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Consider a long, thin wire of charge and length .To calculate the average linear charge density, ¯, of this one dimensional object, we can simply divide the total charge, , by the total length, : ¯ = If we describe the wire as having a varying charge (one that varies as a function of position along the length of the wire, ), we can write: = Each infinitesimal unit of charge, , is equal to ...
An alternative definition: A smooth vector field on a manifold is a linear map : () such that is a derivation: () = + for all , (). [ 3 ] If the manifold M {\displaystyle M} is smooth or analytic —that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields.
A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor, or a tensor, respectively. A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field ...
vector Magnetic field strength: H: Strength of a magnetic field A/m L −1 I: vector field Magnetic flux density: B: Measure for the strength of the magnetic field tesla (T = Wb/m 2) M T −2 I −1: pseudovector field Magnetic moment (or magnetic dipole moment) m: The component of magnetic strength and orientation that can be represented by an ...
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
In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is additionally multiplied or weighted by a power W of the Jacobian determinant of the coordinate transition function or its absolute value.
Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density. Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines.
The linear motion can be of two types: uniform linear motion, with constant velocity (zero acceleration); and non-uniform linear motion, with variable velocity (non-zero acceleration). The motion of a particle (a point-like object) along a line can be described by its position x {\displaystyle x} , which varies with t {\displaystyle t} (time).