View source for Introduction to Mathematical Physics/Differentials and derivatives

==Definitions==

{{IMP/defn|
Let <math>E</math> and <math>F</math> two normed vectorial spaces on <math>R</math> or <math>C</math>.and <math>f</math> a
map defined on an open <math>U</math> of <math>E</math> into <math>F</math>.
<math>f</math> is said differentiable at a point <math>x_0</math> of <math>U</math> if there exists a
continuous linear application <math>g</math> from <math>E</math> into <math>F</math> such that
<math>\|f(x)-f(x_0)-g(x-x_0)\|</math> is negligible with respect to <math>\|x-x_0\|</math>.
}}

The notion of derivative is less general and is usually defined for
function for a part of <math>R</math> to a vectorial space as
follows:
{{IMP/defn|
Let <math>I</math> be an interval of <math>R</math> different from a point and <math>E</math> a
vectorial space normed on <math>R</math>. An application     <math>f</math>     from <math>I</math> to <math>E</math>
admits a derivative at the point <math>x</math> of <math>I</math> if the ratio:
{{IMP/eq|<math>
\frac{f(x+h)-f(x)}{h}
</math>}}
admits a limit as <math>h</math> tends to zero. This limit is then called
derivative of <math>f</math> at point <math>x</math> and is noted <math>f^\prime(x)</math>.
}}
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. 
{{IMP/defn|
Derivative of a distribution <math>T</math> is distribution <math>T'</math> defined by:

{{IMP/eq|<math>\forall \phi \in {\mathcal D}, <T'|\phi>=-<T|\phi'></math>}}

}}
{{IMP/defn|
Let <math>f</math> be a summable function. Assume that <math>f</math> is discontinuous at 
<math>N</math> 
points
<math>a_i</math>, 
and let us note
<math>\sigma_{a_i}=f(a_i^+)-f(a_i^-)</math> 
the jump of 
<math>f</math> at 
<math>a_i</math>.
Assume that 
<math>f'</math> 
is locally summable and almost everywhere defined. It defines
a distribution 
<math>T_{f'}</math>. 
Derivative 
<math>(T_f)'</math> 
of the distribution associated to
<math>f</math> is: 

{{IMP/eq|<math>(T_f)'=T_{f'}+\sum \sigma_{a_i} \delta_{a_i}</math>}}

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 <math>f</math>. It can be noted:

{{IMP/eq|<math>f'=\{f'\}+\sum \sigma_{a_i} \delta_{a_i}</math>}}

}}

===Case of distributions of several variables===
{{IMP/label|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 <math>S</math>:
{{IMP/eq|<math>
\frac{\partial f}{\partial x_i}=\{\frac{\partial f}{\partial
x_i}\}+n_i\sigma_f \delta_S
</math>}}


{{IMP/eq|<math>\mbox{ grad } f=\{\mbox{ grad } f\}+n \sigma_f \delta_S</math>}}

{{IMP/eq|<math>\mbox{ div } a=\{ \mbox{ div } f\}+n \sigma_a \delta_S</math>}}

{{IMP/eq|<math>\mbox{ rot } a = \{\mbox{ rot } f\}+n\wedge \sigma_a \delta_S</math>}}

where <math>f</math> is a scalar function, 
<math>a</math> a vectorial function, 
<math>\sigma</math>
represents the jump of 
<math>a</math> or 
<math>f</math> through surface 
<math>S</math> and 
<math>\delta_S</math>, 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|chaptens]]

{{IMP/exmp|''Electromagnetism'' 
The fundamental laws of electromagnetism are the Maxwell
equations:\index{passage relation} 
{{IMP/eq|<math>\mbox{ rot } E=-\frac{\partial B}{\partial t}</math>}}
{{IMP/eq|<math>\mbox{ rot } H=j+\frac{\partial D}{\partial t}</math>}}
{{IMP/eq|<math>\mbox{ div } D=\rho</math>}}
{{IMP/eq|<math>\mbox{ div } B=0</math>}}
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:

{{IMP/eq|<math>\rho=\rho_v+\rho_s</math>}}

where <math>\rho_v</math> is a volumic charge and 
<math>\rho_s</math> 
a surfacic charge, and a
current distribution defined by:

