Excerpt: Uniform tensors and tensors.

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Thus, if(~ei)is a basis then, for alli, j,

Tr(~ei⊗e^j) =δ^j_i (=δ(e^j, ~ei)). (Q.57) Thus, ifL∈ L(E;E)and[L]_|~_e = [Lⁱ_j](that isL=Pn

i,j=1Lⁱ_j~ei⊗e^j) thenTr(L) =Pn

i,j=1Lⁱ_jTr(~ei⊗e^j) (linearity), that is,

Tr(L) =

n

X

i=1

Lⁱ_i called the trace of the endomorphismL. (Q.58) Thus the trace of an endomorphism is an invariant: IfL∈ L(E;E), if(~ai)and(~bi)are two bases, then

L=

n

X

i,j=1

Aⁱ_j~ai⊗a^j=

n

X

i,j=1

B_jⁱ~bi⊗b^j ⇒ Tr(L) =

n

X

i=1

Aⁱ_i=

n

X

i=1

B_iⁱ. (Q.59) (The valueTr(L)is independent of an observer.) Indeedaⁱ(~aj) =bⁱ(~bj) =δⁱ_j.

Example Q.21 ~v=P

ivⁱ~ei and`=P

j`je^j giveTr(~v⊗`) =P

ivⁱ`i. Example Q.22 The Kronecker tensor δ =Pn

i=1~e_i⊗eⁱ, cf. (Q.54), seen as a endomorphism, has the trace

Tr(δ) =X

i

Tr(~ei⊗eⁱ) =

n

X

i=1

δ_iⁱ=

n

X

i=1

1 =n= Tr(I), (Q.60)

trace of the identityI:E→E.

Exercice Q.23 Let M ∈ L(E;E), let (~a_i) and (~b_i) be bases in E, and let M = Pn

i,j=1M_jⁱ~b_i⊗a^j ∈ L(E;E), that is, M.~aj =Pn

i=1Lⁱ_j~bi. (NB: Here M is not considered to be an endomorphism since two bases have been used for its representation.) Prove: ifP is the change of basis from (~ai) to (~bi), then TrM =P

j(P M)^j_j. Answer. M =P

ijM_jⁱ(P

kP_i^k~ak)⊗a^j=P

kj(P

iM_jⁱP_i^k)~ak⊗a^j=P

kj(P M)^k_j~ak⊗a^j. Q.7.3 The divergence is an invariant

If~visC¹(Ω;Rⁿ)andp∈Ω, thend~v(p)∈ L(Rⁿ;Rⁿ)is an endomorphism. Its trace is called the divergence of~v atp:

div(~v)(p) := Tr(d~v(p)). (Q.61)

E.g., if(~e_i)is a Euclidean basis atp, if~v(p) =Pn

i=1vⁱ(p)~e_i, thend~v(p) =Pn i,j=1

∂vⁱ

∂x^j(p)~e_i⊗dx^j, and div(~v)(p) =

n

X

i=1

∂vⁱ

∂xⁱ(p) value independent of the Euclidean basis. (Q.62)

R Tensors in T

_s^r

(U )

First we need the denition of a (Eulerian) vector eld. Then a rst degree of complexity is introduced with the denition of functions acting on the vector elds, that is, the dierential forms (or one-forms).

Then a second degree of complexity is introduced with the tensors that are functionals acting on vector elds and on dierential forms.

R.1 Introduction and module

Let A and B be any sets, and let F(A;B) be the set of scalar valued functions. The plus interior operation and the dot exterior operation are dened by, for allf, g∈ F(A;B), allλ∈Rand allp∈A,

( (f+g)(p) :=f(p) +g(p), and

(λ.f)(p) :=λ f(p), λ.f^written= λf. (R.1)

Thus (F(A;B),+, .,R) is a vector space on the eldR (easy proof). This is introduced in any classic elementary course.

But the eld R is too small to dene a tensor = an linear object that satises the change of coordinate system rules:

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tiond:w~ ∈C^∞(Ω)→d ~w∈C^∞(Ω). It isR-linear (trivial), but its component on a basis of a coordinate system (the Christoel symbols) do not satisfy the change of coordinate system rules. In fact, the change of coordinate system rules deal with algebra (where no derivation is used), while a derivation is not an algebraic concept but a functional analysis concept (a derivation is not a tensor: It is a spray, see AbrahamMarsden [1]). In fact the problem comes from the derivation rule, with f ∈ C^∞(Ω;R) a regular scalar valued function,

d(f ~w) =f d ~w+df. ~w, so d(f ~w)6=f d(w).~ (R.2) Thus theR-linearity of an operatorT :E→F (like a derivation) is not sucient to characterize a tensor (corresponds to the casef is a constant function), and the requirement for some T to be a tensor will look like:

T(f ~w) =f T(~w), (R.3)

that is,T has to beC^∞(Ω;R)-linear, and not onlyR-linear (the eld Ris to small: It doesn't introduce enough constraints; It has to be enlarged toC^∞(Ω;R)).

