Английская Википедия:Divergence
Шаблон:Short description Шаблон:About Шаблон:Calculus
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.
Physical interpretation of divergence
In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of the field vectors exiting from an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence, and is often called a "source" of the field. A point at which the flux is directed inward has negative divergence, and is often called a "sink" of the field. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point. A point at which there is zero flux through an enclosing surface has zero divergence.
The divergence of a vector field is often illustrated using the simple example of the velocity field of a fluid, a liquid or gas. A moving gas has a velocity, a speed and direction at each point, which can be represented by a vector, so the velocity of the gas forms a vector field. If a gas is heated, it will expand. This will cause a net motion of gas particles outward in all directions. Any closed surface in the gas will enclose gas which is expanding, so there will be an outward flux of gas through the surface. So the velocity field will have positive divergence everywhere. Similarly, if the gas is cooled, it will contract. There will be more room for gas particles in any volume, so the external pressure of the fluid will cause a net flow of gas volume inward through any closed surface. Therefore the velocity field has negative divergence everywhere. In contrast, in a gas at a constant temperature and pressure, the net flux of gas out of any closed surface is zero. The gas may be moving, but the volume rate of gas flowing into any closed surface must equal the volume rate flowing out, so the net flux is zero. Thus the gas velocity has zero divergence everywhere. A field which has zero divergence everywhere is called solenoidal.
If the gas is heated only at one point or small region, or a small tube is introduced which supplies a source of additional gas at one point, the gas there will expand, pushing fluid particles around it outward in all directions. This will cause an outward velocity field throughout the gas, centered on the heated point. Any closed surface enclosing the heated point will have a flux of gas particles passing out of it, so there is positive divergence at that point. However any closed surface not enclosing the point will have a constant density of gas inside, so just as many fluid particles are entering as leaving the volume, thus the net flux out of the volume is zero. Therefore the divergence at any other point is zero.
Definition
The divergence of a vector field Шаблон:Math at a point Шаблон:Math is defined as the limit of the ratio of the surface integral of Шаблон:Math out of the closed surface of a volume Шаблон:Math enclosing Шаблон:Math to the volume of Шаблон:Math, as Шаблон:Math shrinks to zero
where Шаблон:Math is the volume of Шаблон:Math, Шаблон:Math is the boundary of Шаблон:Math, and <math>\mathbf{\hat n}</math> is the outward unit normal to that surface. It can be shown that the above limit always converges to the same value for any sequence of volumes that contain Шаблон:Math and approach zero volume. The result, Шаблон:Math, is a scalar function of Шаблон:Math.
Since this definition is coordinate-free, it shows that the divergence is the same in any coordinate system. However it is not often used practically to calculate divergence; when the vector field is given in a coordinate system the coordinate definitions below are much simpler to use.
A vector field with zero divergence everywhere is called solenoidal – in which case any closed surface has no net flux across it.
Definition in coordinates
Cartesian coordinates
In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field <math>\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}</math> is defined as the scalar-valued function:
- <math>\operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (F_x,F_y,F_z) = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}.</math>
Although expressed in terms of coordinates, the result is invariant under rotations, as the physical interpretation suggests. This is because the trace of the Jacobian matrix of an Шаблон:Math-dimensional vector field Шаблон:Math in Шаблон:Mvar-dimensional space is invariant under any invertible linear transformationШаблон:Clarification needed.
The common notation for the divergence Шаблон:Math is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of the Шаблон:Math operator (see del), apply them to the corresponding components of Шаблон:Math, and sum the results. Because applying an operator is different from multiplying the components, this is considered an abuse of notation.
Cylindrical coordinates
For a vector expressed in local unit cylindrical coordinates as
- <math>\mathbf{F}= \mathbf{e}_r F_r + \mathbf{e}_\theta F_\theta + \mathbf{e}_z F_z,</math>
where Шаблон:Math is the unit vector in direction Шаблон:Math, the divergence isШаблон:Refn
- <math>\operatorname{div} \mathbf F = \nabla \cdot \mathbf{F} = \frac{1}{r} \frac{\partial}{\partial r} \left(r F_r\right) + \frac1r \frac{\partial F_\theta}{\partial\theta} + \frac{\partial F_z}{\partial z}.
