1.12 Euclidean Point Space
In this section a mathematical model of the ordinary three-dimensional Euclidean space will be given. A setEn, whose elements will be denoted by capital lettersA, B,: : :, will be called ann-dimensional affine spaceif a mapping
f W.A; B/2EnEn!f .A; B/!
AB 2En; (1.92)
exists between the pairs of elements ofEnand the vectors of a realn- dimensional vector spaceEn, such that
1.
!AB D ! BA;
!AB D! AC C!
CBI (1.93)
2. for anyO2Enand any vectoru2En, one and only one elementP 2Enexists such that
!OP Du: (1.94)
Then the elements ofEnwill be calledpoints; moreover, the pointsA; BofEnthat correspond to!
AB, are called the initial and final point of!
AB, respectively.
Aframe of reference.O;ei/in ann-dimensional affine spaceEnis the set of a pointO2Enand a basis.ei/ofEn. The pointOis called theoriginof the frame.
For a fixed frame, the relation (1.94) can be written as
!OP Duiei; (1.95)
where the real numbersui denote the contravariant components ofuin the basis .ei/. These components are also called therectilinear coordinatesofP in the frame .O;ei/. In order to derive the transformation formulae of the rectilinear coordinates (Sect.1.1) of a pointP 2 E3for the frame change.O;ei/! .O0;e0i/, we note that
!OP D! OO0C!
O0P : (1.96)
Representing the vectors of (1.96) in both the frames, we have uiei DuiO0eiCu0je0j;
whereuiO0 denote the coordinates ofO0 with respect to.O;ei/. Recalling (1.25), from the previous relation we derive
ui DuiO0CAiju0j: (1.97) Finally, ifEnis a Euclidean vector space, then Enis called a Euclidean point space. In this case, the distance between two points Aand B can be defined as the length of the unique vectorucorresponding to the pair .A; B/. To determine the expression for the distance in terms of the rectilinear coordinates of the above points, we note that, in a given frame of reference.O;ei/, we have:
!ABD! AOC!
OB D!
OB!
OAD.uiBuiA/ei; (1.98) whereuiB,uiAdenote the rectilinear coordinates ofBandA, respectively. Therefore, the distancejABjcan be written as
jABj D q
gij.uiB uiA/.ujB ujA/ ; (1.99) and it assumes the Pitagoric form
jABj D vu utXn
iD1
.uiBuiA/2 (1.100)
when the basis.ei/is orthonormal.
1.13 Exercises
1. In Sect.1.3the vectors of dual basis have been defined by (1.18). Then, it has been proved that
ei ej Dıij: Prove that, this condition, in turn, leads to (1.18).
Hint: If we put ei D ijej, then from the above condition we obtain ijgij Dıij.
2. Prove that (1.38) is equivalent to the following formula
.uv/wDp g
ˇˇˇˇ ˇˇ
u1 u2 u3 v1 v2 v3 w1 w2w3 ˇˇˇˇ ˇˇ:
1.13 Exercises 33 In particular, verify that the elementary volumed V of the three-dimensional Euclidean point space in the coordinates.yi/can be written as follows
d V Dp
gdy1dy2dy3:
3. Prove the invariance under base changes of the coefficients of the characteristic equation (1.64). Use the transformation formulae (1.53) of the mixed compo-nents of a tensor.
4. Using the properties of the cross product, determine the reciprocal basis.ei/of .ei/; iD1; 2; 3.
The vectors of the reciprocal basis.ei/satisfy the condition (1.19) ei ej Dıij;
which constitutes a linear system ofn2 equations in then2 unknowns repre-sented by the components of the vectorsei. Forn D 3, the reciprocal vectors are obtained by noting that, sincee1 is orthogonal to bothe2 ande3, we can write
e1Dk.e2e3/:
On the other hand, we also have
e1e1Dk.e2e3/e1D1;
so that
1
k De1e2e3: Finally,
e1Dke2e3; e2 Dke3e1; e3Dke1e2:
These relations also show thatk D 1and the reciprocal basis coincides with .ei/, when this is orthonormal.
5. Evaluate the eigenvalues and eigenvectors of the tensorTwhose components in the orthonormal basis.e1;e2;e3/are
0
@ 4 21 2 4 1 1 1 3
1
A: (1.101)
The eigenvalue equation is
TuDu:
This has nonvanishing solutions if and only if the eigenvalueis a solution of the characteristic equation
det.TI/D0: (1.102)
In our case, the relation (1.102) requires that
det 0
@4 2 1
2 4 1
1 1 3 1 AD0;
corresponding to the following third-degree algebraic equation in the unknown:
3C11234C24D0;
whose solutions are
1D1; 2D4; 3D6:
The components of the corresponding eigenvectors can be obtained by solving the following homogeneous system:
Tijuj Dıijuj: (1.103)
ForD1, Eq. (1.103) become
3u1C2u2u3 D0;
2u1C3u2Cu3 D0;
u1Cu2C2u3D0:
Imposing the normalization condition uiui D1 to these equations gives
u1 D 1
p3; u2D 1
p3; u3D 1 p3:
Proceeding in the same way forD4andD 6, we obtain the components of the other two eigenvectors:
1
p6; 1 p6; 2
p6
; 1
p2; 1 p2; 0
:
1.13 Exercises 35 From the symmetry ofT, the three eigenvectors are orthogonal (verify).
6. Letuandbe the eigenvectors and eigenvalues of the tensorT. Determine the eigenvectors and eigenvalues ofT1:
Ifuis an eigenvector ofT, then
TuDu:
Multiplying byT1, we obtain the condition T1uD 1
u;
which shows thatT1andThave the same eigenvectors, while the eigenvalues ofT1are the reciprocal of the eigenvalues ofT.