Solution: To replace a vector space build from a eld (likeR) by a module build from a ring (like C^∞(Ω;R)). Reminder: A ring is almost like a eld, except that in a ring some elements don't have an inverse. E.g., a functionϕ∈C^∞(Ω;R)that vanishes at one point doesn't have an inverse∈C^∞(Ω;R).

And the denition of the module is very similar to the denition of vector space, but for the external product where a real is replaced by an element of a ring.

Here for the group(F(A;B),+), and compared to (R.1)2, the external dot product onRis generalized to the external dot product onF(A;R)dened by: For allf ∈ F(A;B), allϕ∈ F(A;R)and allp∈A,

(ϕ.f)(p) :=ϕ(p)f(p), ϕ.f^written= ϕf (R.4) (andf.ϕ:=ϕf). And(F(A;B),+, .,F(A;R))is a module (over the ringF(A;R)).

R.2 Functions and vector elds

R.2.1 Framework

LetU be an open set in an ane spaceE, and let E be an associated vector. (More generallyU is an open set in a dierentiable manifold.)

Classical mechanics: The denition of tensors is done at any (xed) time t.

As before, the approach is rst qualitative, then quantitative, and atp∈ E a basis will then be noted (~e_i(p)), and its dual basis(eⁱ(p)).

R.2.2 Field of functions

Letf ∈ F(U;R)be a function. The associated function eldfeis

fe:

(U →U×R

p →fe(p) := (p;f(p)), (R.5)

andpis called the base point. SoImfe={(p;f(p)) :p∈U}is the graph of f. (A eld of functions is of Eulerian type, not Lagrangian.)

Let T₀⁰(U) be the set of function elds on U, called the set of ⁰₀ type tensor on U, or tensors of order0 on U. In T₀⁰(U)(and for any type tensor), the internal sum and the external multiplication on the ring F(U;R) are dened by, for f ,eeg ∈ T₀⁰(U) with fe(p) = (p;f(p)) and eg(p) = (p;g(p)), and for ϕ∈ F(U;R):

( (fe+eg)(p) := (p; (f+g)(p)) (= (p;f(p) +g(p))), and

(ϕfe)(p) := (p; (ϕf)(p)) (= (p;ϕ(p)f(p))). (R.6) So the base pointpis kept, and (R.1) and (R.4) are applied: (R.6) models the actual computation made by an observer located atp(no gift of ubiquity). Abusive notations:







fe(p)^written= f(p) instead off(p) = (p;e f(p)), and T₀⁰(U)^written= F(U;R).

(R.7) It lightens the notations, but keep the base point in mind (Eulerian functions, no ubiquity gift).

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R.2.3 Vector elds

Classical mechanics: Let w~ ∈ F(U, E) be a vector valued function (suciently regular, at least Lips- chitzian, to get integral curves, cf. CauchyLipschitz theorem). The associated vector eldwe~ is

e~ w:

(U →U×E

p →w(p) = (p;e~ w(p)).~ (R.8)

SoImew~ ={(p;w(p)) :~ p∈U}is the graph ofw~: the vectorw(p)~ is drawn atp(the base point). (A vector eld is of Eulerian type, not Lagrangian.)

Let

Γ(U) := the set of vector elds onU . (R.9)

Abusive notation:

e~

w(p)^written= w(p)~ instead ofw(p) = (p;e~ w(p)).~ (R.10) It lightens the notations, but keep the base point in mind (Eulerian functions, no ubiquity gift).

More precisely, we will use the following full denition of vector elds (see e.g. AbrahamMarsden [1]):

A vector eld is built from tangent vectors to curves. It makes sense on non planar surfaces, and more generally on dierential manifolds.

E.g., see 11.5.2 for a fundamental example in mechanics.

R.3 Dierential forms, covariance and contravariance

Reminder: If f : U → R be C¹, then its dierential df : U → E^∗ is called an exact dierential form. Recall: If p∈ U then df(p)∈ E^∗ =L(E;R) is the linear form dened on E by, for all ~u ∈E, df(p).~u:= limh→0f(p+h~u)−f(p)

h .

And if U is a non planar surface, thendf(p).~u:= lim_h→0^f(c^p^(h))−f(c_h ^p⁽⁰⁾⁾ wherecp:h→cp(h)∈U is a regular curve s.t.cp(0) =pandcp0(h) =~u. (IfU were planar: cp(h) =p+h~u+o(h)).

