Download Lecture, Tuesday April 4 Physics 105C

Lecture, Tuesday April 4
Physics 105C
Continuum mechanics is founded on the “continuum hypothesis,” the assumption that we
can ignore the granularity of matter. (Note that there is an unrelated “continuum hypothesis”
in mathematics. In that case it’s the assumption that no set has size in between the size of
the integers and that of the real numbers.) Since a cubic centimeter of a typical solid material
has around Avogadro’s number, 6 × 1023 , atoms, describing macroscopic forces and responses on
millimeter or even micron scales if effectively averaging over large numbers of atoms.
density
density
The resulting descriptions of behavior are highly classical. Quantum effects can cause a
breakdown of the continuum hypothesis in a few ways. First, studying behavior on very small
length scales, where quantum mechanics plays an important role, means that atomic spacing
need no longer be tiny. On a sufficiently small length scale (below left), the density of a material
as a function of position would be a set of sharp peaks indicating atomic locations. Viewed on
a broader length scale (below right), the density would be averaged into a constant function:
position
position
In other cases, such as electrical conductivity (even for macroscopic materials!), quantum
mechanics is again crucial. It turns out that the Pauli exclusion principle governs the states
occupied by electrons in crystals, building up a “Fermi sea” from individual electron states. This
picture ultimately explains the difference between metals and insulators, and much more. Yet any
discussion of individual electron states acknowledges the ultimate particle nature of materials,
particularly since the electrons move in a periodic potential based on the exact locations of the
crystal atoms. There are rare situations where a continuum description holds despite quantum
influence. One example is superfluid helium, a quantum state that can exist on large (cm or
more) length scales and that simultaneously exhibits fluid behavior.
The course will cover a brief introduction to tensors, the Lagrangian and Eulerian descriptions
of particle motion, elasticity in solids (what happens when we squeeze), constitutive equations
(differential equations such as the continuity equation or diffusion equation), and fluid dynamics
including the Navier-Stokes equation in simple geometries and surface waves.
Today’s class will be on basics of tensor notation. I will in fact say more about tensors than
is strictly necessary for continuum mechanics, since the topic is important and many of you will
encounter it elsewhere.
I’ll use v i , with the i a superscript, to indicate the ith component of a column vector v:
v1


v =  v2 
v3


Similarly, wi with a subscript i is the ith component of a row vector:
w = (w1
w2
w3 )
It is the superscript or subscript that determines whether something is a column or row vector,
not the specific letter used. Thus wi is a column vector and vi is a row vector. Often the
notation for vectors in Cartesian coordinates doesn’t distinguish between column vectors and
row vectors, since as we’ll see it turns out that their transformation properties are identical. For
other coordinate systems though the transformation laws are not identical and it becomes crucial
to distinguish the two types of vectors. I find that it can be helpful to keep the distinction in
mind even for Cartesian vectors and transformations. Here is an example.
The inner product, or dot product, of two vectors is w · v = 3i=1 wi v i . This is exactly
matrix multiplication of a 1 × 3 matrix (row vector) w and an 3 × 1 matrix (column vector) v.
However, the matrix product vw is an entirely different object, a 3 × 3 matrix. (This is called
an “outer product.”) It’s entirely possible that a calculation might involve some outer products
and some inner products, and index notation is immensely convenient for keeping track of them.
Individual components such as wi are simply numbers, which commute with each other, so you
do not need to worry about the order the vectors are multiplied. Instead, you will always know
which is a “row” or “column” vector by whether it has an upper or lower index. Now add
to this a summation convention, where you always calculate the sum over repeated indices, although typically without writing the summation sign explicitly. The inner product then becomes:
P
w·v =
3
X
wi v i = wi v i = v i wi
i=1
where the final equality illustrates my comment about commutativity. The outer product is
represented as wj v i = v i wj , which represents the entry in row i and column j of the resulting
matrix. Since the two indices differ, there is no implied sum in this case. A crucial point about
the notation is that the summation must always be over exactly two identical indices, one upper
and one lower. Effectively that means that you can only take a dot product of a column vector
with a row vector, never with another column vector. (You can multiply two column vectors
through an outer product, but the result is a matrix, with two distinct indices.)
