Introduction to Mathematical Physics/Differentials and derivatives

Definitions 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: 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 sense 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. Derivative of a distribution is distribution defined by: 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 variables 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 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 references])). 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. Fluid mechanics Conservation laws are true at the distribution sense. Using distribution derivatives, so called "discontinuity" relations can be obtained immediately ([#References references]). Let us consider the case of the mass conservation law: As the jump has to be zero on the discontinuity surface. Differentiation of Stochastic processes 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 exponentially with ). We can define using this a probability . So, from the first random variable , we can define another random variable : 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: 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 derivative 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: If then If then . chapretour Comparison of tensor values at different points Expansion of a function in serie about x=a A function admits a expansion in serie at order around if there exists number such that: where tends to zero when tends to zero. 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 quantities Consider two points and of coordinates 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 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 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. 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 antisymmetricIndeed, 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: . exmppartder 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 references]). eqformalder As the quantity is followed during its movement, the speed: 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]). 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 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 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 symbolsI a space with metrics coefficients can expressed as functions of coefficients . are notJust as is not a tensor. However, given by equation eqcovdiff does have the tensors properties 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: 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. 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. From equation eqdefdercov the vectorial derivation formula of equation eqvectderfor can be recovered when: In spaces with metrics, are functions of the metrics tensor . Covariant differential operators 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