The analyzes vectorial is a branch of the Mathématiques which studies the Champ S of Scalaire S and of Vecteur S sufficiently regular of the Euclidean Espaces, i.e. the differentiable applications of a open of an Euclidean space $E$ to values respectively in $\backslash \; mathbb\; R$ and $E$. From the point of view of the mathematician, the vectorial analysis is thus a branch of the differential Géométrie. The latter includes the tensorial Analyze which brings more powerful tools and a more concise analysis inter alia vector fields.

But the importance of the vectorial analysis comes from its intensive use in Physique and in the Engineerings. It is from this point of view that we will present it, and this is why we will generally limit ourselves if $E\; =\; \backslash \; mathbb\; R^3$ is usual space with three dimensions. Within this framework, a Champ of vectors associates with each point space a vector (with three real components), while a Champ of scalars associates a reality with it. Let us imagine for example the water of a lake. The data of its temperature in each point forms a field of scalars, that its speed in each point, a field of vectors. (For a more theoretical approach, to see differential Geometry)

Linear differential main operators of sorting

The Gradient, the divergence and the Rotationnel are the three differential main operators linear first order. That means that they utilize only derivative partial (or Différentielle S) first of the fields, with the difference, for example, Laplacian which utilizes derivative partial of the second order.

• One meets them in particular in Mécanique fluids (Navier-Stokes equations).

• And also in electromagnetism, where they make it possible to express the properties of the electromagnetic Champ. The modern formulation of the Maxwell's equations uses these operators.
• Like in all the Mathematical physics (Propagation, diffusion, Resistance of the materials,…).

The formal operator Nabla

See also: Nabla

The operator Nabla $\backslash \; nabla$ car its name of an ancient quadrant which had the same shape of downwards pointing triangle. It is about a formal operator of $\backslash \; R^3$ defined in Cartesian coordinates by

$$

\ nabla = \begin{pmatrix} \ frac {\ partial} {\ partial X} \ \ \ frac {\ partial} {\ partial there} \ \ \ frac {\ partial} {\ partial Z} \end{pmatrix} . One writes also $\backslash \; vec\; \backslash \; nabla$ to stress that formally, the operator nabla has the characteristics of a vector. It besides is qualified pseudovector . He does not certainly contain scalar values, but one will use his components (which one can see as operations on standby of argument ) very exactly as one would have used the scalar values composing a vector.

The notation nabla provides a convenient means to express the vectorial operators in Cartesian coordinates.

The gradient

See also: Gradient

The Gradient is an operator who applies to a field of scalars and transforms it into field of vectors . Practically, the gradient indicates the direction of the greatest variation of the scalar field, and the intensity of this variation. For example, the gradient of altitude is directed according to the line of greater slope and its standard increases with the slope.

In mathematics, the gradient of the field $f$, supposed continuously differentiable, in the point $a$, is defined by the relation

$\backslash \; mathrm\; D\; F\; (a)\; \backslash \; cdot\; H\; =\; \backslash \; left\; (\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \_a\; F\; \backslash \; right)\; \backslash \; cdot\; h$,

where $\backslash \; mathrm\; D\; F\; (a)\; \backslash \; cdot\; h$ indicates the value on the vector $h$ differential of the function $f$ at the point $a$.

It is thus quite simply the definition of the Linear application tangent of the scalar field F (M) = F (X, there, Z) in M = A. Moreover, for a surface of equation $f\; (X,\; there,\; Z)\; =0$, the normal Vecteur on the surface at the point $a=\; (x\_a,\; y\_a,\; z\_a)$ is given by $\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \_a\; f$, which results easily from what precedes.

It results from it immediately that the derivative of the function in $a$ compared to the vector $v$ is given by

$\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \_a\; F\; \backslash \; cdot\; v.$

In dimension 3 and coordinated Cartesian, the field of gradients checks

$\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; F\; =\; \backslash \; vec\; \backslash \; nabla\; F\; =$

\begin{pmatrix} \ frac {\ partial F} {\ partial X} \ \ \ frac {\ partial F} {\ partial there} \ \ \ frac {\ partial F} {\ partial Z} \end{pmatrix}.