The notation extends beyond products of vectors. For a matrix times a column vector,
Av = aij v j , where there is a sum over j but not i. The result has one upper index (i), as
expected for a column vector. Alternatively, wA = wi aij , where w and the product are both row
vectors. Note that aij wi is exactly the same as wi aij ; the indices show that w is a row vector,
which in our clumsy matrix/vector notation would require w to be on the left but no longer
matters if we just keep proper track of the multiplication formula. A quick note of warning: for
now at least I am being very careful with the typesetting, so that the first matrix index gives
the row and the second gives the column. In other places that distinction won’t be maintained;
if both upper and lower indices are in use they may appear directly above and below each other,
with the assumption that the upper index identifies the row and the lower index the column.
For Cartesian coordinates everything may be written with lower indices, in which case at least
the ordering of the indices will be obvious.
Another multiplication of matrix by vector is aij v k . Here there are no sums since all three
indices used are distinct. Instead, the tensor has 33 = 27 entries, one for each combination of
values of i, j, and k. These could be pictured as arranged in a 3 × 3 × 3 cube.
Vectors are “tensors of rank 1,” and matrices are “tensors of rank 2.” Here rank is the total
number of indices the object has, whether they are upper or lower. The entries of a rank 3
tensor, as in the previous paragraph, could be arranged in a three-dimensional display. Higherrank tensors can easily be defined, although there is no nice visualization for a layout of their
entries. We’ll assume that each index refers to the same underlying vector space, which means
that the tensor entries form a square (or cube, or hypercube). The dimension of a tensor is the
number of rows or columns in a particular direction; it equals the dimension of the underlying
vector space. The rank of a tensor is the number of indices.
So far we’ve looked at rank 2 tensors with one upper and one lower index. These are familiar
matrices, with columns in one direction and rows in the other. Rank 2 tensors could also have
two upper (lower) indices, meaning that for multiplication
purposes BOTH directions would
!
!
1 2
a11 a12
=
function as columns (rows). If A =
and w = (w1 w2 ) = (−1 2),
a21 a22
3 4
!
5
ij
then we can calculate a wi =
. This is the standard multiplication of a matrix by a row
6
vector, although normally we would write the row vector to the left of the matrix (and would
wind up with a row vector, not a column vector). For example, the upper 5 in the result comes
from setting !j = 1 and summing over i: a11 w1 + a21 w2 = −1 + 6 = 5. We can also calculate
3
aij wj =
, which has no analog in standard matrix manipulation. Finally, we can construct
5
aij wk , a rank 3 tensor with 2 sets of “columns” and one set of “rows,” and 8 total entries. Note
that the vector in each case is the same; its name is w, the lower index shows that it is a row
vector rather than a column vector, and the exact variable used for the index shows whether or
not to dot the vector with one of the indices of the rank 2 tensor aij (and if so, which one). Such
a dot product is normally called a “contraction,” since it reduces the total number of indices in
the resulting tensor.
