## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### Appendix B. Some Useful Mathematics

In the theoretical description of liquids, we will encounter functions, matrices and determinants. Moreover, we will encounter scalars, vectors, and tensors. We will also need occasionally coordinate transformations, some transforms and calculus of variations. In the following we will briefly review these concepts, as well as some of the operations between them and some useful general results.

**B.1 Symbols and Conventions**

Often, we will use quantities with subscripts, for example, *A _{ij}*. With respect to these quantities a convenient symbol is the

*Kronecker delta*, denoted by

*δ*and defined by

_{ij}(B.1)

Note that we can write the identity Σ* _{i}δ_{ij}a_{i} = a_{j}*, and therefore

*δ*is also called the

_{ij}*substitution operator*. Applying

*δ*to

_{ij}*A*results in Σ

_{ij}*Σ*

_{ij}A_{ij}δ_{ij}=*.*

_{j}A_{jj}We also introduce the *alternator* *e _{ijk}* for which it holds that

(B.2)

An alternative expression is given by *e _{ijk}* = ½(

*i*−

*j*)(

*j*−

*k*)(

*k*−

*i*).

**B.2 Partial Derivatives**

A function *f* may be dependent on *variables* *x _{i}* and

*parameters*

*p*, denoted by

_{j}*f*(

*x*;

_{i}*p*). Reference to the parameters is often omitted by writing

_{j}*f*(

*x*). In practice, reference to the variable is also often omitted by writing just

_{i}*f*. For a function

*f*, several derivatives exist. If all variables but one, say

*x*

_{1}, are kept constant during differentiation, the derivative of

*f*with respect to

*x*

_{1}is called the

*partial derivative*and is denoted by . Once a choice of independent variables is made, there is no need to indicate, as frequently done, which variables are kept constant. Therefore can be indicated without confusion by ∂

*f*(

*x*)/∂

_{i}*x*

_{1}. The function ∂

*f/*∂

*x*generally is a function of all variables

_{i}*x*and, if continuous, may be differentiated again to yield the

_{i}*second partial derivatives*∂

^{2}

*f*/∂

*x*∂

_{i}*x*(= ∂

_{j}^{2}

*f*/∂

*x*∂

_{j}*x*).

_{i}Example B.1

For *f*(*x*,*y*)* = x*^{2}*y*^{3} one simply calculates

In case the independent variables *x _{i}* increase by d

*x*, the value of the function

_{i}*f*at

*x*+ d

_{i}*x*is given by Taylor's expansion

_{i}__(B.3)__

One can also write symbolically

(B.4)

Expansion of the exponential yields __Eq. (B.3)__. Another way is to write

(B.5)

where the (*nth* *order*) *differential* is given by

(B.6)

Example B.2

Consider again the function *f*(*x*,*y*) *= x*^{2}*y*^{3}. A first-order estimate for *f*(2.1,2.1) is

using *f*(2,2) as the reference value. The actual value is 40.84.

A function *f*(*x*) is *analytic* at *x* = *c* if *f*(*x*) can be written as (a sum of) Taylor series (with a positive convergence radius). If *f*(*x*) is analytic at each point on the open interval I, *f*(*x*) is analytic on the interval I. For a function *w*(*z*) of a complex variable to be analytic, it must satisfy the *Cauchy–Riemann conditions*

(B.7)

where *u*(*x*,*y*) = Re *w*(*z*) and *v*(*x*,*y*) = Im *w*(*z*) denote the real and imaginary parts of *w*, respectively. Moreover, if *u*(*x*,*y*) and *v*(*x*,*y*) have continuous second derivatives the function *w*(*z*) obeys the *Laplace equation* ∂^{2}*u*/∂^{2}*x *+ ∂^{2}*u*/∂^{2}*y* = ∂^{2}*v*/∂^{2}*x *+ ∂^{2}*v*/∂^{2}*y* = 0, and is said to be *harmonic*.

Example B.3

Consider the function *w*(*z*) = e* ^{x}* cos

*y*+ i e

*sin*

^{x}*y*= exp(

*z*). Then it holds that

The derivatives are given by

Hence, Cauchy–Riemann conditions are satisfied and the function is harmonic.

**B.3 Composite, Implicit, and Homogeneous Functions**

If for a function *f* the variables *x _{i}* are themselves a function of

*y*, it is called a

_{j}*composite function*. For the first-order differentials the

*chain rule*applies so that

(B.8)

In many cases the variables *x _{i}* are not independent, that is, a relation exists between them meaning that an arbitrary member, say

*x*

_{1}, can be expressed as a function of

*x*

_{2}, … ,

*x*. Often, this relation is given in the form of an

_{n}*implicit function*, that is,

*f*=

*f*(

*x*) = constant. Of course, if the equation can be solved, the relevant differentials can be obtained from the solution. The appropriate relations between the differentials can also be obtained by observing that d

_{i}*f*= Σ

*(∂*

_{i}*f/*∂

*x*) = 0. If

_{i}*x*

_{1}is the dependent variable, putting d

*x*

_{1}= 0 and division by d

*x*(

_{i}*i*≠1) yields

__(B.9)__

Example B.4

Consider explicitly a function *f* of three variables *x*, *y* and *z*, where *z* is the dependent variable. If we take *x*_{1} = *z*, *x _{i}* =