</math>
The use of local coordinates is vital for the validity of the expression. If we consider Шаблон:Math the position vector and the functions Шаблон:Math, Шаблон:Math, and Шаблон:Math, which assign the corresponding global cylindrical coordinate to a vector, in general <math>r(\mathbf{F}(\mathbf{x}))\neq F_r(\mathbf{x})</math>, <math>\theta(\mathbf{F}(\mathbf{x}))\neq F_{\theta}(\mathbf{x})</math>, and <math>z(\mathbf{F}(\mathbf{x}))\neq F_z(\mathbf{x})</math>. In particular, if we consider the identity function Шаблон:Math, we find that:
- <math>\theta(\mathbf{F}(\mathbf{x})) = \theta \neq F_{\theta}(\mathbf{x}) = 0</math>.
Spherical coordinates
In spherical coordinates, with Шаблон:Mvar the angle with the Шаблон:Mvar axis and Шаблон:Mvar the rotation around the Шаблон:Mvar axis, and Шаблон:Math again written in local unit coordinates, the divergence isШаблон:Refn
- <math>\operatorname{div}\mathbf{F} = \nabla \cdot \mathbf{F} = \frac1{r^2} \frac{\partial}{\partial r}\left(r^2 F_r\right) + \frac1{r\sin\theta} \frac{\partial}{\partial \theta} (\sin\theta\, F_\theta) + \frac1{r\sin\theta} \frac{\partial F_\varphi}{\partial \varphi}.</math>
Tensor field
Let Шаблон:Math be continuously differentiable second-order tensor field defined as follows:
- <math>\mathbf{A} = \begin{bmatrix}
A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix}</math>
the divergence in cartesian coordinate system is a first-order tensor fieldШаблон:Sfn and can be defined in two ways:[1]
- <math>\operatorname{div} (\mathbf{A})
= \cfrac{\partial A_{ik}}{\partial x_k}~\mathbf{e}_i = A_{ik,k}~\mathbf{e}_i = \begin{bmatrix} \dfrac{\partial A_{11}}{\partial x_1} +\dfrac{\partial A_{12}}{\partial x_2} +\dfrac{\partial A_{13}}{\partial x_3} \\ \dfrac{\partial A_{21}}{\partial x_1} +\dfrac{\partial A_{22}}{\partial x_2} +\dfrac{\partial A_{23}}{\partial x_3} \\ \dfrac{\partial A_{31}}{\partial x_1} +\dfrac{\partial A_{32}}{\partial x_2} +\dfrac{\partial A_{33}}{\partial x_3} \end{bmatrix}</math>
- <math>
\nabla\cdot \mathbf A = \cfrac{\partial A_{ki}}{\partial x_k}~\mathbf{e}_i = A_{ki,k}~\mathbf{e}_i = \begin{bmatrix} \dfrac{\partial A_{11}}{\partial x_1} + \dfrac{\partial A_{21}}{\partial x_2} + \dfrac{\partial A_{31}}{\partial x_3} \\ \dfrac{\partial A_{12}}{\partial x_1} + \dfrac{\partial A_{22}}{\partial x_2} + \dfrac{\partial A_{32}}{\partial x_3} \\ \dfrac{\partial A_{13}}{\partial x_1} + \dfrac{\partial A_{23}}{\partial x_2} + \dfrac{\partial A_{33}}{\partial x_3} \\ \end{bmatrix} </math>
We have
- <math>\operatorname{div} (\mathbf{A^T}) = \nabla\cdot\mathbf A</math>
If tensor is symmetric Шаблон:Math then <math>\operatorname{div} (\mathbf{A}) = \nabla\cdot\mathbf A</math>. Because of this, often in the literature the two definitions (and symbols Шаблон:Math and <math>\nabla\cdot</math>) are used interchangeably (especially in mechanics equations where tensor symmetry is assumed).
Expressions of <math>\nabla\cdot\mathbf A</math> in cylindrical and spherical coordinates are given in the article del in cylindrical and spherical coordinates.
