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For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation = where is an arbitrary constant vector. A tensor field of order greater than one may be decomposed into a sum of outer products, and then the following identity may be used: = ().
The image was created by using a pseudocolor plot of the dataset's Mach variable, a Mesh plot of the grid, and Vector plot of a slice through the Velocity field. City rendering : An ESRI shapefile containing a polygonal description of the building footprints was read in and then the polygons were resampled onto a rectilinear grid, which was ...
A (,)-tensor field is a differential -form, a (,)-tensor field is a vector field, and a (,)-tensor field is -vector field. While differential forms are widely studied as such in differential geometry and differential topology , multivector fields are often encountered as tensor fields of type ( 0 , k ) {\displaystyle (0,k)} , except in the ...
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .
If the interior product of a vector field with the symplectic form is an exact form (and in particular, a closed form), then it is called a Hamiltonian vector field. If the first De Rham cohomology group H 1 ( M ) {\displaystyle H^{1}(M)} of the manifold is trivial, all closed forms are exact, so all symplectic vector fields are Hamiltonian.
Compared to other integration-based techniques that compute field lines of the input vector field, LIC has the advantage that all structural features of the vector field are displayed, without the need to adapt the start and end points of field lines to the specific vector field. In other words, it shows the topology of the vector field.
If a vector field has negative divergence in some area, there will be field lines ending at points in that area. The Kelvin–Stokes theorem shows that field lines of a vector field with zero curl (i.e., a conservative vector field, e.g. a gravitational field or an electrostatic field) cannot be closed loops. In other words, curl is always ...
In a nutshell, once a set of measurements of the system state over some period of time has been acquired, one then finds the derivatives of these measurements, which forms a local vector field. They can then determine a global vector field consistent with this local field. This is usually done by a least squares fit to the derivative data.