*x*and

*x*=

_{j}*y*,

__Eq. (B.9)__reads

On the other hand, taking *x _{i}* =

*y*and

*x*=

_{j}*x*for

*x*

_{1}=

*z*results in

Hence, it easily follows that

(B.10)

By cyclic permutation of the variables we obtain

resulting, after substitution in each other, in

(B.11)

Now consider *x*, *y* and *z* to be composite functions of another variable *u*. If *f* is constant, there is a relation between *x*, *y* and *z* and thus also between ∂*x*/∂*u*, ∂*y*/∂*u* and ∂*z*/∂*u*. Moreover, d*f* = 0 and __Eq. (B.9)__ explicitly reads

Further taking *z* as constant, independent of *u*, results in

Comparing with (∂*y*/∂*x*)_{f}_{,}* _{z}* = –(∂

*f*/∂

*x*)

_{y}

_{,}*(∂*

_{z}*f*/∂

*y*)

_{z}

_{,}*one obtains*

_{x}(B.12)

The equations, indicated by •, are frequently employed in thermodynamics.

A function *f*(*x _{i}*) is said to be positively

*homogeneous*of degree

*n*if for every value of

*x*and for every

_{i}*λ*> 0 we have

(B.13)

For such a function we have *Euler's theorem*

to be proven by differentiation with respect to *λ* first and taking *λ* = 1 afterwards.

Example B.5

Consider the function *f*(*x*,*y*) = *x ^{2}* +

*xy*−

*y*. One easily finds

^{2}Consequently, *x*(∂*f*/∂*x*) + *y*(∂*f*/∂*y*) = *x*(2*x* + y) + *y*(*x *− 2*y*) = 2(*x*^{2} + *xy *−* y*^{2}) = 2*f*. Hence, *f* is homogeneous of degree 2.

**B.4 Extremes and Lagrange Multipliers**

For obtaining an extreme of a function *f*(*x _{i}*) of

*n*independent variables

*x*the first variation δ

_{i}*f*has to vanish (see Section B.10). This leads to

(B.14)

and, since the variables *x _{i}* are independent and the variations δ

*x*are arbitrary, to ∂

_{i}*f*/∂

*x*= 0 for

_{i}*i*= 1, …,

*n*. If, however, the extreme of

*f*has to be found when the

*x*are dependent and satisfy

_{i}*r*constraint functions

__(B.15)__

where the parameters *C _{j}* are constants, the variables

*x*must also obey

_{i}(B.16)

Of course, the system can be solved in principle by solving __Eq. (B.15)__ for the independent *n *−* r* variables *x _{i}* as functions of the others, but the procedure is often complex. It can be shown that finding the extreme of

*f*subject to the constraint of

__Eq. (B.15)__is equivalent to finding the extreme of a function

*g*defined by

(B.17)

where now the original variables *x _{i}* and the additional variables

*λ*, called the

_{j}*Lagrange*(

*undetermined*)

*multipliers*, are to be considered independent. Variation of

*λ*leads to

_{j}__Eq. (B.15)__and variation of

*x*to

_{i}__(B.18)__

From __Eq. (B.18)__ the values for *x _{i}* can be determined. These values are still functions of

*λ*but they can be eliminated using

_{j}__Eq. (B.15)__. In physics, chemistry and materials science, the Lagrange multiplier often can be physically interpreted.

Example B.6

One can ask what is the minimum circumference *L* of a rectangle given the area *A*. Denoting the edges by *x* and *y*, the circumference is given by *L* = 2(*x *+ *y*), while the area is given by *A* = *xy*. The equations to be solved are

Hence, the solution is *x* = *y*, and .

**B.5 Legendre Transforms**

In many problems we meet the demand to interchange between dependent and independent variables. If *f*(*x _{i}*) denotes a function of

*n*variables

*x*, we have

_{i}(B.19)

Elimination of *x _{i}* from

*X*(

_{i}*x*) and

_{i}*f*(

*x*) leads to

_{i}*f*=

*f*(

*X*). However, from

_{i}*f*(

*X*) it is impossible to uniquely recover

_{i}*f*(

*x*) by repeating this procedure for

_{i}*f*(

*X*). Now consider the function

_{i}*g*=

*f*−

*X*

_{1}

*x*

_{1}. For the differential we obtain

(B.20)

and we see that the roles of *x*_{1} and *X*_{1} have been interchanged. This transformation can be applied to only one variable, to several variables or to all variables. In the last case we use *g* = *f *− Σ* _{i}X_{i}x_{i}* and obtain d

*g*= –Σ

*d*

_{j}x_{j}*X*(

_{j}*j*= 1, … ,

*n*). This so-called

*Legendre transformation*, if applied to the transform, results in the complete original expression and is often used in thermodynamics. For example, the Gibbs energy

*G*(

*T*,

*P*) with pressure

*P*and temperature

*T*as independent variables results from the internal energy

*U*(

*S*,

*V*) with entropy

*S*and volume

*V*as independent variables using

*G*=

*U*−

*TS*+

*PV*.

