Introduction to Mathematical Physics/Differentials and derivatives

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

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

Definition
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.

Case of distributions of several variables
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 $$S$$:

where $$f$$ is a scalar function, $$a$$ a vectorial function, $$\sigma$$ represents the jump of $$a$$ or $$f$$ through surface $$S$$ and $$\delta_S$$, 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

Differentiation of Stochastic processes
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 $$N$$ times, then the number of possible results is $$\Omega'=6^N$$ (the size of the set $$\Omega$$ grows exponentially with $$N$$). We can define using this $$\Omega'$$ a probability $$P'$$. So, from the first random variable $$X$$, we can define another random variable $$X_t$$: $$X_t$$ is called a stochastic function of $$X$$ or a stochastic process. Generally probability $$P(X_t\in \mathrel{[}x,x+dx\mathrel{[} \mbox{ at } t_i)$$ depends on the history of values of $$X_t$$ before $$t_i$$. One defines the conditional probability $$P(X_{t=t_i}\in \mathrel{[}x,x+dx\mathrel{[}|X_{t\leq t_i})$$ as the probability of $$X_t$$ to take a value between $$x$$ and $$x+dx$$, at time $$t_i$$ knowing the values of $$X_t$$ for times anterior to $$t_i$$ (or $$X_t$$ "history"). A Markov process is a stochastic process with the property that for any set of succesive times $$t_1,\dots,t_n$$ one has:

$$P_{i|j}$$ denotes the probability for $$i$$ conditions to be satisfied, knowing $$j$$ anterior events. In other words, the expected value of $$X_t$$ at time $$t_n$$ depends only on the value of $$X_t$$ at previous time $$t_{n-1}$$. It is defined by the transition matrix by $$P_1$$ and $$P_{1|1}$$ (or equivalently by the transition density function $$f_1(x,t)$$ and $$f_{1|1}(x_2,t_2|x_1,t_1)$$. It can be seen ([#References|references]) that two functions $$f_1$$ and $$f_{1|1}$$ defines a Markov\index{Markov process} process if and only if they verify: A Wiener process\index{Wiener process}\index{Brownian motion} (or Brownian motion) is a Markov process for which:
 * the Chapman-Kolmogorov equation\index{Chapman-Kolmogorov equation}:

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) $$W_t$$. This our second definition of a stochastic process:

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 $$W_t$$ 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 $$f(W_t)$$:

Usually (for newtonian functions), the differential $$df(W_t)$$ is just $$f^{'}(W_t)dW_t$$. But, for a stochastic process $$f(W_t)$$ the second order term $$\frac{1}{2}f^{''}(W_t)(dW_t)^2$$ 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} on a differentiable function}]] Let us here just mention the most basic scheme to integrate stochastic processes using computers. Consider the time integration problem:
 * center | frame |Comparison of a progressive zooming on a brownian motion and

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 derivative
Let $$(\phi)$$ be a functional. To calculate the differential  $$dI(\phi)$$ of a functional $$I(\phi)$$ one express the difference $$I(\phi+d\phi)-I(\phi)$$ as a functional of $$d\phi$$.

The functional derivative of $$I$$ noted $$\frac{\delta I}{\delta \phi}$$ is given by the limit:

where $$a$$ is a real and $$\phi_i=\phi(ia)$$.

Here are some examples:

Expansion of a function in serie about x=a
Note that the reciproque of the theorem is false: $$f(x)=\frac{1}{x^3}\sin(x)$$ is a function that admits a expansion around zero at order 2 but isn't two times derivable.

Non objective quantities
Consider two points $$M$$ and $$M'$$ of coordinates $$x^i$$ and $$x^i+dx^i$$. A first variation often considered in physics is:

The non objective variation is

Note that $$da^i$$ is not a tensor and that equation eqapdai assumes that $$e_i$$ doesn't change from point $$M$$ to point $$M'$$. 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.

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:

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

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

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

Covariant derivative
In this section a derivative that is independent from the considered reference frame is introduced (an objective derivative). Consider the difference between a quantity $$a$$ evaluated in two points $$M$$ and $$M'$$. As at section secderico:

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

Rotation vector depends linearly on the displacement:

Symbols $$\Gamma^{j}_{ik}$$ called Christoffel symbols are not tensors. they connect properties of space at $$M$$ and its properties at point $$M'$$. By a change of index in equation eqchr :

As the $$x^j$$'s are independent variables:

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.

Covariant differential operators
Following differential operators with tensorial properties can be defined: For more details on operators that can be defined on tensors, see ([#References|references]). In an orthonormal euclidian space on has the following relations:
 * Gradient of a scalar:   with $$a_i=\frac{\partial V}{\partial x^i}$$.
 * Rotational of a vector   with $$b_{ik}=\frac{\partial a_k}{\partial x^i}-\frac{\partial  a_i}{\partial x^k}$$.   the tensoriality of the rotational can be shown using the tensoriality of the  covariant derivative:
 * Divergence of a contravariant density:   where $$d=\frac{\partial a^i}{\partial x^i}$$.

and