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52-457: Divergence is a mathematical function that associates a scalar with every point of a vector field. Divergence , divergent , or variants of the word, may also refer to: Divergence 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

104-947: A is the unit vector in direction a , the divergence is The use of local coordinates is vital for the validity of the expression. If we consider x the position vector and the functions r ( x ) , θ ( x ) , and z ( x ) , which assign the corresponding global cylindrical coordinate to a vector, in general r ( F ( x ) ) ≠ F r ( x ) {\displaystyle r(\mathbf {F} (\mathbf {x} ))\neq F_{r}(\mathbf {x} )} , θ ( F ( x ) ) ≠ F θ ( x ) {\displaystyle \theta (\mathbf {F} (\mathbf {x} ))\neq F_{\theta }(\mathbf {x} )} , and z ( F ( x ) ) ≠ F z ( x ) {\displaystyle z(\mathbf {F} (\mathbf {x} ))\neq F_{z}(\mathbf {x} )} . In particular, if we consider

156-432: A 2-form to a 3-form in R . Define the current two-form as It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density ρ = 1 dx ∧ dy ∧ dz moving with local velocity F . Its exterior derivative dj is then given by Coordinate system In geometry , a coordinate system is a system that uses one or more numbers , or coordinates , to uniquely determine

208-421: A commutative ring . The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa ; this is the basis of analytic geometry . The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line . In this system, an arbitrary point O (the origin ) is chosen on a given line. The coordinate of

260-439: A right-handed or a left-handed system. Another common coordinate system for the plane is the polar coordinate system . A point is chosen as the pole and a ray from this point is taken as the polar axis . For a given angle θ , there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from

312-413: A source-free part B ( r ) . 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 with The source-free part, B , can be similarly written: one only has to replace the scalar potential Φ( r ) by a vector potential A ( r ) and the terms −∇Φ by +∇ × A , and

364-417: A Euclidean coordinate system with coordinates x 1 , x 2 , ..., x n , define In the 1D case, F reduces to a regular function, and the divergence reduces to the derivative. For any n , the divergence is a linear operator, and it satisfies the "product rule" for any scalar-valued function φ . One can express the divergence as a particular case of the exterior derivative , which takes

416-399: 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

468-409: 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

520-480: A point P is defined as the signed distance from O to P , where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system . In the plane , two perpendicular lines are chosen and

572-476: A point varies while the other coordinates are held constant, then the resulting curve is called a coordinate curve . If a coordinate curve is a straight line , it is called a coordinate line . A coordinate system for which some coordinate curves are not lines is called a curvilinear coordinate system . Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. A coordinate line with all other constant coordinates equal to zero

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624-404: A second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix , which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of

676-405: Is a homeomorphism from an open subset of a space X to an open subset of R . It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an atlas covering the space. A space equipped with such an atlas is called a manifold and additional structure can be defined on a manifold if the structure

728-421: Is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product : take the components of the ∇ operator (see del ), apply them to the corresponding components of F , and sum the results. Because applying an operator is different from multiplying the components, this is considered an abuse of notation . For a vector expressed in local unit cylindrical coordinates as where e

780-430: 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 ρ = 1 , ρ = r and ρ = r sin θ , respectively. The volume can also be expressed as ρ = | det g a b | {\textstyle \rho ={\sqrt {\left|\det g_{ab}\right|}}} , where g ab

832-413: Is assumed). Expressions of ∇ ⋅ A {\displaystyle \nabla \cdot \mathbf {A} } in cylindrical and spherical coordinates are given in the article del in cylindrical and spherical coordinates . Using Einstein notation we can consider the divergence in general coordinates , which we write as x , …, x , …, x , where n is the number of dimensions of

884-414: Is called a coordinate axis , an oriented line used for assigning coordinates. In a Cartesian coordinate system , all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwise orthogonal . A polar coordinate system is a curvilinear system where coordinate curves are lines or circles . However, one of the coordinate curves

936-417: Is consistent where the coordinate maps overlap. For example, a differentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function. In geometry and kinematics , coordinate systems are used to describe the (linear) position of points and the angular position of axes, planes, and rigid bodies . In the latter case, the orientation of

988-454: Is defined as the scalar -valued function: 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 N -dimensional vector field F in N -dimensional space is invariant under any invertible linear transformation . The common notation for the divergence ∇ · F

1040-501: Is described by coordinate transformations , which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates ( x ,  y ) and polar coordinates ( r ,  θ ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x  =  r  cos θ and y  =  r  sin θ . With every bijection from

1092-428: Is one where only the ratios of the coordinates are significant and not the actual values. Some other common coordinate systems are the following: There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length . These include: Coordinates systems are often used to specify the position of a point, but they may also be used to specify

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1144-418: Is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero. Many curves can occur as coordinate curves. For example, the coordinate curves of parabolic coordinates are parabolas . In three-dimensional space, if one coordinate is held constant and

1196-467: Is symmetric A ij = A ji then div ⁡ ( A ) = ∇ ⋅ A {\displaystyle \operatorname {div} (\mathbf {A} )=\nabla \cdot \mathbf {A} } . Because of this, often in the literature the two definitions (and symbols div and ∇ ⋅ {\displaystyle \nabla \cdot } ) are used interchangeably (especially in mechanics equations where tensor symmetry

1248-813: 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 n = 3 gives ρ = | ∂ ( x , y , z ) ∂ ( x 1 , x 2 , x 3 ) | {\textstyle \rho =\left|{\frac {\partial (x,y,z)}{\partial (x^{1},x^{2},x^{3})}}\right|} . Some conventions expect all local basis elements to be normalized to unit length, as