{{IMP/eq|<math>j=j_v+j_s</math>}}

where 
<math>j_v</math> 
is a volumic current, and 
<math>j_s</math> 
a surfacic current. Using the
formulas of section  [[secdisplu|secdisplu]], one obtains the following passage
relations:
<center><math>\begin{matrix}
n_{12}\wedge(E_2-E_1)&=&0\\
n_{12}\wedge(H_2-H_1)&=&j_s\\
n_{12}(D_2-D_1)&=&\rho_s\\
n_{12}(B_2-B_1)&=&0
\end{matrix}</math></center>
where the coefficients of the Delta surfacic distribution <math>\delta_s</math> have been
identified (see  ([#References|references])).}}

{{IMP/exmp|''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:
{{IMP/eq|<math>U(t)=L\frac{di}{dt}+Ri</math>}}
This equation implies that even if <math>U</math> is not continuous, <math>i</math> does. Indeed ,
if <math>i</math> is not continuous, derivative
<math>\frac{di}{dt}</math> would create a Dirac distribution in the second member.
Consider equation:
{{IMP/eq|<math>
i=\frac{dq}{dt}
</math>}}
This equation implies that <math>q(t)</math> is continuous even if <math>i</math>
is discontinuous.
}}

{{IMP/exmp|''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: 
{{IMP/eq|<math>\frac{\partial \rho}{\partial t}+\mbox{ div }(\rho \vec{u})=0</math>}}
As
{{IMP/eq|<math>
\mbox{ div } \rho {u}=\{\mbox{ div } \rho {u}\}+\sigma_{\rho u},
</math>}}
the jump <math>\sigma_{\rho u}</math> has to be zero on the discontinuity surface.
}}

==Differentiation of Stochastic processes==
{{IMP/label|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 <math>N</math>
times, then the number of possible results is <math>\Omega'=6^N</math> (the size
of the set <math>\Omega</math> grows exponentially with <math>N</math>).
We can define using this <math>\Omega'</math> a probability <math>P'</math>.
So, from the first random variable <math>X</math>, we can define another random
variable <math>X_t</math>:
{{IMP/defn|
Let <math>X</math> a random variable\index{random variable}. A stochastic process
(associated to <math>X</math>) is a 
function of <math>X</math> and <math>t</math>.
}} <math>X_t</math> is called a stochastic function of <math>X</math> or a
stochastic process.
Generally probability <math>P(X_t\in \mathrel{[}x,x+dx\mathrel{[} \mbox{ at  } t_i)</math> depends on
the history of values of <math>X_t</math> before <math>t_i</math>.
One defines the conditional probability 
<math>P(X_{t=t_i}\in \mathrel{[}x,x+dx\mathrel{[}|X_{t\leq t_i})</math> as the probability of <math>X_t</math> to
take a value between <math>x</math> and <math>x+dx</math>, at time <math>t_i</math> knowing the values
of <math>X_t</math> for times anterior to <math>t_i</math> (or <math>X_t</math> "history").
A Markov process is a stochastic process with the property that for
any set of succesive times <math>t_1,\dots,t_n</math> one has:
{{IMP/eq|<math>
P_{1|n-1}(X_{t=t_n}\in \mathrel{[}x,x+dx\mathrel{[}|X_{t_1}\dots X_{t_n})=P_{1|1}(X_{t=t_n}\in \mathrel{[}x,x+dx\mathrel{[}|X_{t_{n-1}})
</math>}}
<math>P_{i|j}</math> denotes the probability for <math>i</math> conditions to be satisfied,
knowing <math>j</math> anterior events.
In other words, the expected value of <math>X_t</math> at time <math>t_n</math> depends only
on the value of <math>X_t</math> at previous time <math>t_{n-1}</math>. It is defined by the
transition matrix by <math>P_1</math> and <math>P_{1|1}</math> (or
equivalently by the transition density function <math>f_1(x,t)</math> and 
<math>f_{1|1}(x_2,t_2|x_1,t_1)</math>.
It can be seen  ([#References|references]) that two functions <math>f_1</math> and <math>f_{1|1}</math>
defines a Markov\index{Markov process} process if and only if they verify:
   
