Vector and Tensor Analysis: Vectors in Euclidean and Riemannian Spaces, Exercises of Algebra

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TENSORS

P. VANICEK

September 1972

TECHNICAL REPORT

NO. 217

LECTURE NOTES

TENSORS

(Third Printing)

Petr Vanicek

Department of Surveying Engineering

University of New Brunswick

P.O. Box 4400

Fredericton, N .B.

Canada

E3B 5A

September 1977

Latest Reprinting December 1993

PREFACE FOR FIRST PRINTING

This cours.e is beip.g o.fi'ered to the post"';graduate students in Su:rveying Engineering. Its aim is: to give a baS;ic knowledge oi' tensor "language" that can be applied i'or solving s-ome problems in photogra:m:rnetry and geodesy. By no :means-, can the course claim any: completeness; the emphasis is on achieving a basic understanding ana, perhaps, a deeper insight into a i'ew i'unda:mental questions oi' dii'i'erential geometry. The course is divided into three parts: The i'irst part is a very brief recapitulation oi' vector algebra ana analysis as taught in the undergraduate courses. Particular attention is paid to the appli- cations of vectors in differential geometry. The second part is :meant to provide a link between the concepts of vectors in the ordinary Eucleidean space and generalized Riemannian space. The third, and the :most extensive of all the three parts, deals with the tensor calculus in the proper sense. The course concentrates on giving the theoretical outline rather than applications. However, a number of solved and :mainly unsolved problems is provided for the students who want to apply the theory to the "real world" of photograrn:metry and geodesy. It is hoped that :mistakes and errors in the lecture notes will be charged against the pressure of time under which the author has worked when writing them. Needless to say that any comment and criticism communi- cated to the author will be highly appreciated.

P. Yani~ek 2/ll/

PREFACE FOR SECOND PRINTING

The second printing of these lecture notes is basically the same as the first printing with the exception of Chapter 4 that has been added. This addition was requested by some of the graduate students who sat on this course. I should like to acknowledge here comments given to me by

Dr. G. Blaha and Mr. T. Wray that have helped in getting rid of some

errors in the first printing as well as in clarifying a few points.

P. Vanlcek^ lv 12/7/

• 1. Vectors in Rectangular Cartesian Coordinates ...•.............
     • 1.1) BasiC4t Definitions
     • 1.2) Vector Algebra · - 1.2.1) Zero Vector r · · ·. · · ~ · ·. · - 1. 2. 2) Unit Vectors - 1.2.3) Summation of Vectors - 1. 2. 4) Multiplication by a Constant • • • - 1.2.5) Opposite Vector - 1.2.6) Multiplication of Vectors · - 1.2.7) Vector Equations ......•....•......................
     • 1. 3. Vector Analysis 1.2.8) Note on Coordinate Transformation .....•.......... ll
          • 1.3.1) Derivative Scalar .Arguments of a Vector Function of One and• Two
          • 1.3.2) Elements of Differential Geometry of Curves
          • 1. 3. 3. Elements of Differential Geometry of Surfaces · · • ·
          • 1. 3 ~.4) Differentiation of Vector and Scalar Fields · · · · · · ·
  • 2. Vectors in Other Coordinate Systems .• ·.·· .•...................
       • 2.1) Vectors in Skew Cartesian Coordinates •······•···········
       • 2.2) Vectors in Curvilinear Coordinates ·········•·············
       • 2.3) Transformation of Vector Components······················
• 3. Tensors · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · - 3.1) Definition of Tenso:tl····················•················ - 3.2) Tensor Field, Tensor Equations········· • · · • · · · · · · · • · · · · • • - 3. 3) Tensor Algebra · · · · · · · · · · · · · · · · · · · · · · • • · • · · · • · · · · · · • · · · · · · - 3.3.1) Zero Tensor - 3. 3. 2) Kronecker 0. - 3.3.3) Summation of Tensors ......•...•.•.......•......... - 3. 3. 4) Multiplication by a Constant .....•••............•. - 3.3.5) Opposite Tensor - 3.3.6) Multiplication of Tensors •. , •••...........•••..... - 3. 3. 8) Tensor Character 3. 3. 7) Contraction • • 42· - 3.3.9) Symmetric and Antisymmetric Tensors ........•...... - 3.3.10)Line3. 3.11) Terminological Element and RemarksMetric ..•...•.•.•..•••.....•.....Tensor •··················· - 3.3.13) Scalar of Indeces Product of Two Vectors, Applications .....• 3.3.12) Associated Metric Tensor, Lowering and Rising - 3.3.14) Levi-Civitl Tensor: •..................•.......... - 3. 3.15) Vector Product of Two Vectors • • •
  • 3. 4. Tensor Analysis · · · · ·
          • 3.4.1) Constant Vector Field
          • 3. 4. 2. Christoffel Symbols ......•...........•..•.•..... ·
          • 3.4.3) Tensor Argument Derivative (Intrinsic with Derivative) Respect to...•...•...••....• Scalar
            • Coordinates Derivatives) (Covariant. and Contravariant 3.4.4) Tensor Derivative with Respect to
          • 3.4.5) V and A Operators in Tensor Notation .........•...
          • 3.4.6) Riemann-Christoffel Tensor ·.·•··· ·····.···•·
          • 3.4.7) Ricci-Einstein and Lam~ Tensors··················
            • ification of Spaces · · · · · · · · · · · · · • · · · · 3.4.8) Gaussian Curvature of a Surface, Class-
    • of Surfaces 4) SGme Applications of Tensors'.oi:rii.Dffferential GeometTy · ·
      • 4.1) First and Second Fundamental Forms of a Surface
      • 4.2) Curvature of a Surface Curve at a Point •...........
• Recommended4. 3) Euler_'References s Equation for Further ..•.............• Reading ..•.....•............ , ......, .••••.•..•.