Example B.7

Consider the function *f*(*x*) = ½*x*^{2}. The dependent variable *X* is given by

which can be solved to yield *x* = *X*. Therefore, the function expressed in the variable *X* reads *f*(*X*) = ½*X*^{2}. For the transform *g*(*X*) one thus obtains

**B.6 Matrices and Determinants**

A *matrix* is an array of numbers (or functions), represented by a roman boldface uppercase symbol, for example, **A**, or by an italic uppercase symbol with indices, for example, *A _{ij}*. In full we write

(B.21)

The numbers *A _{ij}* are called the

*elements*. The matrix with

*m*rows and

*n*columns is called an

*m*×

*n*matrix or a matrix of order (

*m*,

*n*). The

*transpose*of a matrix, indicated by a superscript T, is formed by interchanging rows and columns. Hence

(B.22)

Often, we will use square matrices *A _{ij}* for which

*m = n*(order

*n*). A

*column matrix*(or

*column*for short) is a matrix for which

*n*= 1 and is denoted by a lowercase italic standard symbol with an index, for example, by

*a*, or by a lowercase roman bold symbol, for example,

_{i}**a**. A row matrix is a matrix for which

*m*= 1 and is the transpose of a column matrix and thus denoted by (

*a*)

_{i}^{T}or

**a**

^{T}.

Two matrices of the same order are *equal* if all their corresponding elements are equal. The *sum* of two matrices **A** and **B** of the same order is given by the matrix **C** whose corresponding elements are the sums of the elements of **A**and **B** or

(B.23)

The *product* of two matrices **A** and **B** results in matrix **C** whose elements are given by

(B.24)

representing the *row-into-column* rule. Note that, if **A** represents a matrix of order (*k*,*l*) and **B** a matrix of order (*m*,*n*), the product **BA** is not defined unless *k = n*. For square matrices, we generally have **AB** ≠ **BA**, so that the order must be maintained in any multiplication process. The *transpose* of a *product* (**ABC***..*)^{T} is given by (**ABC***..*)^{T} = ..^{T}**C**^{T}**B**^{T}**A**^{T}.

A *real* matrix is a matrix with real elements only while a *complex* matrix is a matrix with complex elements. The *complex conjugate* of a matrix **A** is the matrix **A***** formed by the complex conjugate elements of **A** or

(B.25)

If a real (complex), square matrix **A** is equal to its transpose (complex conjugate)

(B.26)

then **A** is a *symmetric* (*Hermitian*) matrix. For an *antisymmetric* matrix it holds that

(B.27)

and thus **A** has the form

(B.28)

A *diagonal* matrix has only non-zero entries along the diagonal:

(B.29)

The *unit* matrix **I** is a diagonal matrix with unit elements:

(B.30)

Obviously, **IA ***= ***AI ***= ***A**, where **A** is any square matrix of the same order as the unit matrix.

The *determinant* of a square matrix of order *n* is defined by

__(B.31)__

where the summation is over all permutations of the indices *i*, *j*, *k*, ···, *p*. The sign in brackets is positive (negative) when the permutation involves an even (odd) number of permutations from the initial term *A*_{11}*A*_{22}*A*_{33}*..A _{nn}*.

Example B.8

For a matrix **A** of order 3, __Eq. (B.31)__ yields

Alternatively, it can be written as det **A*** = *Σ_{r,s,t }e_{rst}*A*_{1r}* A*_{2s}* A*_{3t}.

The determinant of the product **AB** is given by

(B.32)

Further, the determinant of a matrix equals the determinant of its transpose, that is

(B.33)

The *inverse* of a square matrix **A** is denoted by **A**^{−1} and it holds that

(B.34)

where **I** is the unit matrix of the same order as **A**. Hence, a square matrix *commutes* with its inverse. The inverse only exists if det **A** ≠ 0. The inverse of the product (**ABC***..*)^{−1} is given by (**ABC**..)^{−1}* = ..*^{−1}**C**^{−1}**B**^{−1}**A**^{−1}. The inverse of a transpose is equal to the transpose of the inverse, that is, (**A**^{T})^{−1}* = *(**A**^{−1})^{T}, often written as **A**^{−T}.

The *cofactor* *α _{ij}* of the element

*A*is (–1)

_{ij}

^{i}^{+j}times the

*minor θ*. The latter is the determinant of a matrix obtained by removing row

_{ij}*i*and column

*j*from the original matrix. The inverse of

**A**is then found from

*Cramers' rule*

(B.35)

Note the reversal of the element and cofactor indices.

Example B.9

Consider the matrix . The determinant is det **A** = –2. The cofactors are given by

The elements of the inverse *A*^{−1} are thus given by

For a *diagonal* matrix **A** the inverse is particularly simple and given by

(B.36)

For an *orthogonal* (*unitary*) matrix it holds that

(B.37)

implying det **A** = ±1. With det **A** = 1, the matrix **A** is a *proper* orthogonal matrix.

**B.7 Change of Variables**

It is also often required to use different independent variables, in particular in integrals. For definiteness consider the case of three “old” variables *x*, *y* and *z* and three “new” variables *u*, *v* and *w*. In this case, we have

where the functions *u*, *v* and *w* are continuous and have continuous first derivatives in some region R*. The transformations *u*(*x*,*y*,*z*), *v*(*x*,*y*,*z*) and *w*(*x*,*y*,*z*) are such that a point (*x*,*y*,*z*) corresponding to (*u*,*v*,*w*) in R* lies in a region R and that there is a one-to-one correspondence between the points (*u*,*v*,*w*) and (*x*,*y*,*z*). The Jacobian matrix *J* = ∂(*x*,*y*,*z*)/∂(*u*,*v*,*w*) is defined by

(B.38)

The determinant, det *J*, should be either positive or negative throughout the region R*. Consider now the integral

(B.39)

over the region R. If the function *F* is now expressed in *u*, *v* and *w* instead of *x*, *y* and *z*, the integral has to be evaluated over the region R* as

(B.40)

where |det *J*| denotes the absolute value of the determinant of the Jacobian matrix^{2)}*J*. The expression is easily generalized to more variables than 3.

