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In vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the ...
The Java Naming and Directory Interface ( JNDI) is a Java API for a directory service that allows Java software clients to discover and look up data and resources (in the form of Java objects) via a name. Like all Java APIs that interface with host systems, JNDI is independent of the underlying implementation.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . [1] A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed ...
Solenoidal vector field. An example of a solenoidal vector field, In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: A common way of expressing this property is to say that ...
In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations : From the vector calculus identity it follows that. that is, that the field v satisfies Laplace's equation . However, the converse is not true ...
In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if is a symplectic manifold with smooth manifold and symplectic form , then a vector field in the Lie algebra is symplectic if its flow preserves the symplectic structure. In other words, the Lie derivative of the vector field must vanish:
Definition. A tensor field of type ( p, q) is a section. where V is a vector bundle on M, V* is its dual and ⊗ is the tensor product of vector bundles. Equivalently, it is a collection of elements Tx ∈ Vx⊗p ⊗ ( Vx*) ⊗q for all points x ∈ M, arranging into a smooth map T : M → V⊗p ⊗ ( V*) ⊗q. Elements Tx are called tensors .
In the vector field formulation, the theorem states that a subbundle of the tangent bundle of a manifold is integrable (or involutive) if and only if it arises from a regular foliation. In this context, the Frobenius theorem relates integrability to foliation; to state the theorem, both concepts must be clearly defined.