General coordinates
Using Einstein notation we can consider the divergence in general coordinates, which we write as Шаблон:Math, where Шаблон:Mvar is the number of dimensions of the domain. Here, the upper index refers to the number of the coordinate or component, so Шаблон:Math refers to the second component, and not the quantity Шаблон:Mvar squared. The index variable Шаблон:Mvar is used to refer to an arbitrary component, such as Шаблон:Math. The divergence can then be written via the Voss-Weyl formula,[5] as:
- <math>\operatorname{div}(\mathbf{F}) = \frac{1}{\rho} \frac{\partial \left(\rho\, F^i\right)}{\partial x^i},</math>
where <math>\rho</math> is the local coefficient of the volume element and Шаблон:Math are the components of Шаблон:Nowrap with respect to the local unnormalized covariant basis (sometimes written as Шаблон:Nowrap. The Einstein notation implies summation over Шаблон:Mvar, since it appears as both an upper and lower index.
The volume coefficient Шаблон:Mvar is a function of position which depends on the coordinate system. In Cartesian, cylindrical and spherical coordinates, using the same conventions as before, we have Шаблон:Math, Шаблон:Math and Шаблон:Math, respectively. The volume can also be expressed as <math display="inline">\rho = \sqrt{\left|\det g_{ab}\right|}</math>, where Шаблон:Math is the metric tensor. The determinant appears because it provides the appropriate invariant definition of the volume, given a set of vectors. Since the determinant is a scalar quantity which doesn't depend on the indices, these can be suppressed, writing <math display="inline">\rho=\sqrt{\left|\det g\right|}</math>. The absolute value is taken in order to handle the general case where the determinant might be negative, such as in pseudo-Riemannian spaces. The reason for the square-root is a bit subtle: it effectively avoids double-counting as one goes from curved to Cartesian coordinates, and back. The volume (the determinant) can also be understood as the Jacobian of the transformation from Cartesian to curvilinear coordinates, which for Шаблон:Math gives Шаблон:Nowrap
Some conventions expect all local basis elements to be normalized to unit length, as was done in the previous sections. If we write <math>\hat{\mathbf{e}}_i</math> for the normalized basis, and <math>\hat{F}^i</math> for the components of Шаблон:Math with respect to it, we have that
- <math>\mathbf{F}=F^i \mathbf{e}_i =
F^i \|{\mathbf{e}_i }\| \frac{\mathbf{e}_i}{\| \mathbf{e}_i \|} = F^i \sqrt{g_{ii}} \, \hat{\mathbf{e}}_i = \hat{F}^i \hat{\mathbf{e}}_i,</math> using one of the properties of the metric tensor. By dotting both sides of the last equality with the contravariant element <math>\hat{\mathbf{e}}^i</math>, we can conclude that <math display="inline">F^i = \hat{F}^i / \sqrt{g_{ii}}</math>. After substituting, the formula becomes:
- <math>\operatorname{div}(\mathbf{F}) = \frac 1{\rho} \frac{\partial \left(\frac{\rho}{\sqrt{g_{ii}}}\hat{F}^i\right)}{\partial x^i} =
\frac 1{\sqrt{\det g}} \frac{\partial \left(\sqrt{\frac{\det g}{g_{ii}}}\,\hat{F}^i\right)}{\partial x^i}.</math>
See Шаблон:Section link for further discussion.
Properties
The following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, the divergence is a linear operator, i.e.,
- <math>\operatorname{div}(a\mathbf{F} + b\mathbf{G}) = a \operatorname{div} \mathbf{F} + b \operatorname{div} \mathbf{G}</math>
for all vector fields Шаблон:Math and Шаблон:Math and all real numbers Шаблон:Math and Шаблон:Math.