1300-406: 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 ρ = | det g | {\textstyle \rho ={\sqrt {\left|\det g\right|}}} . The absolute value

1352-436: 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 x 0 and approach zero volume. The result, div F , is a scalar function of x . 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

1404-523: Is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be dualistic . Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the principle of duality . There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems

1456-409: 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

1508-401: The ( n − 1) -dimensional spaces resulting from fixing a single coordinate of an n -dimensional coordinate system. The concept of a coordinate map , or coordinate chart is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map

1560-432: The curl and reads as follows: or The Laplacian of a scalar field is the divergence of the field's gradient : The divergence of the curl of any vector field (in three dimensions) is equal to zero: If a vector field F with zero divergence is defined on a ball in R , then there exists some vector field G on the ball with F = curl G . For regions in R more topologically complicated than this,

1612-457: The cylindrical coordinate system , a z -coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple ( r ,  θ ,  z ). Spherical coordinates take this a step further by converting the pair of cylindrical coordinates ( r ,  z ) to polar coordinates ( ρ ,  φ ) giving a triple ( ρ ,  θ ,  φ ). A point in

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1664-428: The position of the points or other geometric elements on a manifold such as Euclidean space . The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x -coordinate". The coordinates are taken to be real numbers in elementary mathematics , but may be complex numbers or elements of a more abstract system such as

1716-577: The volume element and F are the components of F = F i e i {\displaystyle \mathbf {F} =F^{i}\mathbf {e} _{i}} with respect to the local unnormalized covariant basis (sometimes written as e i = ∂ x / ∂ x i {\displaystyle \mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}} ) . The Einstein notation implies summation over i , since it appears as both an upper and lower index. The volume coefficient ρ

1768-432: The coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three mutually orthogonal planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n -dimensional Euclidean space. Depending on the direction and order of the coordinate axes , the three-dimensional system may be

1820-473: The divergence at any other point is zero. The divergence of a vector field F ( x ) at a point x 0 is defined as the limit of the ratio of the surface integral of F out of the closed surface of a volume V enclosing x 0 to the volume of V , as V shrinks to zero where | V | is the volume of V , S ( V ) is the boundary of V , and n ^ {\displaystyle \mathbf {\hat {n}} }

1872-460: The domain. Here, the upper index refers to the number of the coordinate or component, so x refers to the second component, and not the quantity x squared. The index variable i is used to refer to an arbitrary component, such as x . The divergence can then be written via the Voss - Weyl formula, as: where ρ {\displaystyle \rho } is the local coefficient of

1924-460: 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

1976-403: 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

2028-408: 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,

2080-467: The identity function F ( x ) = x , we find that: In spherical coordinates , with θ the angle with the z axis and φ the rotation around the z axis, and F again written in local unit coordinates, the divergence is Let A be continuously differentiable second-order tensor field defined as follows: the divergence in cartesian coordinate system is a first-order tensor field and can be defined in two ways: and We have If tensor

2132-523: The last equality with the contravariant element e ^ i {\displaystyle {\hat {\mathbf {e} }}^{i}} , we can conclude that F i = F ^ i / g i i {\textstyle F^{i}={\hat {F}}^{i}/{\sqrt {g_{ii}}}} . After substituting, the formula becomes: See § In curvilinear coordinates for further discussion. The following properties can all be derived from

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2184-559: 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 serves as a nice quantification of the complicatedness of the underlying region U . These are the beginnings and main motivations of de Rham cohomology . It can be shown that any stationary flux v ( r ) that is twice continuously differentiable in R and vanishes sufficiently fast for | r | → ∞ can be decomposed uniquely into an irrotational part E ( r ) and

2236-458: The ordinary differentiation rules of calculus . Most importantly, the divergence is a linear operator , i.e., for all vector fields F and G and all real numbers a and b . There is a product rule of the following type: if φ is a scalar-valued function and F is a vector field, then or in more suggestive notation Another product rule for the cross product of two vector fields F and G in three dimensions involves

2288-448: The origin is r for given number r . For a given pair of coordinates ( r ,  θ ) there is a single point, but any point is represented by many pairs of coordinates. For example, ( r ,  θ ), ( r ,  θ +2 π ) and (− r ,  θ + π ) are all polar coordinates for the same point. The pole is represented by (0, θ ) for any value of θ . There are two common methods for extending the polar coordinate system to three dimensions. In

2340-453: The other two are allowed to vary, then the resulting surface is called a coordinate surface . For example, the coordinate surfaces obtained by holding ρ constant in the spherical coordinate system are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak of coordinate planes . Similarly, coordinate hypersurfaces are

2392-422: The plane may be represented in homogeneous coordinates by a triple ( x ,  y ,  z ) where x / z and y / z are the Cartesian coordinates of the point. This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity . In general, a homogeneous coordinate system

2444-549: The position of more complex figures such as lines, planes, circles or spheres . For example, Plücker coordinates are used to determine the position of a line in space. When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term line coordinates is used for any coordinate system that specifies the position of a line. It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this

2496-421: The source density div v by the circulation density ∇ × v . 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. The divergence of a vector field can be defined in any finite number n {\displaystyle n} of dimensions. If in

2548-399: The space to itself two coordinate transformations can be associated: For example, in 1D , if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more. Given a coordinate system, if one of the coordinates of

2600-556: 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. In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field F = F x i + F y j + F z k {\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} }

2652-403: 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. In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source or a sink at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of

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2704-424: Was done in the previous sections. If we write e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} for the normalized basis, and F ^ i {\displaystyle {\hat {F}}^{i}} for the components of F with respect to it, we have that using one of the properties of the metric tensor. By dotting both sides of

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