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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.
Consider a long, thin rod of mass and length .To calculate the average linear mass density, ¯, of this one dimensional object, we can simply divide the total mass, , by the total length, : ¯ = If we describe the rod as having a varying mass (one that varies as a function of position along the length of the rod, ), we can write: = Each infinitesimal unit of mass, , is equal to the product of ...
Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: = () , where ∇ F is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space).
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.
In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle , called the density bundle .
A linear operator between topological vector spaces and is said to be densely defined if its domain is a dense subset of and if its range is contained within . See also Continuous linear extension . A topological space X {\displaystyle X} is hyperconnected if and only if every nonempty open set is dense in X . {\displaystyle X.}
We come now to the main question of defining how to transport a vector field parallelly along a curve. Suppose that P(t) is a curve in S. Naïvely, one may consider a vector field parallel if the coordinate components of the vector field are constant along the curve.