Example B.10

In many cases the use of cylindrical coordinates is convenient. Here, we consider the Cartesian coordinates *x*_{1}, *x*_{2} and *x*_{3} as “new” variables and the cylindrical coordinates *r*, *θ* and *z* as “old” variables. The relations between the Cartesian coordinates and the cylindrical coordinates (__Figure B.1__) are

while the inverse equations are given by

** Figure B.1** (a) Cylindrical and (b) spherical coordinates.

The Jacobian determinant is easily calculated as |det *J|* = *r*. Similarly, for spherical coordinates^{3)}

and the corresponding inverse equations

In this case the Jacobian determinant becomes |det *J*| = *r*^{2} sin *θ*.

**B.8 Scalars, Vectors, and Tensors**

A *scalar* is an entity with a magnitude. It is denoted by an italic, lowercase or uppercase, roman or greek letter, for example, *a*, *A*, *γ*, or *Γ*.

A *vector* is an entity with a magnitude and direction. It is denoted by a lowercase boldface, italic letter, for example, ** a**. It can be interpreted as an arrow from a point O (origin) to a point P. Let

**be this arrow. Its**

*a**magnitude*(length), equal to the distance OP, is denoted by

*a*= |

**|. A unit vector in the same direction as the vector**

*a***, here denoted by**

*a***, has a length of 1. Vectors obey the following rules (**

*e*__Figure B.2__):

· ** c **=

**+**

*a***=**

*b***+**

*b***(commutative rule)**

*a*· ** a **+ (

**+**

*b***) = (**

*d***+**

*a***) +**

*b***(associative rule)**

*d*· (** a **+

**)·**

*b***=**

*c***·**

*a***+**

*c***·**

*b***(distributive law)**

*c*· ** a **+ (−

**) =**

*a***(zero vector definition)**

*0*· ** a** = |

**|**

*a***, |**

*e***| = 1 (unit vector definition)**

*e*· ** 0**·

**= 0**

*u*· *α a*

**=**

*α*|

**|**

*a***, |**

*e***|**

*e***= 1**

** Figure B.2** Vector properties.

Various products can be formed using vectors. The *scalar* (or *dot* or *inner*) *product* of two vectors ** a** and

**yields a scalar and is defined as**

*b***·**

*a***=**

*b***·**

*b***= |**

*a***|**

*a***|**

**|cos(**

*b**ϕ*), where

*ϕ*is the enclosed angle between

**and**

*a***. From this definition it follows that**

*b**a*= |

**| = (**

*a***·**

*a***)**

*a*^{1/2}. Two vectors

**and**

*a***are orthogonal if**

*b***·**

*a***= 0. The scalar product is commutative (**

*b***·**

*a***=**

*b***·**

*b***) and distributive (**

*a***·(**

*a***+**

*b***) =**

*c***·**

*a***+**

*b***·**

*a***).**

*c*The *vector* (or *cross* or *outer*) *product* of ** a** and

**denotes a vector**

*b***=**

*c***×**

*a***. We define a unit vector**

*b***perpendicular to the plane spanned by**

*n***and**

*a***. The sense of**

*b***is right-handed: rotate from**

*n***to**

*a***along the smallest angle and the direction of**

*b***is given by a right-hand screw. It holds that**

*n***·**

*a***=**

*n***·**

*b***= 0 and |**

*n***|**

*n***= 1. Explicitly,**

**=**

*n***×**

*a***/|**

*b***|**

*a***|**

**|. The vector product**

*b***is equal to**

*c***=**

*c***×**

*a***= −**

*b***×**

*b***= |**

*a***|**

*a***|**

**|sin(**

*b**ϕ*)

**. The length of |**

*n***| = |**

*c***|**

*a***|**

**|sin(**

*b**ϕ*) is numerically equal to the area of the parallelogram whose sides are given by

**and**

*a***. The vector product is anti-commutative (**

*b***×**

*a***= −**

*b***×**

*b***) and distributive (**

*a***× (**

*a***+**

*b***) =**

*c***×**

*a***+**

*b***×**

*a***), but not associative (**

*c***× (**

*a***×**

*b***) ≠ (**

*c***×**

*a***) ×**

*b***).**

*c*The *triple product* is a scalar and given by *d* = ** a**·

**×**

*b***=**

*c***×**

*a***·**

*b***. It yields the volume of the block (or parallelepepid) the edges of which are**

*c***,**

*a***, and**

*b***. Three vectors**

*c***,**

*a***, and**

*b***are independent if from**

*c**α*

**a****+**

*β*

**b****+**

*γ*= 0 it follows that

**c***α*=

*β*=

*γ*= 0. This is only the case if

**,**

*a***and**

*b***are noncoplanar or, equivalently, the product**

*c***·**

*a***×**

*b***≠ 0.**

*c*Finally, we need the *tensor* (or *dyadic*) *product* ** ab**. Operating on a vector

