My main reference is John Lee's Introduction to Riemannian manifolds. The divergence is defined as:
Let $M$ be an $n$-dimensional Riemannian manifold. If $X$ is a tangent vector field and $dV$ is its volume form, then the divergence is $$div X=d(X\lrcorner\ dV)/dV$$ where $X\lrcorner\ dV$ denotes the interior multiplication.
I am trying to compute the divergence of a vector field $X=A\frac{\partial}{\partial x^1}+B\frac{\partial}{\partial x^2}+C\frac{\partial}{\partial x^3}$ in the Euclidean space $\mathbb R^3$, where one has $dV=dx^1\wedge dx^2\wedge dx^3$.
By definition, $$dV=dx^1\wedge dx^2\wedge dx^3\\ =\frac{1}{6}(dx^1\otimes dx^2\otimes dx^3-dx^1\otimes dx^3\otimes dx^2\\ +dx^2\otimes dx^3\otimes dx^1-dx^2\otimes dx^1\otimes dx^3\\ +dx^3\otimes dx^1\otimes dx^2-dx^3\otimes dx^2\otimes dx^1)$$ Then $$X\lrcorner\ dV=\frac{1}{6}(Adx^2\otimes dx^3-Adx^3\otimes dx^2\\ +Bdx^3\otimes dx^1-Bdx^1\otimes dx^3\\ +Cdx^1\otimes dx^2-Cdx^2\otimes dx^1)\\ =\frac{1}{3}(Adx^2\wedge dx^3+Bdx^3\wedge dx^1+Cdx^1\wedge dx^2)$$ and $$d(X\lrcorner\ dV)=\frac{1}{3}(\frac{\partial A}{\partial x^1}+\frac{\partial B}{\partial x^2}+\frac{\partial C}{\partial x^3})dx^1\wedge dx^2\wedge dx^3$$ By the definition above, the divergence would be $\frac{1}{3}(\frac{\partial A}{\partial x^1}+\frac{\partial B}{\partial x^2}+\frac{\partial C}{\partial x^3})$. Why is there an extra coefficient?
Update: In response to one of the comments: how I computed the interior product.
Note that here $M=\mathbb R^3$ and $TM,T^*M\cong\mathbb R^3$ denote the tangent/cotangent spaces with their subscripts omitted rather than the bundles.
First, $dx^1\otimes dx^2\otimes dx^3\in T^*M\otimes T^*M\otimes T^*M\cong L(TM,TM,TM;\mathbb R)$. The isomorphism (to the space of multilinear mappings) is given by $$(dx^1\otimes dx^2\otimes dx^3)(u,v,w)=(dx^1(u))\cdot(dx^2(v))\cdot(dx^3(w))\text{ for }u,v,w\in TM$$ Hence if $X$ is as above we have $$X\lrcorner(dx^1\otimes dx^2\otimes dx^3)(v,w)=dx^1(X)\cdot(dx^2(v))\cdot(dx^3(w))\\=A\cdot(dx^2(v))\cdot(dx^3(w))=Adx^2\otimes dx^3$$ Then by linearity I obtain the expression for $X\lrcorner dV$.