This relation can be useful, in the particular case where it applies, of definition of the gradient. It spreads naturally in unspecified dimension by adding components to the nabla.

Tangent linear application of a field of vectors $\backslash \; vec\; \{F\; (M)\}$

That is to say Me not relocated it a M of $\backslash \; vec\; \{H\}$; then:

$\backslash \; vec\; \{F\}\; (Me)\; -\; \backslash \; vec\; \{F\}\; (M)\; =\; (\backslash \; hat\; \{\backslash \; partial\; \backslash \; vec\; \{F\}\})\; \_M\; \backslash \; cdot\; \backslash \; vec\; \{H\}\; +\; O\; (\backslash |\backslash \; vec\; \{H\}\; \backslash |)$

the linear operator noted by a hat defines to mean that its representation in base is a square matrix, tangent linear application of the vector field F (M) .

The determinant of this operator is Jacobien of the transformation which with M associates F (M) .

Its trace will define (see hereafter) the divergence of the vector field F (M) .

That will make it possible to give rotational vector field F (M) an intrinsic definition.

One will be able to check that symbolically:

$(\backslash \; hat\; \{\backslash \; partial\; \backslash \; vec\; \{F\}\})\; \_M\; \backslash \; cdot\; \backslash \; vec\; \{H\}\; =\; (\backslash \; vec\; \{H\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; nabla\})\; \backslash \; vec\; \{F\}$

The Divergence

See also: Divergence (physical)

The divergence applies to a field of tensors of order N and transforms it into a field of tensors of order n-1 . Practically, the divergence of a vector field expresses its tendency to creep locally out of a small volume surrounding the point M where the divergence is calculated.

In dimension 3 and Cartesian coordinates, if $\backslash \; vec\; F$ is a tensor of order 1, then it is a vector and one can define the divergence by the relation

$\backslash \; mathrm\; \{div\}\; \backslash \; vec\; F\; =\; \backslash \; vec\; \backslash \; nabla\; \backslash \; cdot\; \backslash \; vec\; \{F\}$

\ frac {\ partial F_x} {\ partial X} +

\ frac {\ partial partial F_y} {\ there} + \ frac {\ partial F_z} {\ partial Z} where $\backslash \; vec\; \{F\}\; =\; (F\_x,\; F\_y,\; F\_z)$ indicates the vector field to which the operator divergence is applied. The divergence can be seen, formally, like the scalar product of the operator nabla by the “generic” vector of the field to which it is applied, which justifies the notation $\backslash \; vec\; \backslash \; nabla\; \backslash \; cdot$. Of course, this definition spreads naturally in unspecified dimension.

The definition independent of the choice of the base is:

$\backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{F\}\; =\; \backslash \; mbox\; \{Tr\}\; (\backslash \; hat\; \{\backslash \; partial\; \backslash \; vec\; \{F\}\})$

Another possible definition, more general but more difficult to formalize, consists in defining the divergence of a field of vectors in a point like the local flow of the field around this point.

The rotational one

See also: Rotational

The rotational one transforms a field of vectors into another field of vectors . More difficult to represent itself as precisely as the gradient and the divergence, it expresses the tendency which has a field to turn around a point : its local circulation on a small lace surrounding the point M is nonnull. For example  :

• in a Tornado, the wind turns around the eye of the cyclone and the vector field speed of the wind has rotational not no one around the eye. The rotational one of this field speed (in other words the field of vorticity or field swirl) is all the more intense as one is close to the eye.
• the rotational one of the field speeds $\backslash \; overrightarrow\; \{V\; (M)\}\; =\; \backslash \; vec\; \{\backslash \; Omega\_0\}\; \backslash \; wedge\; \backslash \; vec\; \{OM\}$ of a solid which turns at constant speed $\backslash \; vec\; \{\backslash \; Omega\_0\}$ constant, is directed according to the axis of rotation and is directed so that rotation takes place, compared to him, in the direct direction and is worth simply $2\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; Omega\_0\}$