**, it associates with**

*c***a new vector according to**

*c***·**

*ab***=**

*c***(**

*a***·**

*b***) = (**

*c***·**

*b***)**

*c***. Note that**

*a***operating on**

*ba***yields**

*c***·**

*ba***=**

*c***(**

*b***·**

*a***) = (**

*c***·**

*a***)**

*c***.**

*b*A *tensor* (of rank 2), denoted by an uppercase boldface, italic letter, for example, ** A**, is a linear mapping that associates with a vector

**another vector**

*a***according to**

*b***=**

*b***·**

*A***. Tensors obey the following rules:**

*a*· ** C** =

**+**

*A***=**

*B***+**

*B***(commutative law)**

*A*· ** A **+ (

**+**

*B***) = (**

*C***+**

*A***) +**

*B***(associative law)**

*C*· (** A **+

**)·**

*B***=**

*u***·**

*A***+**

*u***·**

*B***(distributive law)**

*u*· ** A** + (−

**) =**

*A***(zero tensor definition)**

*O*· ** I**·

**=**

*u***(**

*u***unit tensor definition)**

*I*· ** O**·

**=**

*u***(**

*0***zero tensor,**

*O***zero vector)**

*0*· ** A** ·(

*α*)

**u****= (**

*α*)·

**A****=**

*u**α*(

**·**

*A***)**

*u*where *α* is an arbitrary scalar and ** u** is an arbitrary vector. The simplest example of a tensor is the tensor product of two vectors, for example, if

**=**

*A***, the vector associated with**

*bc***is given by**

*a***·**

*A***=**

*a***·**

*bc***= (**

*a***·**

*c***)**

*a***.**

*b*So far we have discussed vectors and tensors using the *direct notation* only, that is using a symbolism, which represents the quantity without referring to a coordinate system. It is convenient though to introduce a coordinate system. In this book we will make use primarily of Cartesian coordinates, which are a rectangular and rectilinear coordinate system with origin O and unit vectors *e*_{1}, *e*_{2} and *e*_{3} along the axes. The set *e** _{i}* = {

*e*_{1},

*e*_{2},

*e*_{3}} is called an

*orthonormal basis*. It holds that

*e*

_{i}

*e**=*

_{j}*δ*. The vector OP =

_{ij}**is called the position of point P. The real numbers**

*x**x*

_{1},

*x*

_{2}and

*x*

_{3}, defined uniquely by the relation

**=**

*x**x*

_{1}

*e*_{1}+

*x*

_{2}

*e*_{2}+

*x*

_{3}

*e*_{3}, are called the (Cartesian)

*components*of the vector

**. It follows that**

*x**x*=

_{i}**·**

*x*

*e**for*

_{i}*i*= 1, 2, 3. Using the components

*x*in equations, we use the

_{i}*index notation*. Using the index notation the scalar product

**·**

*u***can be written as**

*v***·**

*u***=**

*v**u*

_{1}

*v*

_{1}+

*u*

_{2}

*v*

_{2}+

*u*

_{3}

*v*

_{3}= Σ

*. The length of a vector*

_{i}u_{i}v_{i}**, |**

*x***| = (**

*x***·**

*x***)**

*x*^{1/2}, is thus also equal to . Sometimes, it is also convenient to use

*matrix notation*; in this case the components

*x*are written collectively as a column matrix

_{i}**x**. In matrix notation the scalar product

**·**

*u***is written as**

*v***u**

^{T}

**v**. The tensor product

**in matrix notation is given by**

*ab***ab**

^{T}.

Using the alternator *e _{ijk}* the relations between the unit vectors can be written as

*e**×*

_{i}

*e**= Σ*

_{j}

_{k}e_{ijk}

*e**. Similarly, the vector product*

_{k}**=**

*c***×**

*a***can alternatively be written as**

*b***=**

*c***×**

*a***Σ**

*b*=

_{j,k}e_{ijk}

*e**. In components this leads to the following expressions:*

_{i}a_{j}b_{k}(B.41)

The triple product ** a**·

**×**

*b***in components is given by**

*c**e*, while the tensor product

_{ijk}a_{i}b_{j}c_{k}**is represented by**

*ab**a*. A useful relation involving three vectors using the tensor product is

_{i}b_{j}**× (**

*a***×**

*b***) = (**

*c***− (**

*ba***·**

*a***)**

*b***)·**

*I***.**

*c*If *e*_{i}** **= {*e*_{1}, *e*_{2}, *e*_{3}} is a basis, the tensor products *e*_{i}*e** _{j}*,

*i*,

*j*= 1, 2, 3, form a basis for representing a tensor, and we can write

**=**

*A**A*

_{kl}

*e*

_{k}

*e**. The nine real numbers*

_{l}*A*are the (Cartesian) components of the tensor

_{kl}**, and are conveniently arranged in a square matrix. It follows that**

*A**A*=

_{kl}

*e**·(*

_{k}**·**

*A*

*e**), which can be taken as the definition of the components. Applying this definition to the unit tensor, it follows that*

