Introduction to Mathematical Physics/Differentials and derivatives

DefinitionsEdit

Definition:

Let   and   two normed vectorial spaces on   or  .and   a map defined on an open   of   into  .   is said differentiable at a point   of   if there exists a continuous linear application   from   into   such that   is negligible with respect to  .

The notion of derivative is less general and is usually defined for function for a part of   to a vectorial space as follows:

Definition:

Let   be an interval of   different from a point and   a vectorial space normed on  . An application   from   to   admits a derivative at the point   of   if the ratio:

 

admits a limit as   tends to zero. This limit is then called derivative of   at point   and is noted  .

We will however see in this appendix some generalization of derivatives.

Derivatives in the distribution's senseEdit

DefinitionEdit

Derivative\index{derivative in the distribution sense} in the usual function sense is not defined for non continuous functions. Distribution theory allows in particular to generalize the classical derivative notion to non continuous functions.

Definition:

Derivative of a distribution   is distribution   defined by:

 


Definition:

Let   be a summable function. Assume that   is discontinuous at   points  , and let us note   the jump of   at  . Assume that   is locally summable and almost everywhere defined. It defines a distribution  . Derivative   of the distribution associated to   is:

 

One says that the derivative in the distribution sense is equal to the derivative without precaution augmented by the Dirac distribution multiplied by the jump of  . It can be noted:

 


Case of distributions of several variablesEdit

secdisplu

Using derivatives without precautions, the action of differential operators in the distribution sense can be written, in the case where the functions on which they are acting are discontinuous on a surface  :

 


 

 

 

where   is a scalar function,   a vectorial function,   represents the jump of   or   through surface   and  , is the surfacic Dirac distribution. Those formulas allow to show the Green function introduced for tensors. The geometrical implications of the differential operators are considered at next appendix chaptens

Example: Electromagnetism The fundamental laws of electromagnetism are the Maxwell equations:\index{passage relation}

 

 

 

 

are also true in the distribution sense. In books on electromagnetism, a chapter is classically devoted to the study of the boundary conditions and passage conditions. The using of distributions allows to treat this case as a particular case of the general equations. Consider for instance, a charge distribution defined by:

 

where   is a volumic charge and   a surfacic charge, and a current distribution defined by:

 

where   is a volumic current, and   a surfacic current. Using the formulas of section secdisplu, one obtains the following passage relations:

 

where the coefficients of the Delta surfacic distribution   have been identified (see ([#References

Example: Electrical circuits

As Maxwell equations are true in the distribution sense (see previous example), the equation of electricity are also true in the distribution sense. Distribution theory allows to justify some affirmations sometimes not justified in electricity courses. Consider equation:

 

This equation implies that even if   is not continuous,   does. Indeed , if   is not continuous, derivative   would create a Dirac distribution in the second member. Consider equation:

 

This equation implies that   is continuous even if   is discontinuous.

Example: Fluid mechanics Conservation laws are true at the distribution sense. Using distribution derivatives, so called "discontinuity" relations can be obtained immediately ([#References

Differentiation of Stochastic processesEdit

secstoch

When one speaks of stochastic\index{stochastic process} processes ([#References|references]), one adds the time notion. Taking again the example of the dices, if we repeat the experiment   times, then the number of possible results is   (the size of the set   grows exponentialy with  ). We can define using this   a probability  . So, from the first random variable  , we can define another random variable  :

Definition:

Let   a random variable\index{random variable}. A stochastic process (associated to  ) is a function of   and  .

  is called a stochastic function of   or a

stochastic process. Generally probability   depends on the history of values of   before  . One defines the conditional probability   as the probability of   to take a value between   and  , at time   knowing the values of   for times anterior to   (or   "history"). A Markov process is a stochastic process with the property that for any set of succesive times   one has:

 

  denotes the probability for   conditions to be satisfied, knowing   anterior events. In other words, the expected value of   at time   depends only on the value of   at previous time  . It is defined by the transition matrix by   and   (or equivalently by the transition density function   and  . It can be seen ([#References|references]) that two functions   and   defines a Markov\index{Markov process} process if and only if they verify:

  • the Chapman-Kolmogorov equation\index{Chapman-Kolmogorov equation}:

 

eqnecmar

 

A Wiener process\index{Wiener process}\index{Brownian motion} (or Brownian motion) is a Markov process for which:

 

Using equation eqnecmar, one gets:

 

As stochastic processes were defined as a function of a random variable and time, a large class\footnote{This definition excludes however discontinuous cases such as Poisson processes} of stochastic processes can be defined as a function of Brownian motion (or Wiener process)  . This our second definition of a stochastic process:

Definition:

Let   be a Brownian motion. A stochastic process is a function of   and  .

For instance a model of the temporal evolution of stocks ([#References|references]) is

 

A stochastic differential equation

 

gives an implicit definition of the stochastic process. The rules of differentiation with respect to the Brownian motion variable   differs from the rules of differentiation with respect to the ordinary time variable. They are given by the It\^o formula\index{It\^o formula} ([#References|references]). To understand the difference between the differentiation of a newtonian function and a stochastic function consider the Taylor expansion, up to second order, of a function  :

 

Usually (for newtonian functions), the differential   is just  . But, for a stochastic process   the second order term   is no more neglectible. Indeed, as it can be seen using properties of the Brownian motion, we have:

 

or

 

Figure figbrown illustrates the difference between a stochastic process (simple brownian motion in the picture) and a differentiable function. The brownian motion has a self similar structure under progressive zooms. \begin{figure} \begin{tabular}[t]{c c}

\epsffile{b0_3} \epsffile{n0_3}

\epsffile{b0_4} \epsffile{n0_4}

\epsffile{b0_5} \epsffile{n0_5} \end{tabular} | center | frame |Comparison of a progressive zooming on a brownian motion and on a differentiable function}

figbrown

]]

Let us here just mention the most basic scheme to integrate stochastic processes using computers. Consider the time integration problem:

 

with initial value:

 

The most basic way to approximate the solution of previous problem is to use the Euler (or Euler-Maruyama). This schemes satisfies the following iterative scheme:

 

More sofisticated methods can be found in ([#References|references]).