In a space with 3 dimension and in Cartesian coordinates, one can define the rotational one by the relation

$\{\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{belch\}\}\}\; \backslash \; \backslash \; vec\; F\; =\; \backslash \; vec\; \backslash \; nabla\; \backslash \; wedge\; \backslash \; vec\; F$

\begin{pmatrix}

{\ partial partial F_z/\ there} - {\ partial F_y/\ partial Z} \ \ {\ partial F_x/\ partial Z} - {\ partial F_z/\ partial X} \ \ {\ partial F_y/\ partial X} - {\ partial partial F_x/\ there} \end{pmatrix} where $\backslash \; vec\; \{F\}\; =\; (F\_x,\; F\_y,\; F\_z)$ indicates the vector field to which is applied the rotational operator. The formal analogy with a vector product justifies the notation $\backslash \; vec\; \backslash \; nabla\; \backslash \; wedge$.

That can be also written, by abuse notation, using a déterminant :

where $(\backslash \; vec\; I,\; \backslash \; vec\; J,\; \backslash \; vec\; K)$ indicates the canonical base. This last expression is a little more complicated than the preceding one, but it spreads easily with other frames of reference.

• an intrinsic definition (among others) of rotational is following the  :

From the field $\backslash \; vec\; \{F\}$, one can build the field $\backslash \; vec\; \{X\_0\}\; \backslash \; wedge\; \backslash \; vec\; \{F\}$ (where $\backslash \; vec\; \{X\_0\}$ is a uniform vector) which the divergence is a linear form of $\backslash \; vec\; \{X\_0\}$ and thus exprimable by a scalar product $\backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{X\_0\}$, where $\backslash \; vec\; \{K\}$ is the opposite of rotational of $\backslash \; vec\; \{F\}$ :

$\backslash \; mathrm\; \{div\}\; (\backslash \; vec\; \{X\_0\}\; \backslash \; wedge\; \backslash \; vec\; \{F\})\; =\; -\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; vec\; \{X\_0\}$

Another possible definition, more general but more difficult to formalize, consists in defining the rotational one of a field of vectors in a point like the local circulation of the field around this point (see Rotationnel in physics).

Operators of a higher nature

The Laplacian

More used operators of order 2 is the Laplacian , of the name of the Mathématicien Pierre-Simon Laplace. The Laplacian of a field is equal to the sum of the derived seconds of this field compared to each variable.

In dimension 3, he is written:

$\backslash \; Delta=\; \backslash \; nabla^2\; =\; \backslash \; frac\; \{\backslash \; partial^2\}\; \{\backslash \; partial\; x^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\}\; \{\backslash \; partial\; y^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\}\; \{\backslash \; partial\; z^2\}$.

This definition as well has a direction for a field of scalars as for a field of vectors. One respectively speaks about scalar Laplacian and vectorial Laplacian . The scalar Laplacian of a field of scalars is a field of scalars whereas the vectorial Laplacian of a field of vectors is a field of vectors. To distinguish this last, it is noted sometimes $\backslash \; vec\; \backslash \; Delta$.

The other notation of the Laplacian which appears above, $\backslash \; nabla^2$, invites to consider it, formally, like the scalar square of the operator nabla “$\backslash \; nabla$”.

The Laplacian appears in the writing of several partial derivative equations which play a fundamental role in physics.

• simplest is the equation of Laplace $\backslash \; Delta\; F\; =\; 0$. Its solutions (of class $\backslash \; mathcal\; C^2$) are the harmonic functions, whose study is called Théorie of the potential. This name comes from the electric Potentiel, whose behavior (just as that of others Potentiel S in physics) is governed, under certain conditions, by this equation.
• the Laplacian is also used to write:

the Poisson's equation: $\{\backslash \; nabla\}\; ^2\; \backslash \; varphi\; =\; f$

or the equation of the vibrating cords: $\{\backslash \; nabla\}\; ^2\; \backslash \; varphi\; (X,\; there,\; Z,\; T)\; =\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; partial^2\; \backslash \; varphi\; (X,\; there,\; Z,\; T)\}\{\backslash \; partial\; t^2\}$

The vectorial Laplacian

The Laplacian of a vector field $\backslash \; vec\; A$ is a vector defined by the scalar Laplacian of each component of the vector field, thus in Cartesian Coordonnées, it is defined by:

$\backslash \; operatorname\; \{\backslash \; vec\; \{\backslash \; Delta\}\}\; \backslash \; vec\; has\; =\; \backslash \; operatorname\; \{\backslash \; vec\; \backslash \; nabla^2\}\; \backslash \; vec\; has\; =\; \backslash \; begin\; \{bmatrix\}\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_x\}\; \{\backslash \; x^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_x\}\; \{\backslash \; y^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_x\}\; \{\backslash \; z^2\}\; \backslash \; \backslash \; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_y\}\; \{\backslash \; x^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_y\}\; \{\backslash \; y^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_y\}\; \{\backslash \; z^2\}\; \backslash \; \backslash \; \backslash \; frac\; \{\backslash \; partial^2\; A\_z\}\; \{\backslash \; partial\; x^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_z\}\; \{\backslash \; y^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_z\}\; \{\backslash \; z^2\}\; \backslash \; end\; \{bmatrix\}\; =\; \backslash \; begin\; \{bmatrix\}\; \backslash \; Delta\; A\_x\; \backslash \; \backslash \; \backslash \; Delta\; A\_y\; \backslash \; \backslash \; \backslash \; Delta\; A\_z\; \backslash \; end\; \{bmatrix\}$

The vectorial Laplacian is present:

• in the Poisson's equation for the vectorial versions
• in mechanics of the viscous fluids where it appears in the Équations of Navier-Stokes

Some differential formulas

Attention: the following formulas are valid provided that certain assumptions are checked! (the scalar function in the first formula must be $C\_2\; (\backslash \; Omega)$, where $\backslash \; Omega\; \backslash \; subset\; \backslash \; mathbb\; \{R\}$, for example. In the same way, if $\backslash \; vec\; f$ indicates the vector function concerned in the second formula, it is necessary to check $\backslash \; vec\; F\; \backslash \; in\; C\_2\; (\backslash \; Omega)$, $\backslash \; Omega\; \backslash \; subset\; \backslash \; mathbb\; \{R\}\; ^n$.)

• $\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; (\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\})\; =\; \backslash \; vec\; \{0\}$

• $\backslash \; mathrm\; \{div\}\; (\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\})\; =0$

• $\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; (\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\})\; =\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (\backslash \; mathrm\; \{div\})\; -\; \backslash \; vec\; \{\backslash \; Delta\}$ (applied to a vector) (Rotational of rotational the)

• $\backslash \; Delta\; =\; \backslash \; mathrm\; \{div\}\; (\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\})$ (applied to a scalar)

Formulas known as of Leibniz for the products

• $\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (\backslash \; vec\; \{X\_0\}\; \backslash \; cdot\; \backslash \; vec\; \{B\})\; =\; (\backslash \; vec\; \{X\_0\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\})\; \backslash \; vec\; \{B\}\; +\; \backslash \; vec\; \{X\_0\}\; \backslash \; wedge\; \backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{B\}$ (where $\backslash \; vec\; \{X\_0\}$ is a uniform vector) and obviously:

• $\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (\backslash \; vec\; \{has\}\; \backslash \; cdot\; \backslash \; vec\; \{B\})\; =\; (\backslash \; vec\; \{has\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\})\; \backslash \; vec\; \{B\}\; +\; \backslash \; vec\; \{has\}\; \backslash \; wedge\; \backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{B\}\; +\; (\backslash \; vec\; \{B\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\})\; \backslash \; vec\; \{has\}\; +\; \backslash \; vec\; \{B\}\; \backslash \; wedge\; \backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{has\}$

• $\backslash \; nabla\; (\backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; vec\; \{F\})\; =\; 2\; (\backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; nabla)\; \backslash \; vec\; \{F\}\; +\; 2\; \backslash \; vec\; \{F\}\; \backslash \; wedge\; (\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{F\})$ (known as of Bernoulli, in mechanics of the fluids)

• $\backslash \; mathrm\; \{div\}\; (\backslash \; vec\; \{X\_0\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})\; =\; -\; \backslash \; vec\; \{X\_0\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{B\}$ (where $\backslash \; vec\; \{X\_0\}$ is a uniform vector, intrinsic definition of the Rotationnel)

• $\backslash \; mathrm\; \{div\}\; (\backslash \; vec\; \{has\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})\; =\; -\; \backslash \; vec\; \{has\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{B\}\; +\; \backslash \; vec\; \{B\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{has\}$

• $\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; (\backslash \; vec\; \{X\_0\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})\; =\; \backslash \; vec\; \{X\_0\}\; \backslash \; cdot\; \backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{B\}\; -\; (\backslash \; vec\; \{X\_0\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\})\; \backslash \; vec\; \{B\}$ (where $\backslash \; vec\; \{X\_0\}$ is a uniform vector, by definition of the tangent linear application)

• $\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; (\backslash \; vec\; \{has\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})\; =\; \backslash \; vec\; \{has\}\; \backslash \; cdot\; \backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{B\}\; -\; (\backslash \; vec\; \{has\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\})\; \backslash \; vec\; \{B\}\; -\; \backslash \; vec\; \{B\}\; \backslash \; cdot\; \backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{has\}\; +\; (\backslash \; vec\; \{B\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\})\; \backslash \; vec\; \{has\}$

• $\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (fg)\; =\; F\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (G)\; +\; G\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (F)$ (symmetrical out of F and G)

• $\backslash \; mathrm\; \{div\}\; (\backslash \; rho\; \backslash \; cdot\; \backslash \; vec\; \{V\})\; =\; \backslash \; rho\; \backslash \; cdot\; \backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{V\}\; +\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (\backslash \; rho)\; \backslash \; cdot\; \backslash \; vec\; \{V\}$

• $\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; (\backslash \; rho\; \backslash \; cdot\; \backslash \; vec\; \{V\})\; =\; \backslash \; rho\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{V\}\; +\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (\backslash \; rho)\; \backslash \; wedge\; \backslash \; vec\; \{V\}$

• $\backslash \; Delta\; (F\; \backslash \; cdot\; G)\; =\; F\; \backslash \; cdot\; \backslash \; Delta\; G\; +\; 2\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (F)\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (G)\; +g\; \backslash \; cdot\; \backslash \; Delta\; f$

• $\backslash \; mathrm\; \{div\}\; (F\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (G)\; -\; G\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (F))\; =\; F\; \backslash \; Delta\; G\; -\; G\; \backslash \; Delta\; F$

Some useful formulas

• Is F (M) and G (M) two scalar fields, there exists a field of vectors $\backslash \; vec\; \{has\}\; (M)$ such as:

$\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{has\}\; =\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; F\; \backslash \; wedge\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash ,\; g$

• the central field $\backslash \; vec\; \{OM\}\; =\; \backslash \; vec\; \{R\}$ plays a very important part in physics. Also is advisable it to memorize these some obviousnesses:

its tangent linear application is the matrix identity (cf the definition!),

thus $\backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{R\}\; =3$ and $\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; (\backslash \; vec\; \{Xo\}\; \backslash \; wedge\; \backslash \; vec\; \{R\})\; =2\; \backslash \; vec\; \{Xo\}$ (where $\backslash \; vec\; \{X\_0\}$ is a uniform vector)

• In addition $-mg\; \backslash \; vec\; \{K\}\; =\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (mgz)$; that is to say $\backslash \; vec\; \{Xo\}\; =\; \backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (\backslash \; vec\; \{Xo\}\; \backslash \; cdot\; \backslash \; vec\; \{R\})$ (where $\backslash \; vec\; \{X\_0\}$ is a uniform vector). And also:

• $\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; F\; (R)\; =f\text{'}\; (R)\; \backslash \; vec\; \{U\}$ with $\backslash \; vec\; \{U\}\; =\; \backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{R\}$

in particular $\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (r^2)\; =2\; \backslash \; vec\; \{R\}$ (obvious because $d\; (\backslash \; vec\; \{R\}\; \backslash \; cdot\; \backslash \; vec\; \{R\})\; =d\; (r^2)$)

• $\backslash \; Delta\; F\; (R)\; =f$ (R) + \ frac {2} {R} \ cdot f' (R) , except in $r=0$

• the Newtonian field , is $\backslash \; frac\; \{\backslash \; vec\; \{R\}\}\; \{r^3\}$, is very often studied,

because it is the only central field with null divergence (obvious if one thinks in term of Flow) (except $r=0$, where it is worth $4\; \backslash \; pi\; \backslash \; cdot\; \backslash \; delta\; (R)$, theorem of Gauss for the solid Angle)

It results from it that $\backslash \; Delta\; (1/r)\; =\; -\; 4\; \backslash \; pi\; \backslash \; cdot\; \backslash \; delta\; (R)$

• Thus $\backslash \; Delta\; (\backslash \; vec\; \{X\_0\}\; /r)\; =\; -\; 4\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{X\_0\}\; \backslash \; cdot\; \backslash \; delta\; (R)$

(where $\backslash \; vec\; \{X\_0\}$ is a uniform vector)

who breaks up into:

$\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; (\backslash \; mathrm\; \{div\})\; (\backslash \; vec\; \{X\_0\}\; /r)\; =\; -\; 4\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{X\_0\}\; \backslash \; cdot\; \backslash \; delta\; (R)\; \backslash \; cdot\; (1/3)$ (where $\backslash \; vec\; \{X\_0\}$ is a uniform vector), and

$\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; (\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\})\; (\backslash \; vec\; \{X\_0\}\; /r)\; =\; +\; 4\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{X\_0\}\; \backslash \; cdot\; \backslash \; delta\; (R)\; \backslash \; cdot\; (2/3)$ (where $\backslash \; vec\; \{X\_0\}$ is a uniform vector)

what is less obvious (magnetic cf Moment).

• In Mechanical of the fluids, it is necessary to still retain some " évidences" additional, for good to familiarize itself with the vectorial analysis before approaching it.

• the preceding formulas are known as of differential Calculus. It is advisable to associate them with the formulas Integral calculus: formulate of Stokes, theorem of Ostrogradski, etc

• Lastly, it is appropriate not to lose sight of the fact the axial or polar character of the Champ of studied vectors. They are absolutely not the same entities Mathématiques!

Expressions of the operators in various coordinates

Cylindrical coordinates

$\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; f=\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; R\}\; \backslash \; vec\; \{u\_r\}\; +\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; \backslash \; theta\}\; \backslash \; vec\; \{u\_\; \backslash \; theta\}\; +\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; Z\}\; \backslash \; vec\; \{u\_z\}$

$\backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{has\}\; =\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; R\}\; \backslash \; left\; (rA\_r\; \backslash \; right)\; +\; \backslash frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{\backslash \; partial\; A\_\; \backslash \; partial\; theta\}\; \{\backslash \; \backslash \; theta\}\; +\; \backslash \; frac\; \{\backslash \; partial\; A\_z\}\; \{\backslash \; partial\; Z\}$

$\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{has\}\; =\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{\backslash \; partial\; partial\; A\_z\}\; \{\backslash \; \backslash \; theta\}\; -\; \backslash \; frac\; \{\backslash \; partial\; A\_\; \backslash \; theta\}\; \{\backslash \; partial\; Z\}\; \backslash \; right)\; \backslash \; vec\; \{u\_r\}\; +\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; A\_r\}\; \{\backslash \; partial\; Z\}\; -\; \backslash \; frac\; \{\backslash \; partial\; A\_z\}\; \{\backslash \; partial\; R\}\; \backslash \; right)\; \backslash \; vec\; \{u\_\; \backslash \; theta\}\; +\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; R\}\; (rA\_\; \backslash \; theta)\; -\; \backslash \; frac\; \{\backslash \; partial\; partial\; A\_r\}\; \{\backslash \; \backslash \; theta\}\; \backslash \; right)\; \backslash \; vec\; \{u\_z\}$

$\backslash \; Delta\; f=\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; R\}\; \backslash \; left\; (R\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; R\}\; \backslash \; right)\; +\; \backslash \; frac\; \{1\}\; \{r^2\}\; \backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; \backslash \; theta^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; z^2\}$

Spherical coordinates

$\backslash \; vec\; \{\backslash \; mathrm\; \{grad\}\}\; F$

= \ frac {\ partial F} {\ partial R} \ vec {u_r} + \ frac {1} {R} \ frac {\ partial F} {\ partial \ theta} \ vec {u_ \ theta} + \ frac {1} {R \ sin \ theta} \ frac {\ partial F} {\ partial \ varphi} \ vec {u_ \ varphi}

$\backslash \; mathrm\; \{div\}\; \backslash \; vec\; \{has\}$

= \ frac {1} {r^2} \ frac {\ partial} {\ partial R} (r^2A_r) + \ frac {1} {R \ sin \ partial theta} \ frac {\} {\ partial \ theta} (\ sin \ theta A_ \ theta) + \ frac {1} {R \ sin \ theta} \ frac {\ partial partial A_ \ varphi} {\ \ varphi}

$\backslash \; vec\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; vec\; \{has\}$

= \ frac {1} {R \ sin \ theta} \ left (\ frac {\ partial} {\ partial \ theta} (\ sin \ theta A_ \ varphi) - \ frac {\ partial A_ \ partial theta} {\ \ varphi} \ right) \ vec {u_r} + \ left (\ frac {1} {R \ sin \ theta} \ frac {\ partial partial A_r} {\ \ varphi} - \ frac {1} {R} \ frac {\ partial} {\ partial R} (rA_ \ varphi) \ right) \ vec {u_ \ theta} + \ frac {1} {R} \ left (\ frac {\ partial} {\ partial R} (rA_ \ theta) - \ frac {\ partial partial A_r} {\ \ theta} \ right) \ vec {u_ \ varphi}

$\backslash \; Delta\; F$

= \ frac {1} {r^2} \ frac {\ partial} {\ partial R} \ left (r^2 \ frac {\ partial F} {\ partial R} \ right) + \ frac {1} {r^2 \ sin \ partial theta} \ frac {\} {\ partial \ theta} \ left (\ sin \ theta \ frac {\ partial F} {\ partial \ theta} \ right) + \ frac {1} {r^2 \ sin^2 \ theta} \ frac {\ partial^2 F} {\ partial \ varphi^2}

See too

• Theorem of Green
• Theorem of Stokes
• Theorem of flow-divergence
• Theorem of Helmholtz-Hodge
• Operators nabla in the cylindrical and spherical coordinates
• Electrostatic
• Magnetostatic

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