CARTESIAN TENSORS

2. CARTESIAN TENSORS

In Chapter 3, Section 7, we considered the effect of rotations on vectors, and empha- sized active rotations (vector rotated, axes fixed). Now we want to consider passive rotations (vector fixed, axes rotated), in order to find how the components of a dis- placement vector in one coordinate system are related to its components in a rotated system. Let (x, y, z) be a set of rectangular axes and (x ′ ,y ′ ,z ′ ) another set obtained by rotating the axes in any manner keeping the origin fixed (Figure 2.1). In the table (2.1), we list the cosines of the nine angles between the (x, y, z) axes and the (x ′ ,y ′ ,z ′ ) axes.

z ′ l 3 m 3 n 3 Figure 2.1 In the table, l 2 means the cosine of the angle between the x axis and the y ′ axis,

etc. A vector r (Figure 2.1) has components x, y, z or x ′ ,y ′ ,z ′ relative to the two coordinate systems; we want to find the relations between the two sets of components.

Example 1. Let i, j, k be unit basis vectors along the (x, y, z) axes and i ′ ,j ′ ,k ′

be unit basis vectors along the (x ′ ,y ′ ,z ′ ) axes. Then the vector r can be written in terms of either set of components and basis vectors as follows:

r = ix + jy + kz = i ′ x ′ +j ′ y ′ +k ′ z ′ .

Taking the dot product of this equation with i ′ , we get (2.3)

r·i ′ =i·i ′ x+j·i ′ y+k·i ′ z=x ′

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(since i ′ ′ ·i = 1, and i ′ ′ ·j =i ′

·k ′ = 0). Now i · i is the cosine of the angle between

i and i ′ , that is, between the x and x ′ axes, since i and i ′ are unit vectors; thus i·i ′ =l 1 from the table (2.1). Similarly, j · i ′ =m 1 and k · i ′ =n 1 and (2.3) becomes

x ′ =l 1 x+m 1 y+n 1 z.

Similarly, dotting r into j ′ nd k ′ , and using (2.1) we get

The equations (2.4) and (2.5) are called the transformation equations from the coordinate system (x, y, z) to (x ′ ,y ′ ,z ′ ).

Example 2. In the same way, dotting r with i, j, k in turn, we get equations for x, y, z in terms of x ′ ,y ′ ,z ′ :

These transformation equations may be written more concisely in matrix notation. Equations (2.4) and (2.5) become the matrix equation:

or r ′ = Ar, z ′

where r ′ , A, and r stand for the matrices in (2.7). [Compare Chapter 3, equation (7.13) for the two-dimensional case.] Similarly, (2.6) becomes

r=A T r ′

where A T is the transpose of A. Recall from Chapter 3, Sections 7 and 9, that a rotation matrix is an orthogonal matrix, and for an orthogonal matrix, A T =A −1 . Also see Problems 3 and 4.

Equations (2.7) or (2.8) tell us how displacement vectors in a rectangular co- ordinate system transform under a rotation of axes. We now use this result to define Cartesian vectors, that is, vectors which transform in the same way that displacement vectors do under rotations of rectangular (Cartesian) axes. We will then generalize this to define Cartesian tensors of other ranks.

Definition of Cartesian Vectors

A Cartesian vector V consists of a set of three numbers (components) in every rectangular coordinate system; if V x ,V y ,V z are the components in one system and V ′ x ,V ′

y ′ ,V z are the components in a rotated system, these two sets of components are related by an equation similar to (2.7), namely,

where A is the rotation matrix in (2.7). Alternatively, we could use (2.8) and require that V = A T V ′ .

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We can simplify our notation by making the following changes. Replace x, y, z

by x 1 ,x 2 ,x 3

Replace x ′ ,y ′ ,z ′

by x ′ 1 ,x ′ 2 ,x ′ 3

Replace V x ,V y ,V z

by V 1 ,V 2 ,V 3

(2.10) Replace V ′ x ,V ′ ′

y ′ ,V z by V 1 ,V 2 ,V 3

a  11 a 12 a 13 Replace A in (2.7)

by  a 21 a 22 a 23 

a 31 a 32 a 33 In this notation (2.7) and (2.9) become (2.11) and (2.12):