Functional derivativeEdit

Let   be a functional. To calculate the differential   of a functional   one express the difference   as a functional of  .

The functional derivative of   noted   is given by the limit:

 

where   is a real and  .

Here are some examples:

Example:

If   then  

Example:

If   then  .


chapretour

Comparison of tensor values at different pointsEdit

Expansion of a function in serie about x=aEdit

Definition:

A function   admits a expansion in serie at order   around   if there exists number   such that:

 

where   tends to zero when   tends to zero.

Theorem:

If a function is derivable   times in  , then it admits an expansion in serie at order   around   and it is given by the Taylor-Young formula:

 

where   tends to zero when   tends to zero and where   is the   derivative of   at  .


Note that the reciproque of the theorem is false:   is a function that admits a expansion around zero at order 2 but isn't two times derivable.

secderico

Non objective quantitiesEdit

Consider two points   and   of coordonates   and  . A first variation often considered in physics is:

eqapdai

 

The non objective variation is

 

Note that   is not a tensor and that equation eqapdai assumes that   doesn't change from point   to point  . It doesn't obey to tensor transformations relations. This is why it is called non objective variation. An objective variation that allows to define a tensor is presented at next section: it takes into account the variations of the basis vectors.

exmpderr

Example: Lagrangian speed: the Lagrangian description of the mouvement of a particle number   is given by its position   at each time  . If

 

the Lagrangian speed is:

 

Derivative introduced at example exmpderr is not objective, that means that it is not invariant by axis change. In particular, one has the famous vectorial derivation formula:

eqvectderfor

 

Example:

Eulerian description of a fluid is given by a field of "Eulerian"   velocities and initial conditions, such that:

 

where   is the Lagrangian position of the particle, and:

 

Eulerian and Lagrangian descriptions are equivalent.


Example:

Let us consider the variation of the speed field   between two positions, at time  . If speed field   is differentiable, there exists a linear mapping   such that:

eqchampudif

 

  is called the speed field gradient tensor. Tensor   can be shared into a symmetric and an antisymmetric part:

 

Symmetric part is called dilatation tensor, antisymmetric part is called rotation tensor. Now,  . Thus using equation eqchampudif:

 

This result true for vector   is also true for any vector  . This last equation allows to show that

  • The derivative with respect to time of the elementary volume   at the neighbourhood of a particle that is followed in its movement is\footnote{ Indeed

 

eqformvol

 

} :

 

  • The speed field of a solid is antisymmetric[1].


exmppartder

Example: Particulaire derivative of a tensor: The particulaire derivative is the time derivative of a quantity defined on a set of particles that are followed during their movement. When using Lagrange variables, it can be identified to the partial derivative with respect to time ([#References


The following property can be showed ([#References|references]): \begin{prop} Let us consider the integral:

 

where   is a connex variety of dimension   (volume, surface...) that is followed during its movement and   a differential form of degree   expressed in Euler variables. The particular derivative of   verifies:

 

\end{prop} A proof of this result can be found in ([#References|references]).

Example:

Consider the integral

 

where   is a bounded connex domain that is followed during its movement,   is a scalar valuated function continuous in the closure of   and differentiable in  . The particulaire derivative of   is

 

since from equation eqformvol:

 

secandericov

Covariant derivativeEdit

In this section a derivative that is independent from the considered reference frame is introduced (an objective derivative). Consider the difference between a quantity   evaluated in two points   and  .

 

As at section secderico:

 

Variation   is linearly connected to the  's {\it via} the tangent application:

 

Rotation vector depends linearly on the displacement:

eqchr

 

Symbols   called Christoffel symbols[2] are not[3] tensors. they connect properties of space at   and its properties at point  . By a change of index in equation eqchr :

eqcovdiff

 

As the  's are independent variables:

Definition:

The covariant derivative of a contravariant vector   is

eqdefdercov

 


The differential can thus be noted:

 

which is the generalization of the differential:

 

considered when there are no tranformation of axes. This formula can be generalized to tensors.

Remark:

For the calculation of the particulaire derivative exposed at section

secderico the   are the coordinates of the point, but the quantity

to derive depends also on time. That is the reason why a term   appear in equation eqformalder but not in equation

eqdefdercov. 

Remark:

From equation eqdefdercov the vectorial derivation formula of equation

eqvectderfor can be recovered when: 

 

Remark:

In spaces with metrics,   are functions of the metrics tensor  .

Covariant differential operatorsEdit

Following differential operators with tensorial properties can be defined:

  • Gradient of a scalar:

 

with  .

  • Rotational of a vector

 

with  . the tensoriality of the rotational can be shown using the tensoriality of the covariant derivative:

 

  • Divergence of a contravariant density:

 

where  .

For more details on operators that can be defined on tensors, see

([#References|references]). 

In an orthonormal euclidian space on has the following relations:

 

and

 

 

  1. Indeed, let   and   be two position vectors binded to the solid. By definition of a solid, scalar product   remains constant as time evolves. So:

     

     

    So:

     

    As this equality is true for any  , one has:

     

    In other words,   is antisymmetrical. So, from the preceeding theorem:

     

    This can be rewritten saying that speed field is antisymmetrical, {\it i. e.}, one has:

     

  2. I a space with metrics   coefficients   can expressed as functions of coefficients  .
  3. Just as   is not a tensor. However,   given by equation eqcovdiff does have the tensors properties