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  2. Divergence - Wikipedia

    en.wikipedia.org/wiki/Divergence

    The divergence of a vector field extends naturally to any differentiable manifold of dimension n that has a volume form (or density) μ, e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two-form for a vector field on R 3, on such a manifold a vector field X defines an (n − 1)-form j = i X μ obtained by contracting ...

  3. Vector calculus identities - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus_identities

    In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: ⁡ = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.

  4. Covariance and contravariance of vectors - Wikipedia

    en.wikipedia.org/wiki/Covariance_and_contra...

    (For example, for a position vector of length meters, if all Cartesian basis vectors are changed from meters in length to meters in length, the length of the position vector remains unchanged at meters, although the vector components will all increase by a factor of ). The scalar product of a vector and a covector is invariant, because one has ...

  5. Vector algebra relations - Wikipedia

    en.wikipedia.org/wiki/Vector_algebra_relations

    The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.

  6. Vector calculus - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus

    Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.

  7. Parallel transport - Wikipedia

    en.wikipedia.org/wiki/Parallel_transport

    Parallel transport of tangent vectors is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Note that while the vectors are in the tangent space of the manifold, they might not be in the tangent space of the curve they are being transported along.

  8. Differentiable curve - Wikipedia

    en.wikipedia.org/wiki/Differentiable_curve

    According to problem 25 in Kühnel's "Differential Geometry Curves – Surfaces – Manifolds", it is also true that two Bertrand curves that do not lie in the same two-dimensional plane are characterized by the existence of a linear relation a κ(t) + b τ(t) = 1 where κ(t) and τ(t) are the curvature and torsion of γ 1 (t) and a and b are ...

  9. Curl (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Curl_(mathematics)

    In 3 dimensions, a differential 0-form is a real-valued function (,,); a differential 1-form is the following expression, where the coefficients are functions: + +; a differential 2-form is the formal sum, again with function coefficients: + +; and a differential 3-form is defined by a single term with one function as coefficient: .