# Meaning of curl defined as in differential geometry

there is a short remark in the textbook about GR I have been reading, which has been bugging me for the last few hours: My questions are:

(1): In multivariable calculus, curl can be intuitively described as the rotation vector of a vector field. Why is it here a (0,2)-tensor field?

(2): How can one derive the second equation from the first equation?

Best regards.

EDIT:

The second equation only holds for vector fields $$Y,Z \perp X$$.

1. If $$\Gamma^k_{ij} = 0$$ the equation for the curl in local coordinates is $$\text{curl}(X) = \left(\frac{\partial X_j}{\partial{x^i}}-\frac{\partial X_i}{\partial{x^j}}\right) dx^n\wedge dx^m.$$
In $$\mathbb R^3$$ this expression becomes the usual curl if you apply hodge star (identify $$dx\wedge dy$$ with $$dz$$ etc.) and lower the index (identify $$\partial_x$$ with $$dx$$ etc.).
1. You can start with the last formula and plug-in the formula for $$dX^\flat$$ and the definition of $$\wedge$$. You will find a bunch of zeroes because $$X^\flat(Y) = X^\flat(Z) = 0$$, and what remains is $$g(X,[Y,Z])$$. Using that $$\nabla$$ is torsion-free you find that this is equal to $$g(X,\nabla_YZ)-g(X,\nabla_ZY)$$. You complete this by moving the connection across brackets - because $$\nabla g=0$$ and the fields are perpendicular.