Alternatively, we could solve (2.11) for the x coordinates in terms of the x ′ coor-

dinates as in (2.8), to get, in the summation form: x i =

a ji x ′ j , and a similar

j=1

companion formula to (2.12), namely

Since we will occasionally want the transformation formula for a Cartesian vector solved for the unprimed components as in (2.13), you should be sure you understand (2.13). Compare carefully the indices in (2.12) and (2.13). In matrix form (2.12) is

V ′ = AV and (2.13) is V = A T V ′ [see equations (2.7) and (2.8)]. Now element i, j of A T is the same as element j, i of A, so the coefficients in (2.13) are a ji instead of

a ij as they were in (2.12). It is now straightforward to define tensors. Definition of Cartesian Tensors

A tensor of rank zero has one component which is unchanged by a rotation of axes; it is called an invariant or a scalar. Simple examples are the length of a vector, or the dot product of two vectors. A first rank tensor is just a vector. A tensor of second rank has nine components (in three dimensions) in every rectangular coordinate system. If we call the components in one system T ij , the components T ′ kl in a rotated coordinate system are given by (2.14), where the a’s are the direction cosines in the rotation matrix A.

(2.14) T ′ kl =

a ki a lj T ij ,

k, l = 1, 2, 3.

i=1 j=1

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Direct Product We can give a very simple example of a second-rank tensor. Example 3. Let U and V be vectors; we form the following array (in each coordinate

system) from the components U 1 ,U 2 ,U 3 and V 1 ,V 2 ,V 3 of U and V (in that coordinate system):

We can show that these nine quantities are the components of a second-rank tensor which we shall denote by UV. Note that this is not a dot product or a cross product; it is called the direct product of U and V (or outer product or tensor product ). Since U and V are vectors, their components in a rotated coordinate system are, by (2.12):

Hence the components of the second-rank tensor UV are

i,j=1

which is just (2.14) with T ij =U i V j and T ′ kl =U ′ k V l ′ .

Equation (2.14) generalizes immediately. For example, a 4 th -rank Cartesian ten- sor is defined as a set of 3 4 or 81 components T ijkl , in every rectangular coordinate system, which transform to a rotated coordinate system by the equations

T ′ αβγδ =

a αi a βj a γk a δl T ijkl ,

i,j,k,l

where i, j, k, l take the values 1, 2, 3. Note that a 4 th -rank tensor has 4 indices and requires four a’s in its definition. Similarly, an n th -rank tensor has n indices and requires n a’s in its definition. Also we can generalize (2.17) to show, for example, that the direct product of a vector and a 3 rd -rank tensor produces a 4 th -rank tensor, and, in general, the direct product of tensors of ranks n and m is a tensor of rank m + n (see Problem 7).

PROBLEMS, SECTION 2

1. Verify equations (2.6). 2. Show that the sum of the squares of the direction cosines of a line through the origin

is equal to 1 Hint: Let (a, b, c) be a point on the line at distance 1 from the origin. Write the direction cosines in terms of (a, b, c).

3. Consider the matrix A in (2.7) or (2.10). Think of the elements in each row (or col- umn) as the components of a vector. Show that the row vectors form an orthonormal triad (that is each is of unit length and they are all mutually orthogonal), and the column vectors form an orthonormal triad.

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4. Any rotation of axes in three dimensions can be described by giving the nine direction cosines of the angle between the (x, y, z) axes and the (x ′ ,y ′ ,z ′ ) axes. Show that the matrix A of these direction cosines in (2.7) or (2.10) is an orthogonal matrix. Hint: See Chapter 3, Section 9. Find AA T and use Problem 3.

5. Write equations (2.12) out in detail and solve the three simultaneous equations (say by determinants) for x 1 ,x 2 ,x 3 in terms of x ′ 1 ,x ′ 2 ,x ′ 3 to verify equations (2.13). Use your results in Problem 4.

6. Write the transformation equation for a 3 rd -rank tensor; for a 5 th -rank tensor. 7. Following what we did in equations (2.14) to (2.17), show that the direct product

of a vector and a 3 rd -rank tensor is a 4 th -rank tensor. Also show that the direct product of two 2 nd -rank tensors is a 4 th -rank tensor. Generalize this to show that the direct product of two tensors of ranks m and n is a tensor of rank m + n.

8. Write the equations in (2.16) and so in (2.17) solved for the unprimed components in terms of the primed components.