*the Chapman-Kolmogorov equation\index{Chapman-Kolmogorov equation}:  
{{IMP/eq|<math>f_{1|1}(x_3,t_3)=\int f_{1|1}(x_3,t_3|x_2,t_2)f_{1|1}(x_2,t_2|x_1,t_1)  dx_2</math>}}    
*{{IMP/label|eqnecmar}}
{{IMP/eq|<math>f_1(x_2,t_2)=\int f_{1|1}(y_2,t_2|y_1,t_1)f_1(y_1,t_1)dx_1  </math>}}  
A Wiener process\index{Wiener process}\index{Brownian motion} (or
Brownian motion) is a Markov process for which: 
{{IMP/eq|<math>f_{1|1}(x_2,t_2|x_1,t_1)= \frac{1}{\sqrt{2\pi (t_2-t_1)}}e^{-\frac{(x_2-x_1)^2}{2(t_2-t_1)}}</math>}}
Using equation  [[#eqnecmar|eqnecmar]], one gets:
{{IMP/eq|<math>f_1(x,t)=\frac{1}{2\pi}e^{-\frac{x^2}{2t}}</math>}}

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) <math>W_t</math>.
This our second definition of a stochastic process:
{{IMP/defn|
Let <math>W_t</math> be a Brownian motion. A stochastic process is a function
of <math>W_t</math> and <math>t</math>.
}}
For instance a model of the temporal evolution of stocks ([#References|references]) 
is
{{IMP/eq|<math>X_t=e^{(\sigma W_t+(\mu -\frac{1}{2}\sigma^2)t)}</math>}}
A stochastic differential equation 
{{IMP/eq|<math>dX_t=a(t,X_t)dt+b(t,X_t)dW_t</math>}}
gives an implicit definition of the
stochastic process.
The rules of differentiation with respect to the Brownian motion
variable <math>W_t</math> 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
<math>f(W_t)</math>:
{{IMP/eq|<math>f(W_t+dW_t)-f(W_t)=f^{'}(W_t)dW_t+\frac{1}{2}f^{''}(W_t)(dW_t)^2+\dots</math>}}
Usually (for newtonian functions), the differential <math>df(W_t)</math> is just
<math>f^{'}(W_t)dW_t</math>. But, for a stochastic process <math>f(W_t)</math> the second
order term <math>\frac{1}{2}f^{''}(W_t)(dW_t)^2</math> is no more neglectible.
Indeed, as it can be seen using properties of the Brownian motion, we
have:
{{IMP/eq|<math>\int_0^t(dW_s)^2=t</math>}}
or
{{IMP/eq|<math>(dW_t)^2=dt.</math>}}
Figure  [[figbrown|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}{{IMP/label|figbrown}}]]
Let us here just mention the most basic scheme to integrate stochastic
processes using computers. Consider the time integration problem:
{{IMP/eq|<math>dX_t=a(t,X_t)dt+b(t,X_t)dW_t</math>}}
with initial value:
{{IMP/eq|<math>X_{t_0}=X_0</math>}}
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: 
{{IMP/eq|<math>X_{n+1}=X_n+a(\tau_n,Y_n)(\tau_{n+1}-\tau_n)+b(\tau_n,Y_n)(W_{\tau_{n+1}}-W_{\tau_n})</math>}}
More sofisticated methods can be found in  ([#References|references]).

==Functional derivative==

Let <math>(\phi)</math> be a functional.
To calculate the ''differential''   <math>dI(\phi)</math> of a functional
<math>I(\phi)</math> one express the difference <math>I(\phi+d\phi)-I(\phi)</math> as a
functional of <math>d\phi</math>.

The ''functional derivative''  of <math>I</math> noted <math>\frac{\delta I}{\delta \phi}</math>
is given by the limit:
{{IMP/eq|<math>
\frac{\delta I}{\delta \phi}=\lim_{a\rightarrow 0}\frac{\partial
I}{\partial \phi_i}
</math>}}
where <math>a</math> is a real and <math>\phi_i=\phi(ia)</math>.