7. Let
g11Dg22 D1; g12D0;
be the coefficients of the scalar productg with respect to the basis.e1;e2/;
determine the components ofgin the new basis defined by the transformation matrix (1.101) and check whether or not this new basis is orthogonal.
8. Given the basis.e1;e2;e3/in which the coefficients of the scalar product are g11Dg22 Dg33D2; g12Dg13 Dg23D1;
determine the reciprocal vectors and the scalar product of the two vectorsuD .1; 0; 0/andvD.0; 1; 0/.
9. Given the tensorTwhose components in the orthogonal basis.e1;e2;e3/are 0
@1 0 0 0 1 1 0 2 3
1 A;
determine its eigenvalues and eigenvectors.
10. Evaluate the eigenvalues and eigenvectors of a tensorTwhose components are expressed by the matrix (1.101), but use the basis.e1;e2;e3/of the Exercise 6.
11. Evaluate the component matrix of the most general symmetric tensorTin the basis.e1;e2;e3/, supposing that its eigenvectorsuare directed along the vector nD.0;1; 1/or are orthogonal to it; that is, verify the condition
nuD u2Cu3D0:
Hint: Take the basis formed by the eigenvectorse01 D .1; 0; 0/,e02 D .0; 1; 1/
andn:
12. An axial vectorutransforms according to the law ui D ˙Ajiuj:
Verify that the vector productuvis an axial vector when bothuandvare polar and a polar vector when one of them is axial.
13. Let En be a Euclidean n-dimensional vector space. If u is a vector of En, determine the linear map
PWEn!En;
such thatP.v/uD0,8v2En. Determine the representative matrixP ofPin a basis.ei/ofEn.
Hint: The component ofvalonguisuvso thatP.v/Dv.uv/uDvu˝uv.
14. Let S andT be two endomorphisms of the vector space En. LetW be the eigenspace belonging to the eigenvalue of T. If S andTcommute, i.e., if STDTS, prove thatSmapsW into itself. Furthermore,Smaps the kernel of Tinto itself.
Hint: IfT.u/D0, thenTS.u/D ST.u/D 0so thatSmaps kerTinto itself.
Now,W is the kernel ofTI.Scommutes with this mapping if it commutes withT. Recalling what has just been shown,SmapsW into itself.
15. LetSandTbe 2-tensors of the Euclidean vector spaceEn. Suppose that both SandTadmitnsimple eigenvalues. Denote by1; : : : ; nthe eigenvalues of Sand with.ui/the corresponding basis of eigenvectors. Finally, we denote by 1; : : : ; n and.ui/the eigenvalues and the basis of eigenvectors ofT. Prove that the bases.ui/and.ui/coincide if and only ifSandTcommute.
Hint: Ifui Dui,iD1; : : : ; n, we have that .STTS/ui D iSuiiTui D
iiii D0; (1.104)
so that
STTSD0: (1.105)
IfSandTcommute, we can write what follows TSui DSTui DiSui;
andSuiis an eigenvector ofTbelonging to the eigenvaluei. Buti is simple and the corresponding eigenspace is one-dimensional. Therefore,Sui D ui
1.13 Exercises 37 and we conclude that any eigenvector ofTis also an eigenvector ofS. Repeating the above reasoning starting fromSTui, we prove that any eigenvector ofSis also an eigenvector ofT.
16. Taking into account the results of Exercises 14 and 15, extend the result of Exercise 15 when both the 2-tensors S and T on the vector space En are symmetricand they still admit bases of eigenvectors but their eigenvalues are not simple. In other words, prove that when the symmetric 2-tensorsTandS are symmetric and have bases of eigenvectors,TandScommute if and only if they have at least a common basis of eigenvectors.
Hint: We have already proved in the Exercise 15 that ifTandShave a common basis of eigenvectors, then they commute. Denote byWithe vector subspace of eigenvectors belonging to the eigenvaluei ofTand letni be the dimension ofWi. If1; : : : ; r are all the eigenvectors ofT, it isW1C CWr D En. Further, we have already shown that ifTandScommute, then the restriction of StoWiis an endomorphism ofWi. Since this restriction is symmetric, it admits a basis of eigenvectors ofWi, that are also eigenvectors ofTbelonging toi. These results hold for any subspaceWiand the problem is solved.
17. LetEnbe a Euclidean vector space and denote byVmam-dimensional vector subspace ofEn. It is evident that the vector set
W D fw2EnWwu;8u2Vmg
is a vector subspace formed by all the vectors that are orthogonal to all the vectors ofVm. Prove thatEnis the direct sum ofVmandW.
Hint: It has to be proved that
• Vm
TW D0;
• Anyu 2 En can be written asu D vCw, wherev 2 Vnandw 2 W are uniquely determined.
In fact, ifu2Vm
TW, then it isuuD0so thatuD0. Further, denoting by .ei/,iD1; : : : ; m, a basis ofVm, for anyu2Enwe define the vectors
vD Xn iD1
.uei/ei; wDuv (1.106)
where the vectorw2W sincewvD.uv/vD0. If another decomposition ofuexists, we have that
uDvCwDv0Cw0; that is
w0wDvv0:
Butw0w2W andvv02Vm. Since they are equal, they coincide with0.
18. Prove that the map
PWu2En!v2Vm;
wherevis given by (1.106), is linear, symmetric and verifies the condition P2DI:
The mapPis called the projection map ofEnonVm.
1.14 The Program VectorSys Aim of the Program VectorSys
The programVectorSys determines, for any system † of applied vectors, an equivalent vector system†0and, when the scalar invariant vanishes, its central axis.
Moreover, it plots in the space both the system†and†0, as well as the central axis.