^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] <<[[Schutz - 2.12 - Vector Fields and Integral Curves 1|2.12 Vector Fields and Integral Curves]] | [[Schutz - 2.14 - Lie Brackets and Noncoordinate Bases|2.14 Lie Brackets and Noncoordinate Bases]]>> # Exponentiation of the Operator $d/d\lambda$ Suppose we have an analytic $C^\omega$ manifold, then the coordinate values of $x^i(\lambda)$ of points along the integral curves of $\pmb{Y} = d/d\lambda$ analytic functions of $\lambda$ by definition. The coordinates of two points with parameters $\lambda = \lambda_0$ and $\lambda = \lambda_0 + \epsilon$ are related by: $ \begin{align} x^i(\lambda_0 + \epsilon) &= x^i((\lambda_0 + \epsilon) - \lambda_0) + \epsilon \frac{dx^i}{d\lambda}\Bigg\vert_{\lambda_0} + \epsilon^2 \frac{d^2x^i}{d\lambda^2}\Bigg\vert_{\lambda_0} + \ldots \\ &= \left(1 + \epsilon \frac{d}{d\lambda} + \epsilon^2 \frac{d^2x^i}{d\lambda^2} + \ldots\right)x^i \Bigg\vert_{\lambda_0} \\ & = \left(\exp\left(\epsilon \frac{d}{d\lambda}\right)\right)x^i\Bigg\vert_{\lambda_0} \\ & \equiv \exp(\epsilon d/d\lambda) \equiv \exp(\epsilon \pmb{Y}) \end{align} $ Since $\epsilon d/d\lambda$ is an infinitesimal motion along the curve, the exponentiation gives a finite motion along the curve.