Here are some examples:
{{IMP/exmp|
If <math>I(\phi)=\int f(y)\phi^p(y)dy</math> then <math>\frac{\delta I}{\delta
\phi}=pf(x)\phi^{p-1}(x) </math>
}}

{{IMP/exmp|
If <math>I(\phi)=\int V(\phi(y))dy</math> then <math>\frac{\delta I}{\delta
\phi}=V^\prime(\phi(x))</math>.
}}


{{IMP/label|chapretour}}
==Comparison of tensor values at different points==

===Expansion of a function in serie about x=a===


{{IMP/defn|
A function <math>f</math> admits a expansion in serie at order <math>n</math> around <math>x=a</math>
if there exists number <math>(\lambda_1,\dots,\lambda_n)</math> such that:
{{IMP/eq|<math>
f(a+h)=\sum_{k=0}^n \lambda_k h^k+h^n\epsilon(h)
</math>}}
where <math>\epsilon(h)</math> tends to zero when <math>h</math> tends to zero.
}}

{{IMP/thm|
If a function is derivable <math>n</math> times in <math>a</math>, then it admits an expansion in
serie at order <math>n</math> around <math>x=a</math> and it is given by the Taylor-Young
formula:
{{IMP/eq|<math>
f(a+h)=\sum_{k=0}^n \frac{1}{k!}f^{(k)}(a) h^k+h^n\epsilon(h)
</math>}}
where <math>\epsilon(h)</math> tends to zero when <math>h</math> tends to zero and where
<math>f^{(k)}(a)</math> is the <math>k</math> derivative of <math>f</math> at <math>x=a</math>.


}}

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

{{IMP/label|secderico}} 
===Non objective quantities===

Consider two points <math>M</math> and <math>M'</math> of coordinates <math>x^i</math> and <math>x^i+dx^i</math>.
A first variation often considered in physics is:
{{IMP/label|eqapdai}}
{{IMP/eq|<math>d(a^ie_i)=\frac{\partial a^i}{\partial x^j}dx^j e_i
</math>}}
The non objective variation is
{{IMP/eq|<math>
da^i=\frac{\partial a^i}{\partial x^j}dx^j 
</math>}}
Note that <math>da^i</math> is not a tensor and that equation  [[eqapdai|eqapdai]]
assumes that <math>e_i</math> doesn't change from point <math>M</math> to point <math>M'</math>. 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.

{{IMP/label|exmpderr}}
{{IMP/exmp|''Lagrangian speed:''  the Lagrangian
description of the mouvement of a particle number <math>a</math> is given by its position
<math>r_a</math> 
at each time 
<math>t</math>. 
If

{{IMP/eq|<math>r_a(t)=x^i(t)e_i</math>}}

the Lagrangian speed is:

{{IMP/eq|<math>\frac{dr_a}{dt}=\frac{dx^i}{dt}e_i</math>}}
}}
Derivative introduced at example  [[exmpderr|exmpderr]] is not
objective, that means that it is ''not''  invariant by axis change. In particular,
one has the famous vectorial derivation formula:
{{IMP/label|eqvectderfor}}
{{IMP/eq|<math>\frac{dA}{dt}_R=\frac{dA}{dt}_{R_1}+\omega_{R_1/R}\wedge A</math>}}

{{IMP/exmp|
Eulerian description of a fluid is given by a field of "Eulerian" <math>v(x,t)</math>
velocities and initial conditions, such that:
{{IMP/eq|<math>
x=r_a(0)
</math>}}
where <math>r_a</math> is the Lagrangian position of the particle, and:
{{IMP/eq|<math>
v(x,t)=\frac{dr_a}{dt}.
</math>}}
Eulerian and Lagrangian descriptions are equivalent. 
}}


{{IMP/exmp|
Let us consider the variation of the speed field <math>u</math> between two positions, at
time 
<math>t</math>. If speed field <math>u</math> is differentiable, there
exists a linear mapping <math>K</math> such that:
{{IMP/label|eqchampudif}}
{{IMP/eq|<math>u_i(\vec r+d\vec r)-u_i(\vec r)=K.\delta\vec r_j + O(||\vec r||)</math>}}

