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A vector field which has zero divergence everywhere is called solenoidal.
Let x, y, z be a system of Cartesian coordinates on a 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.
The divergence of a continuously differentiable vector field
is defined to be the scalar-valued function
Another common notation for the divergence is ·F, a convenient mnemonic, where the dot denotes something just reminiscent of the dot product: take the components of (see del), apply them to the components of F, and sum the results.
In physical terms, the divergence of a vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. Indeed, an alternative, but logically equivalent definition, gives the divergence as the derivative of the net flow of the vector field across the surface of a small sphere relative to the volume of the sphere. To wit,
where S(r) denotes the sphere of radius r about a point p in R3, and the integral is a surface integral taken with respect to N, the normal to that sphere.
In light of the physical interpretation, a vector field with constant zero divergence is called incompressible – in this case, no net flow can occur across any closed surface.
The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the divergence theorem.
The following facts can all be derived from 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 productIn mathematics, the cross product is a binary operation on vectors in three dimensions. It is also known as the vector product or outer product . It differs from the dot product in that it results in a vector rather than in a scalar. Its main use lies in of two vector fields F and G in three dimensions involves the curlThis article is about curl in mathematics, see also Curl programming language and cURL, the Unix command line tool for transferring files. In vector calculus, curl is a vector operator that shows a vector field's tendency to rotate about a point. A vector and reads as follows:
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 constant zero. Conversely, if you have a vector field F with zero divergence defined on a ball in R3, say, then there exists some vector field G on the ball with F = curl(G). For regions in R3 more complicated then balls, this latter statement is not true anymore. Indeed, the degree of failure of the truth of the statement, measured by the homologyIn mathematics (especially algebraic topology and abstract algebra), homology is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). See homology theory of the chain complexHomological algebra In homological algebra, a chain complex is a sequence of abelian groups or modules A A A''. connected by homomorphisms d : A rarr A such that the composition of any two consecutive maps is zero: d o d 0 for all n''. They tend to be wri
(where the first map is the gradient, the second is the curl, the third is the divergence) serves as a nice quantification of the complicatedness of the underlying region U. These are the beginnings and main motivations of de Rham cohomologyIn mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete repre.
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