The real definition of a tensor relies on transformation laws. First see how a vector transforms
under rotation of a Cartesian coordinate system. If the original basis vectors are e1 and e2 ,
then rotation counterclockwise by θ gives new basis vectors e01 = cos θe1 + sin θe2 and e02 =
− sin θe1 + cos θe2 . (Here I am picturing rotating only the coordinates, not any physical objects
that live in the space.) As a sanity check, note that when θ = 0 the new basis vectors are the
same as the original ones, as they should be under a zero-degree rotation. A column vector can
be written in terms of these basis vectors as v = v 1 e1 + v 2 e2 = v 01 e01 + v 02 e02 . To see how the
primed and unprimed entries of v i are related, we can invert the equations for the new basis
vectors. The easiest way to do so is to note that going from the primed to unprimed basis
is also a rotation, in this case by −θ. Then we should get exactly the same coefficients as in
the original transformation, except with θ replaced everywhere by −θ. Since cos(−θ) = cos θ
and sin(−θ) = − sin θ, this gives e1 = cos θe01 − sin θe02 and e2 = sin θe01 + cos θe02 . Substitute
these into the equation for v i to get v = v 1 (cos θe01 − sin θe02 ) + v 2 (sin θe01 + cos θe02 ) = (v 1 cos θ +
v 2 sin θ)e01 +(−v 1 sin θ +v 2 cos θ)e02 , so v 01 = v 1 cos θ +v 2 sin θ and v 02 = −v 1 sin θ +v 2 cos θ. These
equations have exactly the same form as the transformations of the unit vectors themselves, but
that is a fluke of rotations in Cartesian coordinates. It’s for this reason that the distinction
between upper and lower indices is often ignored in Cartesian coordinates, although it becomes
crucial in other situations. In general vectors with lower indices transform the same way as the
basis vectors, and are therefore called “covariant” vectors. Vectors with upper indices transform
in the opposite way, and are called “contravariant” vectors.
A simple transformation that makes apparent the need for two types of vectors that transform
in different ways is a stretch along one direction. The dot product of a vector with “itself,” v i vi ,
is a scalar, or rank 0 tensor. It’s simply a number and ought to be invariant under any change
of basis. For most of what you’ve seen in previous classes, the components of v i and vi are the
same; the main exception is that if the components are complex, then the components of vi are
the complex conjugates of those of v i . But clearly something is off if, say, v 1 doubles while the
other components remain the same. If v1 also doubles, there’s no way the new value v 0i vi0 will
equal v i vi . (You might want to check that this relation does hold for the rotation described in
the previous paragraph. In that case the magic of sin2 + cos2 = 1 saves the day.)
I’ll save discussion of more general transformations to next time. For now I’ll introduce two
important tensors. One is the Kronecker delta, δ ij = 1 (i = j) or 0 (i 6= j). Written as a matrix,
this is exactly the identity matrix.
The second is the Levi-Civita symbol, εijk , which is an entirely antisymmetric tensor in three
dimensions. “Entirely antisymmetric” means that switching the values of any two indices changes
the sign of the entry. Thus εijk = −εjik (switching the first two indices), εijk = −εikj (switching
the last two), and εijk = −εkji (switching the first and last). If any two indices are equal the
corresponding entry vanishes. For example, setting i = j in the first of the switching equations
gives εiik = −εiik , so εiik = 0. This is a generalization of the diagonal of an antisymmetric matrix
having all 0’s; for a rank 3 object there are diagonal planes with respect to all pairs of indices.
The only nonzero entries have {i, j, k} = {1, 2, 3}. Also all these nonzero entries can be related
to each other and have the same magnitude. So in fact there is only one degree of freedom in
defining an entirely antisymmetric rank 3 tensor in three dimensions; once a single non-zero entry
is chose, the entire tensor is known. The standard choice for the Levi-Civita symbol is ε123 = 1.
Using the Levi-Civita symbol in the combination εijk v j wk = qi gives q1 = ε123 v 2 w3 +
ε132 v 3 w2 = v 2 w3 − v 3 w2 . Ordinarily you’d expect 9 total terms in the sum, since it goes over
all combinations of j and k, but the Levi-Civita symbol vanishes in the 7 cases where j = 1,
k = 1, and/or j = k. Similarly q2 = v 3 w1 − v 1 w3 and q3 = v 1 w2 − v 2 w1 . In fact this is exactly
a cross-product, q = v × w. (I am being sloppy about upper and lower indices in that last
equation, which as usual doesn’t matter in Cartesian coordinates.)