# linear algebra: volume form expressed in local coordinates in Riemannian collector

I am beginning to study the Riemannian geometry, and in particular the shape of the volume in a Riemannian collector $$(M, g)$$. I was first introduced as a differential $$n$$-to form $$dV$$ for which $$dV (e_1, cdots, e_n) = 1$$ for any choice (positive) of orthonormal basis $$e_1, cdots, e_n$$ in a tangent space $$T_p M$$. It's easy to see that in this local framework, with a dual base $$omega ^ 1, cdots, omega ^ n$$, we have:

$$dV = omega ^ 1 wedge cdots wedge omega ^ n$$

That is well defined, as a change of basis to another positive orthonormal framework, say, $$tilde {e} _1, cdots, tilde {e} _n$$ with double base $$tilde { omega} ^ 1, cdots, tilde { omega} ^ n$$ yields:

$$dV = det (A) tilde { omega} ^ 1 wedge cdots wedge tilde { omega} ^ n$$

Where $$A$$ is the change of base matrix, which must satisfy $$det (A) = 1$$ because both bases are orthonormal and positive. However, I have now seen that in local coordinates, the shape of the volume looks like this:

$$dV = sqrt {g_ {ij}} dx ^ 1 wedge cdots wedge dx ^ n$$

And I'm not sure how to move from the previous representation to this one. I can see that we will have:

$$dV = det (B) dx ^ 1 wedge cdots wedge dx ^ n$$

Where $$B$$ is the change of base matrix with components $$omega ^ i ( frac { partial} { partial x ^ j})$$but i'm not sure how to relate $$B$$ in a sensible way to metric. I guess we can also put $$g$$ in local coordinates such as:

$$g = omega ^ i otimes omega ^ i$$

But this does not seem manageable.