An exact dierential form is a particular case of a dierential form:

R.3.1 Dierential forms

The basic concept is that of vector elds. A rst degree of complexity (a rst overlay) is introduced with the dierential forms with are functions dened on vector elds:

Denition R.2 Letα∈ F(U;E^∗)(so, ifp∈U thenα(p)∈E^∗=L(E;R), i.e., α(p)is a linear form at eachp. The associated dierential form (also called a1-form)αeis

αe:

(U →U×E^∗

p →α(p) = (p;e α(p)). (R.11)

SoImαe ={(p;α(p)) :p∈ U} is the graph of α: α(p)is drawn at point p(base point). (A dierential form is of Eulerian type, not Lagrangian.)

Let

Ω¹(U) := the set of dierential formsU . (R.12) Thus, ifαe ∈ Ω¹(U)(dierential form) and we~ ∈Γ(U) (vector eld), thenα.eew~ ∈ T₀⁰(U)(scalar valued), and

α.ewe~ :

(U →U×R

p →(α.eew)(p) = (p,~ (α. ~w)(p)) = (p, α(p). ~w(p)) ∈U ×R. (R.13) Abusive notation:

α(p)e ^written= α(p) instead ofα(p) = (p, α(p)).e (R.14) It lightens the notations, but keep the base point in mind (Eulerian functions, no ubiquity gift).

Remark R.3 Thermodynamic: Let U be the internal energy. Then its dierential dU is a (exact) dierential form (rst principle of thermodynamic). The elementary workw=δW is a dierential form which is not exact in general (it doesn't derive from a potential in general, e.g. because of frictions losses).

And (thus) the elementary heatq=δQ:=dU−δW is a non exact dierential form in general.

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Denition R.4 A vector eldw~ is said to be contravariant.

A dierential form α (a 1-form), which is a function acting on the vector elds w~, is said to be covariant.

(See Misner, Thorne, Wheeler [11] box 2.1: Without it [the distinction between covariance and contravariance], one cannot know whether a vector is meant or the very dierent () object that is a 1-form.)

R.4 Denition of tensors

The basic concept is that of vector elds and of dierential forms (rst degree of complexity). A second degree of complexity (a second overlay) is introduced with the tensors with are functions dened on vector elds and on dierential forms.

We need uniforms tensors T ∈ L^r_s(E), cf. Q.3. Framework of R.2.1.

The following denition of tensors (or tensor elds) enables to exclude the derivation operators which areR-linear but are not tensors, cf. remark R.1.

Denition R.5 (See e.g. AbrahamMarsden [1].) Let r, s ∈ N, r+s ≥ 1. Let T :

(U → L^r_s(E) p →T(p)

)

(soT(p)is a uniform ^r_s

tensor for each p, cf. (Q.3.1)). The associated functionTe

Te:

(U →U× L^r_s(E)

p →Te(p) = (p;T(p)) (R.15)

is a tensor of type ^r_siTisC^∞(U;R)-multilinear (not onlyR-multilinear), that is, for allf ∈C^∞(U;R), for allz1, z2where applicable, for allp∈U,

T(p)(..., f(p)z1(p) +z2(p), ...) =f(p)T(p)(..., z1(p), ...) +T(p)(..., z2(p), ...), (R.16) writtenT(..., f z1+z2, ...) =f T(..., z1, ...) +T(..., z2, ...), or

Te(..., f z1+z2, ...) =fTe(..., z1, ...) +Te(..., z2, ...). (R.17) Remark: ImT ={(p;T(p)) :p∈U}is the graph ofT, andT(p)is drawn at pointp(base point). (A tensor is of Eulerian type, not Lagrangian.)

Let

T_s^r(U) := the set of ^r_stype tensors onU . (R.18) And letT₀⁰(U) :=C^∞(U;R)the set of function elds, cf. (R.5).

Abusive notation:

Te(p)^written= T(p) instead ofT(p) = (p, Te (p)). (R.19) It lightens the notations, but keep the base point in mind (Eulerian functions, no ubiquity gift).

Example R.6 Fundamental counterexample. See remark R.1.

R.5 Example: Type

⁰₁

tensor = dierential forms

LetT ∈T₁⁰(U), soT(p)∈ L⁰₁(E) =L(E;R) =E^∗. ThusT is a dierential form: T₁⁰(U)⊂Ω¹(U). Converse: Does a dierential form α ∈ Ω¹(U) denes a ⁰₁ type tensor on U? That is, is (R.17) veried byα? That is, isα(f ~w) =f α(w)~ for allf ∈ F(U;R)andw~ ∈Γ(U)? That is, is (R.16) veried byα? That is, isα(p)(f(p)~w(p)) =f(p)α(p)(w(p))~ for allf ∈ F(U;R), allw~ ∈Γ(U)and allp∈U? The answer is yes sincef(p)∈R,w(p)~ ∈E andα(p)isR-linear on E(there is no dierentiation involved).

