>>13959385It depends on what you mean by gradient.
First of all, the coordinates are functions on the manifold
Hence taking the covariant derivative wrt to the \nu-th basis field is just the ordinary partial derivative and hence you get the Kronecker delta
This is what you wrote down. And also as you wrote, lifting or lowering an index, gives you the metric or its inverse.
But, the gradient in differential geometry is a destinct object.
For a function f on the manifold you define the gradient of f via
ie the gradient of f is the (unique) vectorfield associated to the exterior derivative of f via the isomorphism between tangent and cotangent space generated by the metric (musical isomorphisms). In components you have
Hence
