In Weintraub's Differential Forms, he gives this property: Let $f$ be a differentiable function defined on a region R of $\mathbb{R}^3$. Let $f_*$ be the derivative of $f$ with respect to some variable.
Let $p_0$ have coordinates $(x_0, y_0, z_0)$ and $v$ have coordinates $(a, b, c)$. First let's evaluate $df(v)$
By definition I.2.1, $df = f_x dx + f_y dy + f_z dz$, so
$$ df(v) = (f_x dx + f_y dy + f_z dz) (a\mathbf{i} + b \mathbf{j} + c\mathbf{k})$$ $$ = f_x (x_0, y_0, z_0) a + f_y(x_0, y_0, z_0)b + f_z (x_0, y_0, z_0)c$$
For one, I couldn't even find a "definition I.2.1" in his textbook. Secondly, he only defines this interaction (without loss of generality): $$ dx(i) = 1, dx(j\ne i) = 0$$ However, I feel he doesn't rigorously make the leap (without loss of generality) that $(f_xdx)(a \mathbf{i}) = f_x(x_0, y_0, z_0)a$. His book also has certain typos and leaps of logic, so it is hard to tell what is intentional and what is sometimes a mistake in syntax. To wit, how do we know that the function $f$ is a map on preimage $(x_0, y_0, z_0)$? I can vaguely try to connect it to his definition of $p_0$, but I would like to know more explicitly how and why he is defining $f$ on this specific point. For anyone who has the textbook or pdf, this is all on page 64. Thank you in advance for you help and clarification.
