EXERCISES
2.1. SCALARS, VECTORS, TENSORS, NOTATION
2.2. Rotation of Axes: Formal Definition
of a Vector 42
2.3. Multiplication of Matrices 44 2.4. Second-Order Tensors 45 2.5. Contraction and Multiplication 47
2.6. Force on a Surface 48
2.7. Kronecker Delta and Alternating
Tensor 50
2.8. Vector, Dot, and Cross Products 51 2.9. Gradient, Divergence, and Curl 52
2.10. Symmetric and Antisymmetric
Tensors 55
2.11. Eigenvalues and Eigenvectors
of a Symmetric Tensor 56
2.12. Gauss’ Theorem 58
2.13. Stokes’ Theorem 60
2.14. Comma Notation 62
Exercises 62
Literature Cited 64
Supplemental Reading 64
C H A P T E R O B J E C T I V E S
• To define the notation used in this text for scalars, vectors, and tensors
• To review the basic algebraic manipulations of vectors and matrices
• To present how vector differentiation is applied to scalars, vectors, and tensors
• To review the fundamental theorems of vector field theory
2.1. SCALARS, VECTORS, TENSORS, NOTATION
The physical quantities in fluid mechanics vary in their complexity, and may involve multiple spatial directions. Their proper specification in terms ofscalars,vectors,and (second
39
Fluid Mechanics, Fifth EditionDOI:10.1016/B978-0-12-382100-3.10002-2 Ó2012 Elsevier Inc. All rights reserved.
order)tensorsis the subject of this chapter. Here, three independent spatial dimensions are assumed to exist. The reader can readily simplify, or extend, the various results presented here for fewer, or more, independent spatial dimensions.
Scalars or zero-order tensors may be defined with a single magnitude and appropriate units, may vary with spatial location, but are independent of coordinate directions. Scalars are typically denoted herein by italicized symbols. For example, common scalars in fluid mechanics are pressurep, temperatureT, and densityr.
Vectors or first-order tensors have both a magnitude and a direction. A vector can be completely described by its components along three orthogonal coordinate directions.
Thus, the components of a vector may change with a change in coordinate system. A vector is usually denoted herein by a boldface symbol. For example, common vectors in fluid mechanics are positionx, fluid velocityu, and gravitational acceleration g. In a Cartesian coordinate system with unit vectorse1,e2, ande3, in the three mutually perpendicular direc- tions, the position vectorx, OP inFigure 2.1, may be written
x¼e1x1þe2x2þe3x3, (2.1) wherex1,x2, andx3are the components ofxalong each Cartesian axis. Here, the subscripts of edonotdenote vector components but rather reference the coordinate axes 1, 2, and 3; hence, thees are vectors themselves. Sometimes, to save writing, the components of a vector are denoted with an italic symbol having one indexdsuch asi,j,orkdthat implicitly is known to take on three possible values: 1, 2, or 3. For example, the components ofxcan be denoted byxiorxj(orxk, etc.). For algebraic manipulation, a vector is written as a column matrix; thus, (2.1) is consistent with the following vector specifications:
x¼ 2 4x1
x2 x3
3
5 where e1¼ 2 41
0 0
3
5, e2¼ 2 40
1 0
3
5, and e3¼ 2 40
0 1
3 5:
Thetransposeof the matrix (denoted by a superscript T) is obtained by interchanging rows and columns, so the transpose of the column matrixxis the row matrix:
xT¼ ½x1 x2 x3:
2 3
1
O
x1 x2
x3
e1 e2 e3
P x
FIGURE 2.1 Position vector OP and its three Cartesian components (x1,x2,x3). The three unit vectors for the coordinate directions aree1,e2, ande3. Once the coordinate system is chosen, the vectorxis completely defined by its components,xiwherei¼1, 2, or 3.
2. CARTESIAN TENSORS
40
However, to save space in the text, the square-bracket notation for vectors shown here is typi- cally replaced by triplets (or doublets) of values separated by commas and placed inside ordi- nary parentheses, for example,x¼(x1,x2,x3).
Second-order tensors have a component for eachpairof coordinate directions and there- fore may have as many as 33¼9 separate components. A second-order tensor is some- times denoted by a boldface symbol. For example, a common second-order tensor in fluid mechanics is the stresss. Like vector components, second-order tensor components change with a change in coordinate system. Once a coordinate system is chosen, the nine compo- nents of a second-order tensor can be represented by a 33 matrix, or by an italic symbol having two indices, such assijfor the stress tensor. Here again the indicesiandjare known implicitly to separately take on the values 1, 2, or 3. Second-order tensors are further dis- cussed inSection 2.4.
A second implicit feature of index-based or indicial notation is the implied sum over a repeated index in terms involving multiple indices. This notational convention can be stated as follows:Whenever an index is repeated in a term, a summation over this index is implied, even though no summation sign is explicitly written. This notational convention saves writing and increases mathematical precision when dealing with products of first- and higher-order tensors. It was introduced by Albert Einstein and is sometimes referred to as the Einstein summation convention. It can be illustrated by a simple example involving the ordinary dot product of two vectorsaandbhaving componentsaiandbj, respectively. Their dot product is the sum of component products,
a,b¼a1b1þa2b2þa3b3¼X3
i¼1
aibihaibi, (2.2)
where the final three-linedefinitionequality (h) follows from the repeated-index implied- sum convention. Since this notational convention is unlikely to be comfortable to the reader after a single exposure, it is repeatedly illustrated via definition equalities in this chapter before being adopted in the remainder of this text wherever indicial notation is used.
Both boldface (aka,vectorordyadic) and indicial (aka,tensor) notations are used throughout this text. With boldface notation the physical meaning of terms is generally clearer, and there are no subscripts to consider. Unfortunately, algebraic manipulations may be difficult and not distinct in boldface notation since the productabmay not be well defined nor equal to ba when a and b are second-order tensors. Boldface notation has other problems too; for example, the order or rank of a tensor is not clear if one simply calls ita.
Indicial notation avoids these problems because it deals only with tensor components, which arescalars. Algebraic manipulations are simpler and better defined, and special atten- tion to the ordering of terms is unnecessary (unless differentiation is involved). In addition, the number of indices or subscripts clearly specifies the order of a tensor. However, the phys- ical structure and meaning of terms written with index notation only become apparent after an examination of the indices. Hence, indices must be clearly written to prevent mistakes and to promote proper understanding of the terms they help define. In addition, the cross product involves the possibly cumbersome alternating tensor3ijk as described inSections 2.7 and 2.9.
2.1. SCALARS, VECTORS, TENSORS, NOTATION 41