2.Lie derivative of a vector field 下面我们求向量场 Y 沿着X 的李导数,首先我们关注坐标基向量场 \partial_μ \mathcal{L}_X\partial_{\mu}|_p=\lim_{t \rightarrow 0}\frac{(\phi_t^*\partial_{\mu})|_p-\partial_{\mu}|_p}{t} 之前我们已经定义了如何对向量场做pull back,对于任意...
定义3.20(矢量场的李导数(Lie Derivatives of Vector Fields0))。设 X 和 Y 是流形 M 上的光滑矢量场。我们定义 X 沿 Y 的李导数( Lie derivative)为: $$ {\left( {\mathcal{L}}_{X}Y\right) }_{p} \mathrel{\text{:=}} {\left. \frac{d}{dt}\right| }_{t = 0}{\left( {\Phi ...
calculate the Lie derivative of a vector field, differential form, tensor, or connection with respect to a vector field Calling Sequence Parameters Description Examples Calling Sequence LieDerivative(X, T) Parameters X - a vector field on a manifold M or a vector in a Lie algebra A ...
The Lie derivative of a connectionnabla_Y(Z)is the type(1, 2)tensor fieldS = LieDerivative(X, nabla), defined (when viewed as mapping from pairs of vector fields to vector fields) byS(Y, Z) = LieDerivative(X, nabla_Y(Z)) - nabla_{LieDerivative(X, Y)}(Z) - nabla_X(LieDerivat...
Lie DerivativesThis chapter is devoted to the study of a particularly important construction involving vector fields, called the Lie derivative. This is a method of computing the "directional derivative" of a vector field with respect to another vector field....
Lie derivatives take into account the change of a function along the flow of another function, while partial derivatives only measure the change in one direction. Additionally, the Lie derivative of a function along a vector field does not depend on the choice of coordinates, while partial deriva...
The chapter then looks at the Lie derivative of a vector field and of a differential form. The Lie derivative of a differential form is defined in a similar way to the Lie derivative of a vector field, but the chapter uses the pullback instead of the pushforward to compare nearby values...
Lie derivativeCovariant derivativeTo say that a constitutive model has to verify "the principle of material objectivity" to ensure its frame-indifference has... E Rouhaud,B Panicaud,R Kerner - 《Computational Materials Science》 被引量: 20发表: 2013年 Affine and projective vector fields on spray...
When defining the directional derivative of a tensor fieldF(\mathcal{P})along the tangent vector\overrightarrow{A} = d/d\zetato a curve\mathcal{P}(\zeta)(denoted asA\nabla F), one is sensitive only to first-order changes of quantities, not second, so the parallel transport used in ...
为了解决这一问题,我们引入了李导数(Lie Derivative)的概念,使用流动映射 \Phi_t 把一个点的向量场“推进”到另一个点,保证它们位于同一位置,可以进行比较。 李导数的思想是借助流动映射 \Phi_t 将 Y(\gamma(t + \delta)) 推回到 Y(\gamma(t)) 的位置,然后进行相减,以定义向量场的变化。练习...