1) VECTORS IN RECTANGULAR CARTESIAN COORDINATES

1.1) Basic Definitions

The Cartesian power E3 , where Eisa set of real numbers, is called the System of Coordinates in three-dimensional space (futher only 3D-space). Any element 1EE3^ is said to describe a point in the space, the elements ~~being obviously ordered triplets of real numbers. It is usual to denote them thus:

""±: = (x, y, z)

If the distance of any two points, +r^1 and ~r 2 say, is given by

where

then the system of coordinates is known as Rectangular Cartesian. This distance metricizes (measures) the space and this particular distance (metric) is known as the Eucleidean metric. The appropriate metric space (called usually just simply space) is called the Eucleidean space. The graphical interpretation given here is well known from the elementary geometry.

z

y

direction of +A. Note that every one of the three above expressions is dependent on the other two. Squaring the equation for the absolute value and dividing it by A^2 we get

(^ .A 2.)^2 + (:;[)A^2 + (2.)A^2 l A A A =.

This can always be done if A is different from zero and A ~ 0 if and only if at least one of the components is different from zero. This leads to a statement, that a vector of zero length has got an undeter- mined direction. Further, we can see that the point +r can be regarded as a special case of a vector, whose argument is always the center of coordinates C:

• rc (^) = (0, 0, 0) = +o.

It is therefore also called the position vector or the radius-vector of the point. Hence we talk about the triplet of real functions +A as vector function of vector argument. A triplet of constant functions (real numbers) is called free vector, meaning that its absolute value and direction (as well as its components) are independent or free from the argument (point). On the

I

I I

other hand, if we have a vector function of &<>Vector argument defined for each point in a certain region RCE 3 of our space we say that there is a vector field defined in R. Thus obviously a free vector can be regarded as constant vector field and we shall refer to it as such.

4

It is useful to extend the definition of a field to one-valued real functions of a vector argument as well. If we have in a certain region

RCE 3 of our space a real function t of the position vector defined then

we say that

is a scalar field in R. We thus note that vector field is a vector function of a vector variable the scalar field is a scalar function of a vector variable. z One^ more^ useful^ quantitYcan^ be also defined here and that is a 4 ·o (^) vector function of a scalar i·l (^) variable, i.e. three-valued real

·- 17'2 (^) o·o functions^ of^ one^ real^ variable. This qua:ntityis often used whenever ·- 3'(; it is necessary to consider a X (^) varying parameter (real variable) in the space. This parameter can be time, length of a curve, etc. 7 Hence^ we^ may^ have,^ for^ instance, a vector defined along a curve K as a function of its length as shown on the diagram. The more or less trivial extension of this concept is the scalar function of a scalar variable or the well known real function of one real variable-known from the fundamentals of mathematical analysis.

6

The geometrical interpretation of the summation is shown on z (^) the diagram. Evidently the summation^ is^ commuta~ tive and associative, i.e.

• A+B=B+A + + + -~ As (^) (A++^ +B)^ + +C^ =A++^ (B+^ +C).+ The absolute value^ of^ the sum +C of the two vectors X y (^) Aand Bis given by ,., C = I(A 2 + B^2 + 2AB cos('lf~.- A13)) ~ where by AB we (^) ·~··denote -. ··-. ,·· .·... the an[:l:le between Aand B. The proo1· is left to the reader. Convention - From now on we shall be denot:tng x by x 1 , y by x (^2)

and z by x 3 • The corresponding components of a vector +A will accordingly

be A 1 , A 2 , A 3.

1.2.4 Multiplication of a Vecto~~X a Constant Vector +B^ is cal~ed the product of vector +A^ with. a constant k if and only if

Obviously

and

BX = kAX , By = kAy , BZ = kA (^) Z or B.l = kA.l

kA^ +^ = +Ak

B = kA. The direction of +B^ is identical to the direction of +A.

i = 1, 2, 3.

7

1.2.5) Opposite Vector

Vector +B^ is known as opposite vector to +A if and only if

7 A + +B :::: + 0 •

It is usual to denote the opposite vector to +A^ by -A+ because

• B (^) = (-1) A+

1.2.6 Multiplication of Vectors

i) Scalar Product +A^ · +B^ of two vectors +A^ and +B is the real number

(scalar) k given by

Scalar product is obviously commutative, (^) i.e. A·B = +B^ • +A, and it is not associative, i.e. +A^ "'^ (+B^ • +C)^ ':/: (A+^ • +B)^ '+C. The proof of the latter is left to the reader. The reader is also advised to show that

and (A^ +^ B)^ ·^ c^ =^ A·^ c^ +^ B

. c.^ + Obviously, the absolute value of a vector A can be written as A = I(J.. A).

