In the book Schutz, Geometrical Methods of Mathematical Physics, eq. (2.28) is $$ \tilde{d}x^i ( \frac{\partial}{\partial x^j}) \equiv \frac{\partial x^i}{\partial x^j} = \delta^i_j $$ and the author writes "the second equality follows from the ordinary properties of partial derivatives". My question is about the second equality, $\frac{\partial x^i}{\partial x^j} = \delta^i_j$.
First, the book is using tensor notation, so to relate it to regular calculus, I think we can move the indices down. I think this is the same thing: $$ \frac{\partial x_i}{\partial x_j} = \delta^i_j $$
Now, my very very basic question: consider the case where $x_i$ and $x_j$ are non-orthogonal coordinates. Each coordinate has a direction of maximum change, and following that direction traces a line or curve. The relation between the "coordinate lines" for $i$ and $j$ is arbitrary, and in a degenerate case they could be parallel at some point.
A. At a point where the coordinate lines for $x_i$ and $x_j$ are parallel, making a change in $x_i$ must make a change in $x_j$, so how can the equation above be valid?
B. Even if degenerate cases are excluded, the coordinate lines can be arbitrarily close to being parallel, without being exactly parallel. If one moves along the coordinate line for $x_i$, $x_j$ will change as well, unless the coordinate system is Cartesian.
I am completely misunderstanding something basic, or else am looking for a philosophical answer. Suppose that one was doing numerical programming -- one would never base a test in the code on exact equality. How can calculus be founded on something so "fragile"?
I feel like the relation above should instead involve the dot product of the coordinate lines, thereby reflecting their angle somehow... though, that may be a key to my misunderstanding, because $x_i$ and $x_j$ are not vectors, so there is no dot product between them.