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Chapter 1
Basic calculus and algebra review
This section provides a few brief notes on math notation and concepts needed for this text. Not all concepts and formula are presented in a mathematically rigorous way, and you should refer to something like a Math for Engineers text for a more complete treatment. For most of this text, it will be assumed that the reader is familiar with the material treated in this chapter.
1.1 Calculus
1.1.1 Full and partial derivatives
In calculus, we are interested in the change or dependence of some quantity, e.g. u, on small changes in some variable x. If u has value u 0 at x 0 and changes to u 0 + δ u when x changes to x 0 + δ x, the incremental change can be written as
δ u =
δ u δ x
(x 0 ) δ x. (1.1)
The δ (or sometimes written as capital ∆ ) here means that this is a small, but finite quan- tity. If we let δ x get asymptotically smaller around x 0 , we of course arrive at the partial derivative, which we denote with ∂ like
lim δ x→ 0
δ u δ x
(x 0 ) =
∂ u ∂ x
The limit in eq. (1.2) will work as long as u doesn’t do any funny stuff as a function of x, like jump around abruptly. When you think of u(x) as a function (some line on a plot) that depends on x, ∂ u/ ∂ x is the slope of this line that can be obtained by measuring the change δ u over some interval δ x, and then making the interval progressively smaller. We call ∂ ∂ ux (we also write in shorthand ∂ xu(x) or u′(x); if the variable is time, t, we also use ˙u(t) for ∂ u/ ∂ t) the partial derivative, because u might also depend on other variables, e.g. y and z. If this is the case, the total derivative du at some {x 0 , y 0 , z 0 } (we will drop (i.e.
not write down) the explicit dependence on the variables from now on) is given by the sum of the changes in all variables on which u depends:
du =
∂ u ∂ x
dx +
∂ u ∂ y
dy +
∂ u ∂ z
dz. (1.3)
Here, dx and similar are placeholders for infinitesimal changes in the variables. This means that eq. (1.3) works as long as dx is small enough that a linear relationship between δ u and δ x still holds. In fact, we can perform a Taylor approximation on any u(x) around x 0 by
u(x) = u(x 0 ) +
∂ u ∂ x
(x 0 )(x − x 0 ) +
∂^2 u ∂ x^2
(x 0 )
(x − x 0 )^2 2!
∂^3 u ∂ x^3
(x 0 )
(x − x 0 )^3 3!
Here, ∂
(^2) u ∂ x^2 is the second derivative, the change of the change of^ u^ with^ x.^ n! denotes the factorial, i.e. n! = 1 × 2 × 3 ×... n. (1.5)
So, as long as dx = x − x 0 is small, the derivative will work (for well behaved u). For ex- ample, if better approximations are needed, e.g.when the strain tensor is not infinitesimal anymore, quadratic and higher terms like the one that goes with the second derivative in the series eq. (1.4) and so on need to be taken into account. Finite difference methods essentially use Taylor approximations to approximate derivatives, as we will see later.
How to compute derivatives Here are some of the most common derivatives of a few functions:
function f (x) derivative f ′(x) comment
xp^ pxp−^1 special case: f (x) = c = cx^0 → f ′(x) = 0 where c, p are constants exp(x) = ex^ ex^ that’s what makes e so special ln(x) 1/x sin(x) cos(x) cos(x) − sin(x) tan(x) sec^2 (x) = 1/ cos^2 (x)
If you need to take derivatives of combinations of two or more functions, here called f , g, and h, there are four important rules (with a and b being constants):
Chain rule (inner and outer derivative):
If f (x) = h(g(x)) (1.6) f ′(x) = h′(g(x))g′(x), (1.7)
i.e. derivative of nested functions are given by the outer times the inner derivative.