Two non-zero vectors -+A,^ 4-B whose 3Calar product equals to zero are

perpendicular because AB cos AB^ <'1 = 0 implies that

cos AB = 0 and

iv) Mixed Product [ABC]+++^ of three vectors +A,^ +B,^ and +C is a scalar k defined as k =~~. (B X c) = [A Bc J.J Its value can be seen to be

k = A B C sin B~ sin AA'

where^ +t A^ is the projection of +A^ on the plane +B^ +C (see the diagram). But this equals to the volume of the parallelepiped given by the three

• 'I

13xcl

I I I

'^ '•, II^ '

'•r-',.. I^ '^ '

vectors. Evidently, the volume of this body equals to zero if ~· (^) and only if all the three vectors are coplanar (lay in one plane) or at least one of them is the zero-vector. Hence the necessary and sufficient condition for three non-zero vectors to be coplanar is that their mixed product equals to zero:

[A^ +++ B C]^ = 0 =+·+ A, B,^ +C E K

(K denotes a plane). It is left to the reader to prove that [ABC]^ +++ = [BCA] = [CAB] =- [ACB] =- [BAC] =- [CBA]. Problem: Show that a vector given as a linear combination of two other vectors +A^ and +B,^ i.e. +C^ = k 1 +A^ + k 2 +B,^ is coplanar with the vectors +A^ and +B.

1.2.7) Vector Equations Equations involving vectors are known as vector eguations. In three-dimensional geometric applications they invariably describe properties of various objects in 3D-space. For example, the vector equation

• B (^) = kA+ tells us that vectors +A and B^ +^ are parallel and vector +B is k-times longer than +A. Or the vector equation GOS (AB)^11 = +A^ • +B I (AB) determines the angle of two non-zero vectors +A^ and +B. If we decide, for some reason, to change the coordinate system, i.e. trans~orm the original coordinate system to another, the geometric properties of the objects do not change. Two straight lines remain parallel or perpendicular in any system of coordinates. Similarly one vector remains k-times longer than another whatever the system of coordinates may be. This fact is usually expressed by the statement that vector equations are invariant in any transformation of coordinate system. This is the basic reason why we prefer using vectors - and by this we mean here the described compact notation for the triplets of functions - when dealing with properties of objects in space. Another possibility would be to use coordinates instead but in that case formulae would be valid only in the one coordinate system and would not be invariant.

the original one). All the transformation matrices possessing these properties are known as constituting the group of Cartesian transformation matrices. Moreover, when talking about the invariance of lengths, we have to require that

Idet M! = l.

Obviously, the Cartesian transformation matrices are something very special. Later, we shall deal with a more general group of trans- formations. However, it is not considered the aim of this course to deal with transformations in detail.

1.3) Vector Analysis

1.3.1 Derivative of a Vector Function of One and Two Scalar Variables

The quantity

lim

tm + 0

• A(u (^) + Au) ~--~ +A(u) (^) = dA+ _^ Au^ ,....du _....... is called the derivative of vector +A with respect to its scalar argu- ment u. (^) It is sometimes denoted by -r, A. Geometrically, this derivative has important applications in differential geometry of curves as we shall see later. A vector function Aof two scalar arguments u and v has got two partial derivatives. These are defined completely analogously to the above case: dA^ +^ +A(u^ + Au, v (^) = const.) A(u, v = const.) -= au lim Au+ 0 Au -=^ ax^ lim +A(u^ =^ const.^ ,^ v^ +^ b.v)^ -^ A(u^ =^ const., v)

av b.v + 0 Av

Geometrically, these derivatives have a~~lications in differential geometry of surfaces and are somet:tmes denoted by +,A , +,A • Obviously, all

u v

the defined derivatives are again vectors. The rules for differentiation are very much the same as those for the differentiation of real functions. Particularly we have

d +^ + du CA^ +B)^ =-+-dAdu^ dBdu d (^) (kA) kdA^ + du = du

ddu CA • B) = A. .£!?.du + dAdu • i

d -? + + dB^ +^ dA+ + du (.A.^ X^ B)^ =^ A^ X^ - du^ + -du^ X^ B If A = const. then dA^ + • + du^ A

The ~roof of this theorem is left tb the reader. The rules ~or ~artial differentiation are analogous.

1.3.2) Elements of Differential Geometry of Curves If for all U€< a, b > a position-vector +r^ = +r(u) is defined, we say that +r describes a curve (s~atial curve in 3D-s~ace). The real variable u is called the parameter of the curve. Let us assume that

• r is in < a, b > a:,continuous £'litllct;lon and we shall hence talk about

. (^) Tf +r is in <a, b>> continuous, we can define another scalar function of u: