The Lie derivatives quantifies the evolution of a tensor field along the flow generated by a vector field. Algebraically, the Lie derivative $L_X$ along $X$ satisfies the following four properties:

1. $L_X f = Xf$ on functions
2. $L_X (S \otimes T) = (L_X S) \otimes T + S \otimes (L_X T)$
3. $[L_X, d] = 0$
4. $L_X (T(Y_1, \ldots, Y_n)) = (L_X T)(Y_1, \ldots, Y_n) + T(L_X Y_1, \ldots, Y_n) + \cdots + T(Y_1, \ldots, L_X Y_n)$

These axioms imply that, for example, $L_X Y = [X,Y]$ for $Y$ a vector field.

On forms there is also the Cartan 'magic formula' $L_X = d \iota_X + \iota_X d$.

Let $\alpha$ be any form on $M$ and $X$ any vector field; let $\nabla$ be a metric-compatible connection. We compute \begin{align*} L_X (|\alpha|^2 vol) &= (X |\alpha|^2) vol + |\alpha|^2 L_X vol\tag{by axioms 1, 2} \\ &= \langle (2\nabla_X + \mathrm{div}(X)) \alpha, \alpha \rangle vol \end{align*} Thus we may express the Lie derivative in terms of the connection and divergence. A typical choice of $X$ is a normal vector field to a hypersurface; the divergence of $X$ is then the trace of the second fundamental form of the embedding. For example, in Euclidean space it follows that $$\frac{1}{2} L_{\partial/\partial r} (|\alpha|^2 vol) = \langle (\nabla_{\partial/\partial r} + n/2)\alpha, \alpha\rangle vol.$$

Also, writing $g$ for the metric then $$2 \langle \nabla_X \alpha, \alpha \rangle = X |\alpha|^2 = L_X (g(\alpha, \alpha)) = (L_X g)(\alpha, \alpha) + 2g(L_X \alpha, \alpha).$$