Gradient

In Physical, in vectorial Analysis, one defines the gradient as a vector quantity which indicates how a physical size varies in space. In Mathematical, the gradient is a quantity representing the variation of a function depending on several parameters compared to the variation of these various parameters.

It is current, according to the way of noting vectors, of writing the gradient of a function $f$ as follows:

$\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; ~f$ or $\backslash \; overrightarrow\; \{\backslash \; nabla\}\; f$

Often, in typography, one prefers to boldface a character to post his vectorial character:

$\backslash \; mathrm\; \{grad\}\; ~f$ or $\backslash \; nabla\; f$.

The gradient being of an major importance in physics, where it was initially employed. It can be interesting to see certain examples of them before a more mathematical definition.

The variation in temperature

See also: adiabatic Heat gradient

Gradient in only one direction

Let us imagine that we measure the temperature of a solid, a liquid, a gas in only one direction (height, length, thickness). It proves that the temperature $T$ depends on the place $x$ where it is taken. One defines then a function $T\; (X)$. One can seek, for a small variation of $x$ ($\backslash \; mathrm\; dx$), which would be the temperature variation (dT). That Ci is written $\backslash \; mathrm\; dT\; =\; T\; (X\; +\; \backslash \; mathrm\; dx)\; -\; T\; (X)$.

If one seeks to which average variation that corresponds, it is necessary to calculate

$\backslash \; frac\; \{\backslash \; mathrm\; dT\}\; \{\backslash \; mathrm\; dx\}\; =\; \backslash \; frac\; \{T\; (X\; +\; \backslash \; mathrm\; dx)\; -\; T\; (X)\}\{\backslash mathrm\; dx\}$.

It is what is called commonly the variation in temperature.

In a very pragmatic way, one can then imagine that the temperature is a linear function of displacement, the variation in temperature becomes quite simply the average variation of temperature according to the place

$\backslash \; frac\; \{\backslash \; Delta\; T\}\; \{\backslash \; Delta\; X\}$

There but some will recognize the rate of increase in the temperature $T$ according to the place and will be able to notice that, for $\backslash \; mathrm\; dx$ “very small”, this quotient approaches derived from the temperature according to the place, derived noted in mathematics $T\text{'}\; (X)$ and in physics $\backslash \; tfrac\; \{\backslash \; mathrm\; dT\}\; \{\backslash \; mathrm\; dx\}$. One calls then variation in temperature this derivative.

Variation in temperature in three different directions

Actually, the temperature thus varies according to a displacement in space according to $x,\; y$ and $z$. It is then about a function $T$ dependant on three variables $x,\; there,\; z$. A displacement in one of the three directions, induced a temperature variation which one can like previously, to quantify by

$\backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; X\}$, $\backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; there\}$, $\backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; Z\}$.

A vector then is created

$\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; (T)\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; (T)\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; X\},\; \backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; there\},\; \backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; Z\}\; \backslash \; right)$

again called variation in temperature.

Assessment: we had started from a function of $\backslash \; R^3$ in $\backslash \; R$ and we end to a vector function of $\backslash \; R^3$ in $\backslash \; R\; ^3$.

Knowing the temperature at the place $(x\_0,\; y\_0,\; z\_0)$, it is possible to determine the temperature in a point $(x\_0\; +\; \backslash \; mathrm\; dx,\; y\_0\; +\; \backslash \; mathrm\; Dy,\; z\_0\; +\; \backslash \; mathrm\; dz)$

$T\; (x\_0+\; \backslash \; mathrm\; dx,\; y\_0+\; \backslash \; mathrm\; Dy,\; z\_0+\; \backslash \; mathrm\; dz)\; =\; T\; (x\_0,\; y\_0,\; z\_0)\; +\; \backslash \; frac\; \{\backslash \; mathrm\; dT\}\; \{\backslash \; mathrm\; dx\}.\; \backslash \; mathrm\; dx\; +\; \backslash \; frac\; \{\backslash \; mathrm\; dT\}\; \{\backslash \; mathrm\; Dy\}.\; \backslash \; mathrm\; Dy\; +\; \backslash \; frac\; \{\backslash \; mathrm\; dT\}\; \{\backslash \; mathrm\; dz\}.\; \backslash mathrm\; dz$

In condensed writing, that gives

$T\; (\backslash \; vec\; \{r\_\; \{0\}\}\; +\; \backslash \; mathrm\; D\; \backslash \; vec\; \{R\})\; =\; T\; (\backslash \; vec\; \{r\_\; \{0\}\})\; +\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; (T)\; (\backslash \; vec\; \{r\_0\})\; \backslash \; cdot\; \backslash \; mathrm\; D\; \backslash \; vec\; \{R\}$

where the point represents the scalar product of the two vectors

Introduction by the differential elements

As for the Differential of which it is an alternative, the gradient can be introduced with the vocabulary of the differential elements. As example one examines the problem of the variation of the surface of a rectangle.