is a velocity field, then the divergence (grad dot product) operation on a vector field
div v = ∇ · v (1.16)
is equivalent to finding the dilatancy (volumetric) strain-rate 4 ˙ from the strain-rate tensor components because
∆ ˙V
V
= tr( ε˙) = (^) ∑ i
ε ˙ii = ε ˙ 11 + ε ˙ 22 + ε ˙ 33 =
∂ v 1 ∂ x 1
∂ v 2 ∂ x 2
∂ v 3 ∂ x 3
= ∇ · v. (1.17)
Here V is volume, and ∆ ˙V volume rate-change and, mind you, the strain-rate tensor, ε˙, is defined as
ε˙ = ε ˙ij =
∂ vi ∂ xj
∂ vj ∂ xi
In complete analogy, if the vector field are displacements u ( x ), then ∇ · u yields the dila- tancy, i.e. the trace of the strain tensor, ε,
ε = ε ij =
∂ ui ∂ xj
∂ uj ∂ xi
Eq. (1.17) illustrates that the divergence has to do with sinks and sources, or volu- metric effects. The volume integral over the divergence of a velocity field is equal to the surface integral of the flow normal to the surface. (An electro-magnetics example: For the magnetic field: div B = 0 because there are no magnetic monopoles, but for the electric field: div E = q, with electric charges q being the “source”.) If we take the vector instead of the dot product with the grad operator, we have the curl or rot operation curl v = ∇ ∧ v. (1.20)
The curl is a rotation vector just like ω. Indeed, if the velocity field is that of a the rigid body rotation, v = ω ∧ r , one can show that ∇ ∧ v = ∇ ∧ ( ω ∧ r ) = 2 ω. Second derivatives enter into the Laplace operator which appears, e.g. in the diffusion equation:
∇^2 T =
∂^2 T
∂ x^2
∂^2 T
∂ y^2
∂^2 T
∂ z^2
Some rules for second derivatives:
curl(grad T) = ∇ × (∇T) = 0 (1.22) div(curl v ) = ∇ · ∇ × v = 0 (1.23)
1.1.3 Integrals
Taking an integral
F(x) =
∫ f (x)dx, (1.24)
in a general (indefinite) sense, is the inverse of taking the derivative of a function f ,
F
∂ f (x) ∂ x
= f (x) + c (1.25)
∂ ∂ x
F
∂ f (x) ∂ x
∂ x
( f (x) + c) = f ′(x). (1.26)
Any general integration of a derivative is thus only determined up to an integration con- stant, here c, because the derivative, which is the reverse of the integral, of a constant is zero. Graphically, the definite (with bounds) integral over f (x) ∫ (^) b
a
f (x)dx = F(b) − F(a) (1.27)
along x, adding up the value of f (x) over little chunks of dx, from the left x = a to the right x = b corresponds to the area under the curve f (x). This area can be computed by subtracting the analytical form of the integral at b from that at a, F(b) − F(a). If f (x) = c (c a constant), then
F(x) = cx + d (1.28) F(b) = cb + d (1.29) F(a) = ca + d (1.30) F(b) − F(a) = c(b − a), (1.31)
the area of the box (b − a) × c. Here are the integrals (anti derivatives) of a few common functions, all only deter- mined up to an integration constant C
function f (x) integral F(x) comment
xp^ x
p+ 1 p+ 1 +^ C^ special case:^ f^ (x) =^ c^ =^ cx
(^0) → F(x) = cx + C ex^ ex^ +C 1/x ln(|x|) + C sin(x) − cos(x) + C cos(x) sin(x) + C
There are also a few very helpful definite integrals without closed-form anti derivatives, e.g. ∫ (^) ∞
0
e−x
2 dx =
π 2
1.2 Linear algebra
TO BE ADDED: matlab conventions for mathematical operations such as dot and cross products.
1.2.1 The dot product
We will make use of the dot product, which is defined as
c = a · b =
n ∑ i= 1
aibi, (1.43)
where a and b are vectors of dimension n (n-dimensional, geometrical objects with a di- rection and length, like a velocity) and the outcome of this operation is a scalar (a regular number), c. In eq. (1.43), (^) ∑ni= 1 means “sum all that follows while increasing the index i from the lower limit, i = 1, in steps of of unity, to the upper limit, i = n”. In the examples below, we will assume a typical, spatial coordinate system with n = 3 so that
a · b = a 1 b 1 + a 2 b 2 + a 3 b 3 , (1.44)
where 1, 2, 3 refer to the vector components along x, y, and z axis, respectively (ADD FIGURE HERE). In the “Einstein summation” convention, we would rewrite (^) ∑ni= 1 aibi simply as aibi, where summation over repeated indices is implied, i.e. the (^) ∑ is not written. When we write out the vector components, we put them on top of each other
a =
a 1 a 2 a 3
ax ay az
or in a list, maybe with curly brackets, like so: a = {a 1 , a 2 , a 3 }. Here, we will use a bold font a to denote vectors as opposed to scalar a, but another common form is ~a, and on a blackboard, you might also see vectors written as a because that’s easier. We can write the amplitude (or: length, L 2 norm) of a vector as
| a | =
n ∑ i
a^2 i =
a^21 + a^22 + a^23 =
a^2 x + a^2 y + a^2 z. (1.46)
For instance, all of the basis vectors defining the Cartesian coordinate system, e x, e y, and e z have unity length by definition, | e i| = 1. Those e i vectors point along the respective axes of the Cartesian coordinate system so that we can assemble a vector from its compo- nents like a = {ax, ay, az} = ax e x + ay e y + az e z. (1.47)
For a spherical system, the e r, e θ , and e φ unity vectors can still be used to express vectors but the actual Cartesian components of e i depend on the coordinates at which the vectors are evaluated.