There is a product rule of the following type: if Шаблон:Mvar is a scalar-valued function and Шаблон:Math is a vector field, then
- <math>\operatorname{div}(\varphi \mathbf{F}) = \operatorname{grad} \varphi \cdot \mathbf{F} + \varphi \operatorname{div} \mathbf{F},</math>
or in more suggestive notation
- <math>\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi (\nabla\cdot\mathbf{F}).</math>
Another product rule for the cross product of two vector fields Шаблон:Math and Шаблон:Math in three dimensions involves the curl and reads as follows:
- <math>\operatorname{div}(\mathbf{F}\times\mathbf{G}) = \operatorname{curl} \mathbf{F} \cdot\mathbf{G} - \mathbf{F} \cdot \operatorname{curl} \mathbf{G},</math>
or
- <math>\nabla\cdot(\mathbf{F}\times\mathbf{G}) = (\nabla\times\mathbf{F})\cdot\mathbf{G} - \mathbf{F}\cdot(\nabla\times\mathbf{G}).</math>
The Laplacian of a scalar field is the divergence of the field's gradient:
- <math>\operatorname{div}(\operatorname{grad}\varphi) = \Delta\varphi.</math>
The divergence of the curl of any vector field (in three dimensions) is equal to zero:
- <math>\nabla\cdot(\nabla\times\mathbf{F})=0.</math>
If a vector field Шаблон:Math with zero divergence is defined on a ball in Шаблон:Math, then there exists some vector field Шаблон:Math on the ball with Шаблон:Math. For regions in Шаблон:Math more topologically complicated than this, the latter statement might be false (see Poincaré lemma). The degree of failure of the truth of the statement, measured by the homology of the chain complex
- <math>\{ \text{scalar fields on } U \} ~ \overset{\operatorname{grad}}{\rarr} ~ \{ \text{vector fields on } U \} ~ \overset{\operatorname{curl}}{\rarr} ~ \{ \text{vector fields on } U \} ~ \overset{\operatorname{div}}{\rarr} ~ \{ \text{scalar fields on } U \}</math>
serves as a nice quantification of the complicatedness of the underlying region Шаблон:Math. These are the beginnings and main motivations of de Rham cohomology.
Decomposition theorem
Шаблон:Main It can be shown that any stationary flux Шаблон:Math that is twice continuously differentiable in Шаблон:Math and vanishes sufficiently fast for Шаблон:Math can be decomposed uniquely into an irrotational part Шаблон:Math and a source-free part Шаблон:Math. Moreover, these parts are explicitly determined by the respective source densities (see above) and circulation densities (see the article Curl):
For the irrotational part one has
- <math>\mathbf E=-\nabla \Phi(\mathbf r),</math>
with
- <math>\Phi (\mathbf{r})=\int_{\mathbb R^3}\,d^3\mathbf r'\;\frac{\operatorname{div} \mathbf{v}(\mathbf{r}')}{4\pi\left|\mathbf{r}-\mathbf{r}'\right|}.</math>
The source-free part, Шаблон:Math, can be similarly written: one only has to replace the scalar potential Шаблон:Math by a vector potential Шаблон:Math and the terms Шаблон:Math by Шаблон:Math, and the source density Шаблон:Math by the circulation density Шаблон:Math.
This "decomposition theorem" is a by-product of the stationary case of electrodynamics. It is a special case of the more general Helmholtz decomposition, which works in dimensions greater than three as well.
In arbitrary finite dimensions
The divergence of a vector field can be defined in any finite number <math>n</math> of dimensions. If
- <math>\mathbf{F} = (F_1 , F_2 , \ldots F_n) ,</math>
in a Euclidean coordinate system with coordinates Шаблон:Math, define
- <math>\operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} = \frac{\partial F_1}{\partial x_1} + \frac{\partial F_2}{\partial x_2} + \cdots + \frac{\partial F_n}{\partial x_n}.</math>
In the 1D case, Шаблон:Math reduces to a regular function, and the divergence reduces to the derivative.
For any Шаблон:Math, the divergence is a linear operator, and it satisfies the "product rule"
- <math>\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi (\nabla\cdot\mathbf{F})</math>
for any scalar-valued function Шаблон:Mvar.
Relation to the exterior derivative
One can express the divergence as a particular case of the exterior derivative, which takes a 2-form to a 3-form in Шаблон:Math. Define the current two-form as
- <math>j = F_1 \, dy \wedge dz + F_2 \, dz \wedge dx + F_3 \, dx \wedge dy .</math>
It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density Шаблон:Math moving with local velocity Шаблон:Math. Its exterior derivative Шаблон:Math is then given by
- <math>dj = \left(\frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z} \right) dx \wedge dy \wedge dz = (\nabla \cdot {\mathbf F}) \rho </math>
where <math>\wedge</math> is the wedge product.
Thus, the divergence of the vector field Шаблон:Math can be expressed as:
- <math>\nabla \cdot {\mathbf F} = {\star} d{\star} \big({\mathbf F}^\flat \big) .</math>
Here the superscript Шаблон:Music is one of the two musical isomorphisms, and Шаблон:Math is the Hodge star operator. When the divergence is written in this way, the operator <math>{\star} d{\star}</math> is referred to as the codifferential. Working with the current two-form and the exterior derivative is usually easier than working with the vector field and divergence, because unlike the divergence, the exterior derivative commutes with a change of (curvilinear) coordinate system.
