On May 14, 12:04 pm, The TimeLord wrote:

Also, for a covariant tensor, the expression above would be a difference
instead of a sum, but the same conclusions would apply; the first term is
what you actually observe.

I hope this helps.

Thanks, I understand your explanation. My question is just a
mathematical one,
though.

Given a path parameterized by 's' with functions of the coordinates
x0(s), x1(s),
x2(s), x3(s), the derivatives of these with respect to 's' is a
contravariant tensor,
the four-velocity vector. This is not a vector field though, right?
That is,
the vector is not defined at all points, only points along the path.
So in taking
the covariant derivative of this vector, as I have seen, it is not
clear to me
what the term (partial derivative) d[dx0/ds]/dx0 means, for example.
The textbook I have contracts this tensor with the tangent vector dX/
ds to get
the absolute derivative, and continues without further consideration.

Does taking the partial here with respect to x0 mean to take the
derivative
of [dx0/ds] while keeping x1, x2, and x3 fixed?

Thanks,
-Mike