_{l}*δ*are the components of the unit tensor, that is,

_{kl}**=**

*I**δ*

_{kl}

*e*

_{k}

*e**. If*

_{l}**=**

*v***·**

*A***, we also have**

*u***= (**

*v**A*

_{kl}

*e*

_{k}

*e**)·*

_{l}**=**

*u*

*e**. Tensors, like vectors, can form different products. The inner product*

_{k}A_{kl}u_{l}**·**

*A***of two tensors (of rank 2)**

*B***and**

*A***yields another tensor of rank 2 and is defined by (**

*B***·**

*A***)·**

*B***=**

*u***·(**

*A***·**

*B***) = Σ**

*u**wherefrom it follows that (*

_{p,m}A_{kp}B_{pm}u_{m}**·**

*A***)**

*B**= Σ*

_{km}*, representing conventional matrix multiplication. The expression*

_{p}A_{kp}B_{pm}**:**

*A***denotes the double inner product, yields a scalar and is given in index notation by Σ**

*B**. Equivalently,*

_{i,j}A_{ij}B_{ij}**:**

*A***= tr**

*B***AB**

^{T}= tr

**A**

^{T}

**B**.

Recall that the components of a vector ** a** can be transformed to another Cartesian frame by in index notation, or

**a**′ =

**Ca**in matrix notation. Since a tensor

**of rank 2 can be interpreted as the tensor product of two vectors**

*A***and**

*b***, that is,**

*c*the transformation rule for the components of a tensor ** A** obviously is

or, in matrix notation^{4)}

If ** A**′ =

**and thus**

*A***A**′ =

**A**, then

**is an**

*A**isotropic*(or

*spherical*)

*tensor*. Further, if the component matrix of a tensor has a property which is not changed by a coordinate axes rotation that property is shared by

**′ and**

*A***. Such a property is called an**

*A**invariant*. An example is the transpose of a tensor of rank 2: If

**A**′ =

**CAC**

^{T}, then

**A**′

^{T}=

**CA**

^{T}

**C**

^{T}. Consequently, we may speak of the transpose

*A*^{T}of the tensor

**, and we may define the symmetric parts**

*A*

*A*^{(s)}and antisymmetric

*A*^{(a)}parts by

While originally a distinction in terminology is made for a scalar, a vector and a tensor, it is clear that they all transform similarly under a coordinate transformation. Therefore, a scalar is sometimes denoted as a tensor of rank 0 and a vector as a tensor of rank 1. Expressed in components, all tensors obey the same type of transformation rules, for example, *A _{i…j}* = Σ

*, where the transformation matrix*

_{p,…,q}C_{ip}…C_{jq}A_{p…q}**C**represents the rotation of the coordinate system. Scalars have no index, a vector has one index, a tensor of rank 2 has two indices, while a tensor of rank 4 has four indices. Their total transformation matrix contains a product of respectively 0, 1, 2, and 4 individual transformation matrices

*C*. Obviously, if we define (Cartesian) tensors as quantities obeying the above transformation rules

_{ij}__, extension to any order is immediate.__

^{5)}Finally, we have to mention that, like scalars, vectors and tensors can be a function of one or more variables *x _{k}*. The appropriate notation is

*f*(

*x*),

_{k}*a*(

_{i}*x*) and

_{k}*A*(

_{ij}*x*) or, equivalently,

_{k}*f*(

**),**

*x***(**

*a***) and**

*x***(**

*A***). If**

*x**x*represent the coordinates,

_{k}*f*(

**),**

*x***(**

*a***) and**

*x***(**

*A***) are referred to as a**

*x**scalar, vector,*and

*tensor field*, respectively.

Example B.11

Consider the vectors ** a**,

**and**

*b***the matrix representations of which are**

*c***B.9 Tensor Analysis**

In this section we consider various differential operators, the divergence and some other theorems and their representation in cylindrical and spherical coordinates. Consider *a _{i}* as a typical representative of tensors of rank 1 and take the partial derivatives ∂

*a*/∂

_{i}*x*. Such a derivative transforms like a tensor of rank 2, is called the

_{j}*gradient*and is denoted by grad

*a*in index notation or as ∇

_{i}**in direct notation. The gradient can operate on any tensor thereby increasing its rank by 1.**

*a*Summing over an index, known as *contraction*, decreases the rank of a tensor by 2. If we apply contraction to a tensor *A _{ij}* we calculate Σ

*and the result is known as the*

_{i}A_{ii}*trace*, written in direct notation as tr

**. Contraction of a gradient Σ**

*A**(∂*

_{i}*a*/∂

_{i}*x*) yields a scalar, called

_{i}*divergence*and denoted in direct notation by div

**or ∇·**

*a***.**

*a*Another operator is the *curl* (or *rot* as abbreviation of rotation) of a vector ** a**, in direct notation written as ∇ ×

**, which is a vector with components ∂**

*a**∂*

_{k,j}e_{ijk}*a*/∂

_{k}*x*. It is defined for 3D space only.

_{j}Finally, we have the *Laplace operator.* This can act on a scalar *a* or on a vector ** a**, is denoted by ∇

^{2}or Δ, and is defined by ∇

^{2}

*a*= Δ

*a*= ∂

*a*

^{2}

*/*∂

^{2}

*x*or ∇

_{i}^{2}

**= Δ**

*a***= ∂**

*a**a*

_{j}^{2}

*/*∂

^{2}

*x*.