Let us consider in the plan $(xOy)$ a rectangle on side $x$ and $y$. Its surface is equal to $xy$ and depends on the coordinates $x$ and $y$ on the point $M$. While following an intuitive step, one agrees to note by $\backslash \; mathrm\; dx$ a very small variation of the variable $x$. When one subjects the point $M$ a very weak displacement, surface will change and one can write that:

$S+\; \backslash \; mathrm\; dS=\; (x+\; \backslash \; mathrm\; dx)\; \backslash \; cdot\; (y+\; \backslash \; mathrm\; Dy)\; =x\; \backslash \; cdot\; there\; +x\; \backslash \; cdot\; \backslash \; mathrm\; dy+y\; \backslash \; cdot\; \backslash \; mathrm\; dx\; +\; \backslash \; mathrm\; dx\; \backslash \; cdot\; \backslash \; mathrm\; dy$

One from of deduced easily that

$\backslash \; mathrm\; dS=\; there\; \backslash \; cdot\; \backslash \; mathrm\; dx+x\; \backslash \; cdot\; \backslash \; mathrm\; dy+\; \backslash \; mathrm\; dx\; \backslash \; cdot\; \backslash \; mathrm\; dy$

A simple numerical application where $x$ and $y$ would be meters and $\backslash \; mathrm\; dx$ and $\backslash \; mathrm\; dy$ of the centimetres famous that $\backslash \; mathrm\; dx.\; \backslash \; mathrm\; dy$ is negligible compared to the other sizes

One can give a precise mathematical statute to the notations $\backslash \; mathrm\; dx$ and $\backslash \; mathrm\; dy$ (which is differential forms), and to the quantity $\backslash \; mathrm\; dx.\; \backslash \; mathrm\; dy$ which is then of the second order . Preceding calculation is in fact a calculation of Développement limited to order 1, utilizing the derivative first function $xy$ compared to the two variables.

One thus writes:

$\backslash \; mathrm\; dS=\; (x+\; \backslash \; mathrm\; dx)\; \backslash \; cdot\; (y+\; \backslash \; mathrm\; Dy)\; -\; X\; \backslash \; cdot\; there\; =y\; \backslash \; cdot\; \backslash \; mathrm\; dx\; +\; X\; \backslash \; cdot\; \backslash \; mathrm\; Dy\; =\; (there,\; X)\; \backslash \; cdot\; (\backslash \; mathrm\; dx,\; \backslash \; mathrm\; Dy)\; =\; \backslash \; overrightarrow\; \backslash \; nabla\; S\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; dOM\}$

$\backslash \; overrightarrow\; \backslash \; nabla\; S\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; dOM\}\; =\; (there\; \backslash \; vec\; I\; +x\; \backslash \; vec\; J)\; \backslash \; cdot\; (\backslash \; mathrm\; dx\; \backslash \; vec\; i+\; \backslash \; mathrm\; Dy\; \backslash \; vec\; J)\; =\; \backslash \; left\; (\backslash \; frac\{\backslash \; partial\; (xy)\}\{\backslash \; partial\; X\}\; \backslash \; vec\; I\; +\; \backslash \; frac\; \{\backslash \; partial\; (xy)\}\{\backslash \; partial\; there\}\; \backslash \; vec\; J\; \backslash \; right)\; \backslash \; cdot\; (\backslash \; mathrm\; dx\; \backslash \; vec\; i+\; \backslash \; mathrm\; Dy\; \backslash \; vec\; J)$

All these equalities are various ways of writing… a scalar product of two vectors:

$\backslash \; mathrm\; dS=\; (x+\; \backslash \; mathrm\; dx)\; \backslash \; cdot\; (y+\; \backslash \; mathrm\; Dy)\; -\; X\; \backslash \; cdot\; there\; =y\; \backslash \; cdot\; \backslash \; mathrm\; dx\; +\; X\; \backslash \; cdot\; \backslash \; mathrm\; Dy\; =\; \backslash \; mathrm\; \{\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\}\; (xy)\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; dOM\}\; =\; \backslash \; overrightarrow\; \backslash \; nabla\; (xy)\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; dOM\}$ where $\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; (xy)\; =\; (there,\; X)$

The interest of the introduction of these vectors to express the variation of a function of several parameters is to visualize the fact that the function will vary more in the direction of the vector gradient and that it will not vary for any change of the parameters in a direction perpendicular to the gradient.

$(there\; \backslash \; vec\; I\; +x\; \backslash \; vec\; J)\; \backslash \; cdot\; (\backslash \; mathrm\; dx\; \backslash \; vec\; i+\; \backslash \; mathrm\; Dy\; \backslash \; vec\; J)\; =0$ for: $there\; \backslash \; mathrm\; dx\; +\; X\; \backslash \; mathrm\; Dy\; =\; 0$ in our example of the rectangle.

This will give in electrostatics the curves in the same way potential: the “equipotential ones”.

Mathematical definition

Gradient of a function of $n$ variable

That is to say $U$ open of $\backslash \; R^n$. That is to say $f:\; U\; \backslash \; to\; \backslash \; R$ a differentiable function. Either $a\; \backslash \; in\; U$, then the Differential in $a$, $\backslash \; mathrm\; df\; \backslash \; left\; (\backslash \; right\; has)$, is a linear form on $\backslash \; R^n$.