<math>K_{ij}=u_{i,j}</math> is called the speed field gradient tensor.
Tensor <math>K</math> can be shared into a symmetric and an antisymmetric part:
{{IMP/eq|<math>
K=
\left( \begin{array}{ccc}
e_{11}&e_{12}&e_{13}\\
           e_{21}&e_{22}&e_{23}\\
           e_{31}&e_{32}&e_{33}\\
\end{array} \right)
+
\left( \begin{array}{ccc}
0&-s_3&s_2\\
         .&0&-s_1\\
         .&.&0\\
\end{array} \right)
</math>}}
Symmetric part is called dilatation tensor, antisymmetric part is called
rotation tensor.
Now, <math>u_i(\vec r+d\vec r)-u_i(\vec r)=\frac{d\delta\vec{r}}{dt}</math>. Thus using
equation  [[eqchampudif|eqchampudif]]:
{{IMP/eq|<math>
\frac{d\delta\vec r}{dt}=K\delta\vec r
</math>}}
This result true for vector <math>\delta\vec r</math> is also true for any vector <math>\vec
a</math>. This last equation allows to show that
   
*The derivative with respect to time of the elementary volume <math>dv</math> at the  neighbourhood of a particle that is followed in its movement   is\footnote{ Indeed  

{{IMP/eq|<math>  d(\delta v)=d(\delta x)\delta y\delta z+d(\delta y)\delta x\delta z+d(\delta z)\delta x\delta y</math>}}  

{{IMP/label|eqformvol}}
{{IMP/eq|<math>\frac{d(\delta v)}{dt}=\mbox{ div } u \delta v  </math>}}  } :  {{IMP/eq|<math>  \frac{d(dv)}{dt}=\mbox{ div } u dv  </math>}}    
*The speed field of a solid is antisymmetric<ref>Indeed, let  <math>u</math> and <math>v</math> be two position vectors binded to the solid. By  definition of   a solid, scalar product <math>uv</math> remains constant as time evolves. So:  {{IMP/eq|<math>  \frac{d(uv)}{dt}=0  </math>}}  {{IMP/eq|<math>  \frac{du}{dt}v+u\frac{dv}{dt}=0  </math>}}  So:  {{IMP/eq|<math>  K_{ij}u_jv_i+u_iK_{ij}v_J=0  </math>}}  As this equality is true for any <math>u,v</math>, one has:  

{{IMP/eq|<math>  K_{ij}=-K_{ji}  </math>}}  

In other words, <math>K</math> is antisymmetrical. So, from the preceeding theorem:  {{IMP/eq|<math>  \frac{dPQ}{dt}=\Omega_{i,j}(PQ)_j  </math>}}  This can be rewritten saying that speed field is antisymmetrical, {\it i. e.},  one has:   
{{IMP/eq|<math>  V_P=V_O+\Omega\wedge(OP)  </math>}}  
</ref>.  

}}
{{IMP/label|exmppartder}}
{{IMP/exmp|''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]).
{{IMP/label|eqformalder}}
{{IMP/eq|<math>da=(\frac{\partial a^i}{\partial t}dt  + \frac{\partial a^i}{\partial
x^k}dx^k) e_i
</math>}}
As the quantity is followed during its movement, <math>\frac{dx}{dt}=u_i</math> the speed:
{{IMP/eq|<math>
\frac{d}{dt}a^i=\frac{\partial a^i}{\partial x^i}u^i+\frac{\partial
a^i}{\partial t}
</math>}}
}}


The following property can be showed  ([#References|references]):
\begin{prop}
Let us consider the integral:
{{IMP/eq|<math>
I=\int_V\omega
</math>}}
where <math>V</math> is a connex variety of dimension <math>p</math> (volume,
surface...) that is followed during its movement and <math>\omega</math> a differential
form of degree <math>p</math> expressed in Euler variables. The particular derivative of
<math>I</math> verifies:
{{IMP/eq|<math>
\frac{d}{dt}\int_V \omega = \int_V \frac{d \omega}{dt}
</math>}}
\end{prop}
A proof of this result can be found in  ([#References|references]).
{{IMP/exmp|
Consider the integral
{{IMP/eq|<math>
I=\int_V C(x,t) dv
</math>}}
where <math>D</math> is a bounded connex domain that is followed during its movement, <math>C</math>
is a scalar valuated function continuous in the closure of <math>D</math> and
differentiable in <math>D</math>. The particulaire derivative of <math>I</math> is
{{IMP/eq|<math>
\frac {dI}{dt}=\int_D\{\frac{\partial
C}{\partial t}+\mbox{ div }(C\vec{u})\}dv,
</math>}}
since from equation  [[eqformvol|eqformvol]]:
{{IMP/eq|<math>
\frac{d}{dt}(dv)=\mbox{ div }\vec{u} dv.
</math>}}
}}

