Note: if someone knows a better title, please let me know so I adjust it.
I am reading calculus of several variables by Serge Lang, 3rd ed. (pages 77-79) and need some help with derivative of a function of several variables. The author after talking about vectors, dot product, defined the gradient algebraically only. Now he tries to differentiate a function of several variables as follows: $ \frac{f(X+H)- f(X)}{H}$ where both $H$ and $X$ are vectors, with $\|H\|$ being small. He says it is meaningless to divide by a vector, so he does the following(all in single variable and not in vector):
Let $ \phi(x) = \frac{f(x+h)- f(x)}{h} - f'(x)$ as the limit of h approaches zero, $\phi(x)$ also approaches zero. then:
$h\phi(x) + hf'(x) = f(x+h) - f(x)$
then he defines $ g(x) = -\phi(x)$ if $h < 0 $
and $g(x) = \phi(x)$ if $h > 0 $
I don't understand what is the point of defining $g(x)$ here?
nevertheless, the equation now becomes:
$ f(x+h) - f(x) = hf'(x) + \|h\|g(x) $ saying that this form has an advantage of no division by $h$ thus, he will use this form to define differentiability for function of several variables.
but without pretty much adding anything to that he does the following:
$f(X + H) - f(X) = f(x + h, y + k) - f(x,y)\\ f(x+h, y+k) - f(x, y)= \frac{\partial f}{\partial x}h + \frac{\partial f}{\partial y}k + \|H\|g(H)$
I don't understand where this expression is coming from. for one, multiplying a vector is not defined or talked about, there are at least 4 ways that I know of to multiply 2 vectors (dot, cross, outer and Hammard) I don't know why he changed the expression $ h f'(x) $ to be $\nabla f \cdot H$ multiplication is not the dot product (why would it be changed to be that?), another issue is that the derivative should account for change in all components and not just one component at a time. honestly, that expression gave me so many questions, I would appreciate if someone can explain it.
I attach below photos for the 3 pages, just in case someone wants to read the original pages, in case they think I made a mistake or missed something of importance. 