We can restate eq. (1.43) and give another definition of the dot product,
a · b = | a || b | cos θ (1.48)
where θ is the angle between vectors a and b. The meaning of this is that if you want to know what component of vector a is parallel to b , you just take the dot product. Say, you have a velocity v and want the normal velocity vn along a vector n with | n | = 1 that is oriented at a 90◦^ angle (perpendicular) to some plate boundary, you can use vn = v · n. Also, eq. (1.47) only works because the basis vectors e i of any coordinate system are, by definition, orthogonal (at right angle, perpendicular, at θ = 90 ◦) to each other and e i · e j = 0 for all i 6 = j. Likewise, e i · e i = 1 for all i since a · a = | a |^2 , and basis vectors have unity length by definition. Using the Kronecker δ
δ ij = 1 for i = j, and δ ij = 0 for i 6 = j, (1.49)
we can write the conditions for the basis vectors as
e i · e j = δ ij. (1.50)
1.2.2 Vector or cross product
This operation is written as a × b or a ∧ b and its result is another vector
c = a ∧ b (1.51)
that is at a right angle to both a and b (hence the right-hand-rule, with thumb, index, and middle finger along a , b , and c , respectively). vector c ’s length is given by
| c | = | a ∧ b | = | a || b | sin θ , (1.52)
that is, c is largest when a and b are orthogonal, and zero if they are parallel. Compare this relationship to eq. (1.48). In 3-D,
c = a ∧ b =
aybz − azby azbx − axbz axby − aybx
(note that there is no i component of a or b in the i component of c , this is the aforemen- tioned orthogonality property). An example for a cross product is the velocity v at a point with location r in a body spinning with the rotation vector ω , v = ω ∧ r. The rotation vector ω is different from, e.g., r in that ω has a spin (a sense of rotation) to it (the other right-hand-rule, where your thumb points along the vector and your fingers indicate the counter-clockwise motion). ADD FIGURE
Trace The trace of a n × n matrix A is the sum of its diagonal elements
tr(A) =
n ∑ i= 1
aii. (1.60)
Determinant The determinant for a 2 × 2 matrix is computed as
det(A) = a 11 a 22 − a 12 a 21 (1.61)
and is a measure of area change. For 3 × 3,
det(A) = a 11 (a 22 a 33 − a 23 a 32 ) (1.62) − a 12 (a 21 a 33 − a 23 a 31 )
- a 13 (a 21 a 32 − a 22 a 31 )
(note how the 3 × 3 determinant is assembled from a pattern of 2 × 2 determinants; for n > 3, a correspondingly more complicated formula applies. ADD FIGURE
Vector cross product based on the determinant The cross product c = a ∧ b (eq. 1.53) can also be written as the determinant of the matrix
e x e y e z ax ay az bx by bz
Invariants The trace IA = tr(A) = aii (1.64)
and determinant I I IA = det(A) (1.65)
of a matrix A are two of the three invariants, i.e. properties of a tensor (expressed as a matrix) that are independent of a coordinate system. The third is the “second invariant”,
I IA = a 11 a 22 + a 11 a 33 + a 22 a 33 − a^212 − a^213 − a^223. (1.66)
These expressions arise when finding the eigenvectors and values of a tensor, eq. (1.71).
Transpose of a matrix (AT^ )ij = aijT = aji, i.e. the transpose has all elements flipped by
row and column.
Inverse of A , A−^1 : The inverse of a matrix is defined via
A−^1 A = AA−^1 = I. (1.67)
If the inverse exists, then (A−^1 )−^1 = A, (AT^ )−^1 = (A−^1 )T, and (AB)−^1 = B−^1 A−^1. The inverse only exists if det(A) 6 = 0.
Orthogonal or rotation matrices: For those matrices,
AAT^ = ATA = (1.68)
holds.
Eigenvalues and eigen vectors: Any n × n symmetric matrix A has n eigen vectors v i that correspond to real eigenvalues λ i such that
A v i = λ i v i (1.69)
An example is the stress matrix which can be written in the principal axes system, where the eigen vectors of the Cartesian representation of the stress matrix are the principal axes. Eigenvalues can be found using
det(A − λ I) = 0 (1.70)
and eigen vectors subsequently by using the first property, which leads to
det(A − λ I) = − λ^3 + IA λ^2 − I IA λ + I I IA = 0. (1.71)
If a symmetric matrix A is transformed into the principal axes system, A′, there are no off-diagonal elements
A =
axx axy axz ayx ayy ayz azx azy azz
→ A′^ =
a 1 0 0 0 a 2 0 0 0 a 3
where the a 1 , a 2 , and a 3 correspond to the three eigenvalues λ i. (The coordinate system reference of A′^ is then contained in the orientation of the eigen vectors v i.) For a matrix in the principal axis system, the invariants are very easily computed:
tr(A′) = IA′ = IA = a 1 + a 2 + a 3 (1.73) I IA′ = I IA = a 1 a 2 + a 1 a 3 + a 2 a 3 (1.74) det(A′) = I I IA′ = I I IA = a 1 a 2 a 3. (1.75)
See also sec. ?? for definitions of invariatns using deviators, such as for the deviatoric stress tensor.
Bibliography