{{IMP/label|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 <math>a</math> evaluated in two points <math>M</math> and <math>M'</math>. 
 {{IMP/eq|<math>
da=a(M')-a(M)=da^i e_i+a^i de_i
</math>}}
As at section  [[secderico|secderico]]:
{{IMP/eq|<math>
da^i e_i=\frac{\partial a^i}{\partial x^j}dx^j e_i
</math>}}
Variation <math>de_i</math> is linearly connected to the <math>e_j</math>'s {\it via} the tangent
application: 
{{IMP/eq|<math>
de_i=d\omega^j_ie_j
</math>}}
Rotation vector depends linearly on the displacement:
{{IMP/label|eqchr}}
{{IMP/eq|<math>de_i=\Gamma^{j}_{ik}dx^ke_j</math>}}

Symbols <math>\Gamma^{j}_{ik}</math> called Christoffel symbols<ref>
I a space with metrics <math>g_{ij}</math>
coefficients <math>\Gamma^{i}_{hk}</math> can expressed as functions of coefficients
<math>g_{ij}</math>. 
</ref>
are not<ref>Just as <math>\frac{\partial a^i}{\partial x^j}</math> is not a
tensor. However, <math>d(a^ie_i)</math> given by equation  [[eqcovdiff|eqcovdiff]] does
have the tensors properties</ref>
tensors. they
connect properties of space at <math>M</math> and its properties at point <math>M'</math>. By a
change of index in equation  [[#eqchr|eqchr]] :
{{IMP/label|eqcovdiff}}
{{IMP/eq|<math>da^i e_i=\frac{\partial a^i}{\partial x^j}dx^j e_i+a^k\Gamma^i_{kj}
dx^je_i 
</math>}}
As the <math>x^j</math>'s are independent variables:
{{IMP/defn|
The covariant derivative of a contravariant vector <math>a^i</math> is

{{IMP/label|eqdefdercov}}
{{IMP/eq|<math>\frac{Da^i}{Dx^j}=\frac{\partial a^i}{\partial x^j}+a^k\Gamma^i_{kj}</math>}}

}}
The differential can thus be noted:
{{IMP/eq|<math>
da^i=\frac{Da^i}{Dx^j}dx^j,
</math>}}
which is the generalization of the differential:
{{IMP/eq|<math>
da_i=\frac{\partial a_i}{\partial x^j}dx^j
</math>}}
considered when there are no tranformation of axes.
This formula can be generalized to tensors.
{{IMP/rem|
For the calculation of the particulaire derivative exposed at section
 [[secderico|secderico]] the <math>x^j</math> are the coordinates of the point, but the quantity
to derive depends also on time. That is the reason why a term <math>\frac{\partial
  x^j}{\partial t}</math> appear in equation  [[eqformalder|eqformalder]] but not in equation
 [[eqdefdercov|eqdefdercov]]. 
}}
{{IMP/rem|
From equation  [[eqdefdercov|eqdefdercov]] the vectorial derivation formula of equation
 [[eqvectderfor|eqvectderfor]] can be recovered when: 
{{IMP/eq|<math>
de_i=\omega_i^j dt e_j
</math>}}
}}
{{IMP/rem|
In spaces with metrics, <math>\Gamma^i_{kj}</math> are functions of the metrics
tensor <math>g^{ij}</math>.
}}
===Covariant differential operators===

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

{{IMP/eq|<math>
\mbox{ rot }(\mbox{ grad }\phi)=0
</math>}}
and
{{IMP/eq|<math>
\mbox{ div }(\mbox{ rot }(a))=0
</math>}}
{{IMP/eq|<math>
\nabla\wedge (\nabla\wedge c)=\nabla(\nabla.c)-\nabla^2a
</math>}}

<references/>

{{BookCat}}