Search results
Results From The WOW.Com Content Network
valid for any vector fields X and Y and any tensor field T.. Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation ...
In other words, the covariant derivative is linear (over C ∞ (M)) in the direction argument, while the Lie derivative is linear in neither argument. Note that the antisymmetrized covariant derivative ∇ u v − ∇ v u , and the Lie derivative L u v differ by the torsion of the connection , so that if a connection is torsion free, then its ...
Generally the convective derivative of the field u·∇y, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field u·(∇y), or as involving the streamline directional derivative of the field (u·∇) y, leading to the same result. [10]
There are many ways to understand the gauge covariant derivative. The approach taken in this article is based on the historically traditional notation used in many physics textbooks. [1] [2] [3] Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection.
The covariant derivative is such a map for k = 0. The exterior covariant derivatives extends this map to general k. There are several equivalent ways to define this object: [3] Suppose that a vector-valued differential 2-form is regarded as assigning to each p a multilinear map s p: T p M × T p M → E p which is completely anti-symmetric.
In fact in the above expression, one can replace the covariant derivative with any torsion free connection ~ or locally, with the coordinate dependent derivative , showing that the Lie derivative is independent of the metric. The covariant derivative is convenient however because it commutes with raising and lowering indices.
The exterior derivative is an operation on differential forms that, given a k-form , produces a (k+1)-form . This operation extends the differential of a function (a function can be considered as a 0 -form, and its differential is d f ( x ) = f ′ ( x ) d x {\displaystyle df(x)=f'(x)dx} ).
The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by D. Given a smooth curve γ on (M, g) and a vector field V along γ its derivative is defined by = ˙ (). Formally, D is the pullback connection γ*∇ on the pullback bundle γ*TM.