In curvilinear coordinates
The appropriate expression is more complicated in curvilinear coordinates. The divergence of a vector field extends naturally to any differentiable manifold of dimension Шаблон:Math that has a volume form (or density) Шаблон:Mvar, e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two-form for a vector field on Шаблон:Math, on such a manifold a vector field Шаблон:Math defines an Шаблон:Math-form Шаблон:Math obtained by contracting Шаблон:Math with Шаблон:Mvar. The divergence is then the function defined by
- <math>dj = (\operatorname{div} X) \mu .</math>
The divergence can be defined in terms of the Lie derivative as
- <math>{\mathcal L}_X \mu = (\operatorname{div} X) \mu .</math>
This means that the divergence measures the rate of expansion of a unit of volume (a volume element) as it flows with the vector field.
On a pseudo-Riemannian manifold, the divergence with respect to the volume can be expressed in terms of the Levi-Civita connection Шаблон:Math:
- <math>\operatorname{div} X = \nabla \cdot X = {X^a}_{;a} ,</math>
where the second expression is the contraction of the vector field valued 1-form Шаблон:Math with itself and the last expression is the traditional coordinate expression from Ricci calculus.
An equivalent expression without using a connection is
- <math>\operatorname{div}(X) = \frac{1}{\sqrt{\left|\det g \right|}} \, \partial_a \left(\sqrt{\left|\det g \right|} \, X^a\right),</math>
where Шаблон:Mvar is the metric and <math>\partial_a</math> denotes the partial derivative with respect to coordinate Шаблон:Math. The square-root of the (absolute value of the determinant of the) metric appears because the divergence must be written with the correct conception of the volume. In curvilinear coordinates, the basis vectors are no longer orthonormal; the determinant encodes the correct idea of volume in this case. It appears twice, here, once, so that the <math>X^a</math> can be transformed into "flat space" (where coordinates are actually orthonormal), and once again so that <math>\partial_a</math> is also transformed into "flat space", so that finally, the "ordinary" divergence can be written with the "ordinary" concept of volume in flat space (i.e. unit volume, i.e. one, i.e. not written down). The square-root appears in the denominator, because the derivative transforms in the opposite way (contravariantly) to the vector (which is covariant). This idea of getting to a "flat coordinate system" where local computations can be done in a conventional way is called a vielbein. A different way to see this is to note that the divergence is the codifferential in disguise. That is, the divergence corresponds to the expression <math>\star d\star</math> with <math>d</math> the differential and <math>\star</math> the Hodge star. The Hodge star, by its construction, causes the volume form to appear in all of the right places.
The divergence of tensors
Divergence can also be generalised to tensors. In Einstein notation, the divergence of a contravariant vector Шаблон:Mvar is given by
- <math>\nabla \cdot \mathbf{F} = \nabla_\mu F^\mu ,</math>
where Шаблон:Math denotes the covariant derivative. In this general setting, the correct formulation of the divergence is to recognize that it is a codifferential; the appropriate properties follow from there.
Equivalently, some authors define the divergence of a mixed tensor by using the musical isomorphism Шаблон:Music: if Шаблон:Math is a Шаблон:Math-tensor (Шаблон:Math for the contravariant vector and Шаблон:Math for the covariant one), then we define the divergence of Шаблон:Mvar to be the Шаблон:Math-tensor
- <math>(\operatorname{div} T) (Y_1 , \ldots , Y_{q-1}) = {\operatorname{trace}} \Big(X \mapsto \sharp (\nabla T) (X , \cdot , Y_1 , \ldots , Y_{q-1}) \Big);</math>
that is, we take the trace over the first two covariant indices of the covariant derivative.Шаблон:Efn The <math>\sharp</math> symbol refers to the musical isomorphism.
See also
Notes
Citations
References
External links
- Шаблон:Springer
- The idea of divergence of a vector field
- Khan Academy: Divergence video lesson
- Шаблон:Cite webШаблон:Cbignore
Шаблон:Calculus topics Шаблон:Authority control
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