The gradient in $a$ of $f$ is then the single element of $\backslash \; R^n$, noted $\backslash \; nabla\; f$ such as:

$\backslash \; forall\; \backslash \; vec\; \{H\}\; \backslash \; in\; \backslash \; R^n\; \backslash \; qquad\; \backslash \; mathrm\; df\; (A)\; \backslash \; bigl\; (\backslash \; vec\; \{H\}\; \backslash \; bigr)\; =\; \backslash \; bigl\; (\backslash \; nabla\; F\; \backslash \; big|\; \backslash \; vec\; \{H\}\; \backslash \; bigr)$.

Here, (•|•) the scalar Produit indicates.

Thus, the gradient of $f$ at the point $a$ can be written, in the canonical base , in the form:

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

\begin{pmatrix} \ frac {\ partial F} {\ partial x_1} \ left (\ right has) \ \ \ vdots \ \ \ frac {\ partial F} {\ partial x_p} \ left (\ right has) \end{pmatrix}.

During a basic change, through a $C^1$-diffeomorphism of $\backslash \; R^n$, the expression of the gradient is modified: for example if one uses

• the cylindrical Coordonnées

$\backslash \; nabla\; F$

\ frac {\ partial F} {\ partial R} \ mathbf {E} _r

+ \ frac {1} {R} \ frac {\ partial F} {\ partial \ phi} \ mathbf {E} _ {\ phi} + \ frac {\ partial F} {\ partial Z} \ mathbf {E} _ {Z}

• spherical Coordinated

$\backslash \; nabla\; F$

\ frac {\ partial F} {\ partial R} \ mathbf {E} _r

+ \ frac {1} {R} \ frac {\ partial F} {\ partial \ theta} \ mathbf {E} _ {\ theta} + \ frac {1} {R \ sin \ theta} \ frac {\ partial F} {\ partial \ phi} \ mathbf {E} _ {\ phi}

More general definition

More generally, if $E$ is an Euclidean space and $f$ a differentiable function on open of $E$, in actual values, one can define the gradient by the formula

$\backslash \; forall\; \backslash \; vec\; \{H\}\; \backslash \; in\; E\; \backslash \; qquad\; \backslash \; mathrm\; df\; (A)\; \backslash \; bigl\; (\backslash \; vec\; \{H\}\; \backslash \; bigr)\; =\; \backslash \; bigl\; (\backslash \; nabla\; F\; \backslash \; big|\; \backslash \; vec\; \{H\}\; \backslash \; bigr)$,

since any linear form on $E$ can be regarded as the scalar application of product by a vector of $E$.

One can still extend this definition to a differentiable function definite on a Variété riemannienne $(M,\; G)$. The gradient of $f$ in $a$ is then a tangent vector with the variety in $a$, defined by

$\backslash \; forall\; \backslash \; vec\; \{H\}\; \backslash \; in\; T\_aM\; \backslash \; qquad\; \backslash \; mathrm\; df\; (A)\; \backslash \; bigl\; (\backslash \; vec\; \{H\}\; \backslash \; bigr)\; =g\; \backslash \; bigl\; (\backslash \; nabla\; F\; \backslash \; big|\; \backslash \; vec\; \{H\}\; \backslash \; bigr)$.

Lastly, if $f$ is a scalar field independent of the frame of reference, it is a Tenseur of order 0, and its partial derivative is equal to its Dérivée covariante: $(\backslash \; nabla\; F)\; \_i\; =\; \backslash \; partial\_i\; F\; =\; f\_\; \{,\; I\}\; =\; f\_\; \{;\; I\}$. In coordinates contravariantes, one calculates the field of vectors called gradient of $f$:

$(\backslash \; nabla\; F)\; ^i\; =\; g^\; \{ij\}\; ~f\_\; \{;\; J\}$

This formula allows, once established the metric Tenseur, to easily calculate the gradient in an unspecified frame of reference.

Vectorial relations

In vectorial Analysis, the gradient can be combined with other operators. That is to say $f$ a function describing a scalar field, that one supposes of class $C^2$ compared to each parameter, then:

$\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; T\}\; \backslash \; left\; (\backslash \; nabla\; F\; \backslash \; right)\; =\; \backslash \; nabla\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; T\}$;

$\backslash \; mathrm\; \{\backslash \; mathrm\; div\}\; \backslash \; left\; (\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; F\; \backslash \; right)\; =\; \backslash \; Delta\; F$;

$\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash \; left\; (\backslash \; mathrm\; \{\backslash \; mathrm\; div\}\; F\; \backslash \; right)\; =\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; left\; (\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{belch\}\}\; F\; \backslash \; right)\; +\; \backslash \; Delta\; f$;

See too

• Theorem of the gradient
• vectorial Analysis
• Descent of gradient

References

• Fundamentals off Differential Geometry , Serge Lang, Springer

• Elementary Differential Geometry, Revised 2nd Edition, Second Edition , Barrett O' Neill

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