Therefore

T₁⁰(U) = Ω¹(U). (R.20)

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R.6 Example: Type

¹₀

tensor = identied to a vector eld

LetT ∈T₁⁰(U), soT(p)∈ L¹₀(E) =L(E^∗;R) =E^∗∗ for allp∈U. Thus, thanks to J, cf. (Q.11),T(p) can be identied to a vector, thusT₁⁰(U)can be identied to a subset of Γ(U).

Converse: Does a vector eld w~ ∈ Γ(U)denes a ¹₀ type tensor on U? That is, is (R.17) veried byw~? That is, withw=J(w)~ , w(f α) =f w(α)for allf ∈ F(U;R)andα∈Ω¹(U)? That is, is (R.16) veried byw~? That is, is w(p)(f(p)α(p)) =f(p)w(p)(α(p))for all f ∈ F(U;R), allα∈Ω¹(U)and all p∈U? That is, isf(p)(α(p). ~w(p)) =f(p)α(p). ~w(p)? Yes!

Therefore (identication)

T₀¹(U)'Γ(U). (R.21)

R.7 Example: A metric is a type

⁰₂

tensor

LetT ∈T₂⁰(U), soT(p)∈ L⁰₂(E) =L(E^∗;R) =E^∗∗.

Denition R.7 A metricsgonU is a ⁰₂type tensor onU such that, for allp∈E,g(p) =^writtengp(·,·) is a dot product onE.

R.8 Example: Type

¹₁

tensor...

LetT ∈T₁¹(U), soT(p)∈ L¹₁(E)andT(p)(α(p), ~w(p))∈Rfor allα∈Ω¹(U)and allw~ ∈Γ(U).

R.9 ... and identication with elds of endomorphisms

LetL: p∈U →L(p)∈ L(E;E), so L(p)is an endomorphism. The associated eld of endomorphisms onU is

Le:

(U →U× L(E;E)

p →L(p) = (p, L(p)).e (R.22)

(SoL(p) =^writtenLpis an endomorphism inE for anyp∈E.)

Abusive notation: L(p) =e ^writtenL(p)instead ofL(p) = (p;e L(p)), to lighten the notations.

And we have the natural (i.e. independent of an observer) canonical isomorphismJ₂, cf. (Q.44). So we can identify a eld of endomorphismsLand the ¹₁tensorT_L=J₂(L): for allp∈U,

∀`p∈E^∗, ∀w~p∈E, TL_p(`p, ~wp) =`p.(Lp. ~wp), and TL_p =J2(Lp)^written= Lp. (R.23) Example R.8 Letw~ be a vector eld onU. ThenL=d ~wis a eld of endomorphisms onU. Reminder:

ifp∈U and w~ ∈C¹, thenL(p) =d ~w(p)∈ L(E;E)is dened by d ~w(p).~u= lim_h→0^w(p+h~^~ ^u)−_h ^w(p)^~ ∈E, for all~u∈E. AndL=d ~wis identied to the ¹₁tensorL_T =J₂(L).

Andd ~wis a tensor∈T₁¹(U). Indeedd ~w(p).(f(p)~u1(p) +~u2(p)) =f(p)d ~w(p).~u1(p) +d ~w(p).~u2(p)since L(p) =d ~w(p) isR-linear by denition of a dierential. (We don't deal with the derivation d, but with the resultd ~wof the derivation.)

So, if(~ei)is a basis, ifw_|jⁱ :=eⁱ.d ~w.~ej, then

d ~w=

n

X

i,j=1

w_|jⁱ~e_i⊗e^j, i.e. d ~w.~e_j =

n

X

i=1

wⁱ_|j~e_i ∀j (R.24)

thanks to the contraction rule (Q.23). (Einstein convention is satised.)

R.10 Example: Type

²₀

tensor...

Same steps to dene ²₀tensors, or more generally ^r_stensors.

R.11 Unstationary tensor

Let t ∈ [t1, t2] ⊂ R. Let (Tet)_t∈[t₁_,t₂_] be a family of ^r_s tensors, cf. (R.15). Then Te : t → Te(t) := Tet

is called an unstationary tensor. And the set of instationary tensors is also notedT_s^r(U). Example R.9 A Eulerian velocity eld is a ¹₀

unstationary vector eld.