_{i}Example B.12

If a vector field ** a**(

**) represented by a**

*x*^{T}(

**x**) = (3

*x*

^{2}+ 2

*y*,

*x*+

*z*

^{2},

*x*+

*y*

^{2}), then

and .

Introducing now some general theorems, let us first recall that if ** X** = ∇

*x*with

*x*a scalar, we have

__(B.42)__

Second, without proof we introduce the divergence theorem. Therefore, we consider a region of volume *V* with a piecewise smooth surface *S* on which a single-valued tensor field ** A** or

*A*is defined. The body may be either convex or nonconvex. The components of the exterior normal vector

_{ij}**of**

*n**S*are denoted by

*n*. The

_{i}*divergence theorem*or the

*theorem of Gauss*states that

__(B.43)__

The divergence theorem connects a volume integral (integrating over d*V*) to a surface integral (integrating over ** n**d

*S*) and is mainly used in theoretical work. Applying the divergence theorem to a scalar

*a*, a vector

**or a tensor Σ**

*a**∂*

_{k,j}ε_{ijk}*a*/∂

_{k}*x*we obtain in direct notation

_{j}(B.44)

Third, one can also derive *Stokes' theorem*

__(B.45)__

The theorem connects a surface (integrating over ** n**d

*S*) and line integral (integrating over d

**) and implies that the surface integral is the same for all surfaces bounded by the same curve. By the way, the surface element**

*r***d**

*n**S*is also often written as d

**.**

*S*From __Eq. (B.42)__, __(B.43)__ and __(B.45)__ one can derive many transformations of integrals. We only mention *Green's first identity* (using ** n**d

*S*= d

**)**

*S*(B.46)

and *Green's second identity*

(B.47)

The above operations can also be performed in other coordinate systems. Often, one considers systems where the base vectors *locally* still form an orthogonal basis, although the orientation may differ through space. These systems are normally addressed as *orthogonal curvilinear coordinates*. Cylindrical and spherical coordinates form examples with practical importance. Using the relations of __Example B.10__ one can show that the unit vectors for cylindrical coordinates are

so that the only non-zero derivatives are

Using the chain rule for partial derivatives, one may show that the gradient becomes

(B.48)

The divergence of a vector ** a** becomes

(B.49)

while the Laplace operator acting on a scalar *a* is expressed by

(B.50)

Using again the relations of __Example B.10__, one can show that the unit vectors for spherical coordinates are

so that the only non-zero derivatives are

The gradient operator becomes

(B.51)

The divergence of a vector ** a** becomes

(B.52)

while the Laplace operator acting on a scalar *a* is expressed by

(B.53)

**B.10 Calculus of Variations**

One of the chief applications of the calculus of variations is to find a function for which some given integral has an extreme. We treat the problem essentially as one-dimensional, but extension to more than one dimension is straightforward.

Suppose we wish to find a path *x* = *x*(*t*) between two given values *x*(*t*_{1}) and *x*(*t*_{2}) such that the *functional*__ ^{6)}__ of some function with is an extremum. Let us assume that

*x*

_{0}(

*t*) is the solution we are looking for. Other possible curves close to

*x*

_{0}(

*t*) are written as

*x*(

*t*,

*α*) =

*x*

_{0}(

*t*) +

*αη*(

*t*), where

*η*(

*t*) is any function that satisfies

*η*(

*t*

_{1}) =

*η*(

*t*

_{2}) = 0. Using such a representation, the integral

*J*becomes a function of

*α*,

(B.54)

and the condition for obtaining the extremum is (d*J*/d*α*)_{α}_{=0} = 0. We obtain

__(B.55)__

Through integration by parts the second term of the integral evaluates to

(B.56)

At *t*_{1} and *t*_{2}, *η*(*t*) = ∂*x*/∂*α* vanishes and we obtain for __Eq. (B.55)__

If we define the *variations* δ*J* = (d*J*/d*α*)_{α}_{=0}d*α* and δ*x* = (d*x*/d*α*)_{α}_{=0}d*α*, we find

(B.57)

and since *η* must be arbitrary

__(B.58)__

Once this so-called *Euler condition* is fulfilled an extremum is obtained. It should be noted that this extremum is not necessarily a minimum. Finally, we note that in case the variations at the boundaries do not vanish, that is, the values of *η* are not prescribed, the boundary term evaluates, instead of to zero, to

(B.59)

If we now require δ*J* = 0 we obtain in addition to __Eq. (B.58)__ also the boundary condition at *t* = *t*_{1} and *t* = *t*_{2}.

Example B.13

Let us calculate the shortest distance between two points in a plane. An element of an arc length in a plane is

and the total length of any curve between two points 1 and 2 is

The condition that the curve is the shortest path is

Since and , we have or , where*c* is a constant. This solution holds if where *a* is given by *a* = *c*/(1 + *c*^{2})^{1/2}. Obviously, this is the equation for a straight line *y* = *ax* + *b*, where *b* is another constant of integration. The constants *a*and *b* are determined by the condition that the curve should go through (*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}).

