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C# 3.0 introduced type inference, allowing the type specifier of a variable declaration to be replaced by the keyword var, if its actual type can be statically determined from the initializer. This reduces repetition, especially for types with multiple generic type-parameters , and adheres more closely to the DRY principle.
The flow field around an airplane is a vector field in R 3, here visualized by bubbles that follow the streamlines showing a wingtip vortex. Vector fields are commonly used to create patterns in computer graphics. Here: abstract composition of curves following a vector field generated with OpenSimplex noise.
A property, in some object-oriented programming languages, is a special sort of class member, intermediate in functionality between a field (or data member) and a method.The syntax for reading and writing of properties is like for fields, but property reads and writes are (usually) translated to 'getter' and 'setter' method calls.
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: = ().
A vector field is a vector-valued function that, generally, has a domain of the same dimension (as a manifold) as its codomain, Conservative vector field, a vector field that is the gradient of a scalar potential field; Hamiltonian vector field, a vector field defined for any energy function or Hamiltonian
A smooth vector field on a manifold, at a point, induces a local flow - a one parameter group of local diffeomorphisms, sending points along integral curves of the vector field. The local flow of a vector field is used to define the Lie derivative of tensor fields along the vector field.
Consider a vector field v on a surface S, that is, for each r = (x, y, z) in S, v(r) is a vector. The integral of v on S was defined in the previous section. Suppose now that it is desired to integrate only the normal component of the vector field over the surface, the result being a scalar, usually called the flux passing through the surface.
A generalization of this theorem is the Helmholtz decomposition theorem, which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field. By analogy with the Biot-Savart law , A ″ ( x ) {\displaystyle \mathbf {A''} (\mathbf {x} )} also qualifies as a vector potential for v ...