) 7−→ det

can

(ξξ

ξξ

, . . . ,ξξ

ξξ

) =



. . . ξ



which is an exterior k-form.

Example 17.11 For a ﬁxed vector u ∈ R

, the map ω : ( x, y) 7→ det( x, y, u) on R

×R

an exterior 2-form.

DEFINITION 17.12 The vector space of exterior k-forms on E is denoted

∗k

(E).

We will now ﬁnd a basis of Λ

∗k

(E).

To generalize the corresponding result of the preceding section for Λ

∗2

(E),

we introduce the following deﬁnition:

DEFINITION 17.13 Let k ∈ N and let i

, . . . , i

∈ [[1, n]]. Then the exterior

k-form dx

∧···∧dx

is deﬁned by the formula

∧···∧ dx

def

σ∈S

ǫ(σ) dx

σ(i

)

⊗···⊗dx

σ(i

)

Example 17.14 Consider the space R

and denote by x, y, z, t the coordonnates in this spac e.

Then we have

dx ∧dy ∧dz = dx ⊗ dy ⊗dz −dx ⊗dz ⊗dy + dy ⊗dz ⊗dx −dy ⊗dx ⊗dz

+ dz ⊗dx ⊗ dy −dz ⊗dy ⊗dx.

THEOREM 17.15 The family



∧···∧dx

; 1 ¶ i

< ··· < i

¶ n



is a basis of Λ

∗k

), the dimension of which is equal to the binomial coefﬁcient





Hence any exterior k-form (i.e., any antisymmetric





-tensor) can be written in the

form

<···<i

,...,i

∧···∧ dx

In the remainder of th is cha pter, Einstein’s summation conventions will

be extended to exterior forms, so we will write

ω = ω

,...,i

∧···∧dx

instead of ω =

<···<i

,...,i

∧···∧ dx