**B.11 Gamma Function**

The *gamma function* is defined by

For this function it generally holds that *Γ*(*t* + 1) = *tΓ*(*t*). For integer *n* it is connected to the more familiar *factorial function* *n*! by *Γ*(*n* + 1) = *n*! = *n*(*n *− 1)(*n *− 2)···(2)(1). Using *x* = *y*^{2}, we obtain

which results by setting *t* = ½ in . Consequently,

**B.12 Dirac and Heaviside Function**

The *Dirac* (*delta*) *function* *δ* (*x*) is in one dimension defined by

(B.60)

where *a* > 0 and *a* = ∞ is included and which selects the value of a function *f* at the value of variable *t* from an integral expression. Alternatively, *δ* (*x*) is defined by

(B.61)

Some properties of the delta function for *a* > 0 are

The derivative^{7)}*δ*′(*x*) is related to *f*′(*x*) as follows

This leads further to

Related is the *Heaviside* (*step*) *function* *h*(*x*), defined by

(B.62)

For *a*, *b* > 0 we have

so that the step function can be considered as the integral of the delta function.

**B.13 Laplace and Fourier Transforms**

The *Laplace transform* of a function *f*(*t*), defined by

(B.63)

transforms *f*(*t*) into where *s* may be real or complex. The operation is linear, that is,

The *convolution theorem* states that the product of two Laplace transforms *L*[*f*(*t*)] and *L*[*g*(*t*)] equals the transform of the convolution of the functions *f*(*t*) and *g*(*t*)

(B.64)

Since the Laplace transform has the property

it can transform differential equations in *t* to algebraic equations in *s*. Generalization to higher derivatives is straightforward and reads

Similarly, for integration it is found that

Some useful transforms are given in __Table B.1__.

** Table B.1** Laplace transform pairs.

The *Fourier transform* of a function *f*(*t*) and its inverse are defined by

(B.65)

The normalization constants *N*^{(–)} and *N*^{(+)} can take any value as long as their product is (2π)^{−1}. If *N*^{(–)} = *N*^{(+)} = (2π)^{−1/2} is taken, the transform is called symmetric. The Fourier transform is a linear operation for which the convolution theorem holds.

Since for the delta function *δ*(*t*) it holds that

we have as a representation of the delta function

Using

we can represent *δ*(*t*) as

Similarly, for the 3D delta function *δ*(** t**) for a vector

**we have**

*t*Finally, we note that by the Gauss theorem applied to a sphere with radius *r*

since ∇^{2}(1/*r*) = 0 for *r* ≠ 0 and ∇^{2}(1/*r*) = ∞ for *r* = 0. Therefore, we have

Applying the inverse transform we obtain or

**B.14 Some Useful Integrals and Expansions**

Several integrals and expansions, given without further comment, are useful throughout.

*Integrals*

*Bi- and Multinomial Expansion*

*Sine Cosine and Tangent*

*Exponential and Logarithm*

*Reversion of series*

*Euler McLaurin formula*

Denoting the derivative d* ^{n}f*(

*x*)/d

*x*|

^{n}

_{x}_{=a}by

*f*(

_{n}*a*), the Euler–McLaurin expression reads

with Bernoulli numbers *B*_{1} = 1/6, *B*_{2} = 1/30, *B*_{3} = 1/42, *B*_{4} = 1/30, … .

Notes

__1)__ An even (odd) permutation is the result of an even (odd) number of binary interchanges. The character (even or odd) of a permutation is independent of the order of the binary interchanges.

__2)__ In the literature the name Jacobian sometimes indicates the Jacobian determinant instead of the matrix of derivatives. To avoid confusion, we use Jacobian matrix and Jacobian determinant explicitly.

__3)__ Unfortunately in the usual convention for spherical coordinates the angle *φ* corresponds to the angle *θ* in cylindrical coordinates.

__4)__ Obviously if the transformation is interpreted as a rotation of the tensor instead of the frame, we obtain **A**′ = **C**^{T}**AC**. This is the conventional definition of an orthogonal transformation.

__5)__ This transformation rule is only appropriate for proper rotations of the axes. For improper rotations, which involve a reflection and change of handedness of the coordinate system, there are two possibilities. If the rule applies we call the tensor *polar*; if an additional change of sign occurs for an improper rotation we call the tensor *axial*. Hence, generally *L _{i..j}* = (det

**C**)

^{p}*C*, where

_{ik}…C_{jl}L_{k..l}*p*= 0 for a polar tensor and

*p*= 1 for an axial tensor. Since det

**C**= 1 for a proper rotation and det

**C**= –1 for an improper rotation, this results in an extra change of sign for an axial tensor under improper rotation. It follows that the inner and outer product of two polar or two axial tensors is polar, while the product of a polar and an axial tensor is axial. The permutation tensor

*e*is axial since

_{ijk}*e*

_{123}= 1 for both right-handed and left-handed systems. Hence, the vector product of two polar vectors is axial. If one restricts oneself to right-handed systems, the distinction is irrelevant.

__6)__ A function maps a number on a number. A functional maps a function on a number.

__7)__ A prime denotes differentiation with respect to its argument.

Further Reading

Adams, R.A. (1995) *Calculus*, 3rd edn, Addison-Wesley, Don Mills, ON.

Jeffreys, H. and Jeffreys, B.S. (1972) *Methods of Mathematical Physics*, Cambridge University Press, Cambridge.

Kreyszig, E. (1988) *Advanced Engineering Mathematics*, 6th edn, John Wiley & Sons, Inc., New York.

Margenau, H. and Murphy, G.M. (1956) *The Mathematics of Physics and Chemistry*, 2nd edn, van Nostrand, Princeton, NJ.