Well I think a calculus book might help, I don't know exactly what are you looking for.
A vector function is a function that is also a vector (tautology).
An example is the force in the 2nd Newton's law:
[itex] \vec{F}(t) = m \vec{a}(t)= m a_x(t) \hat{x} + m a_y(t) \hat{y} + m a_z(t) \hat{z}[/itex]

Or another example is the ballistic trajectory of a cannon ball that moves into the gravitational field (which is a parabola), when thrown at an angle [itex]\theta[/itex] with respect to the horizontal axis with some initial velocity [itex]u_0[/itex]:
[itex] \vec{x}(t) = t u_0 \cos \theta \hat{x} + \big(t u_0 \sin \theta - \frac{1}{2} gt^2\big) \hat{y}[/itex].

If you have an opportunity look pdf page 22 Neighborhoods and then vector functions.Vector function is a tye of function describes curves.Let ##a,b,c## be a vectors then vector function is ##f(t)=f_1(t)e_1+f_2(t)e_2+f_3(t)e_3## this is called vector function .It defines curves

did you read my editted x(t) example? for the curve that describes the trajectory of a cannon ball?
[itex]u_0, \theta[/itex] are just numbers, and the parametrization of the curve is done by the parameter [itex]t[/itex] (the time here)

You parametrize the trajectory curve (blue dots) with the elapsed time [itex]t[/itex] (for different [itex]t[/itex] your vector function [itex]\vec{x}(t)[/itex] will point to some point along the curve as I show some examples for different values of [itex]t[/itex]).

PS. Sorry for the ugly sketch, but I am not an artist when it comes to using the paint with a touchpad.

I'm not clear on what exactly you need to learn, so I can only say that this will be a good time to wipe the dust off your old calculus books. Some general advice:

Make sure that you understand the concept of "function" (="map") and associated concepts perfectly. I'm talking about concepts such as domain, codomain and range, and also injective, surjective, bijective, preimage, etc., as well as notations such as ##f:X\to Y##.

Never refer to f(x) as a function (unless it really is). Typically f is the function, and f(x) is an element of its codomain. (When you're doing physics, you can usually ignore this advice without causing too much confusion. But if you ignore this advice when you're doing differential geometry, you will confuse the hell out of yourself).

Make sure that you understand the difference between "x is a function of t" and "x is a function". (The former statement is typically used when the symbols x and t both represent real numbers, or when x represents a vector and t a real number. It means that the value of x, i.e. the number or vector represented by the symbol x, is completely determined by the value of t. If that's the case, then there's a function f such that x=f(t). That function is sometimes denoted by x, especially in physics books. This contributes to students not understanding the difference between the two phrases).

Make sure that you understand partial derivatives.

Make sure that you understand the definitions of kf, f+g and fg, when k is a real number and f,g are real-valued functions on some set S. Make sure that you understand that the first two of these definitions turn the set of real-valued functions on S into a vector space.

Actually This differantial geometry book is really though.I dont know I can understand the next chapters.The book designed for graduate students but I am in high school.

I understand I guess.I made examples and I can solve them.I dont want to give up but Is there any chance to me understand further chapters (considering my situation )

I see. I think it will be impossible for you to understand the proofs and definitions that involve topology (open sets, limits of sequences, continuity, etc.) I'm not familiar with this particular book, but books on differential geometry usually contain a lot of proofs of that sort. If you insist on trying to study differential geometry now, you will have to skip the parts that involve topology and then be prepared to study differential geometry again a few years from now, when you're better prepared. But if you have a solid understanding of the concepts I mentioned (this would make you a very unusual high school student), then you may still be able to understand the big picture and learn enough to do many different types of problems and enough to be able to read books on general relativity.

It depends on what you define as "tough" ...
The + (for your level) of this book is that it explains things through examples and illustrations rather than definitions and proofs (that can be a - for someone who wants to get deeper into the maths).
The - (I think for you) is that it explains some things in a very simple way (thinking that the reader understands what is going on) while for your level some of those things may not appear so "natural". It's a matter of practice for many people...you should spend more time thinking/working over the things that appear "non-trivial" to you.
Calculus may help, however I don't know of a good book for you. I never had a good book in calculus
Well, I still have a feeling you are going too fast...maybe you were in a hurry to reach chapters 8+...spend more time in understanding how to work with the first chapters (which can also help you prepare for the first year of your uni studies).
That's because differential geometry can help you work with things that are not only associated with General Relativity, but also classical mechanics (as the ballistic trajectory I mentioned in previous post). Especially when studying curves (associated with trajectories), and getting the positions (those "vector functions"), velocities (which will be the "vector functions" derivatives/tangent to the curves) etc, things that he starts building in chapter 4 and things like TNB frames: https://en.wikipedia.org/wiki/Frenet–Serret_formulas

I will gonna spend more time on thinking.I will gonna look again chapter 1 and chapter 2.

This is also good for me.

This is %90 true for chapter one and chapter two.I can see the big picture.But I didnt understand the concepts nearly and of chapter two.I will gonna look them again
I will gonna finish this book.
The book ask sometimes "proof this thing" I can do half of them in chapter one.I work 5-6 hours to come this chapter.In high school we learned chapter one so it didnt took me a so much time to work on.In second chapter again.I know plane equation and other stuff.The textbook is very very good. I understand the basic concept.Example there says lets take a four point on a plane and lets ask us the equation of plane.This will be [(x-a)(b-a)(c-a)]=0 why, cause If we make (b-a)x(c-a) We will find a vector which its both perpendicular to b-a and c-a vectors and dot product of x-a and (b-a)x(c-a) will give us zero cause they are perpendicular.I dont know why but I felt that I need to proof myself.But you are right ChrisVer I am hurrying this my curse.I can be patient in my life but I cant be patient when te thing comes "learning".I want to know everthing without effort.I know this is not possible.
In next chapters I need to work so hard cause they look like so tough.But again I will gonna look again firt two chapters to understand concept better.Maybe I am a smart guy who knows ? (I dont thing thats true )
Thanks for advice.
Later In homework section I will gonna mention the questions which I couldnt.I will give you a total number of wrong answers or empty questions so you can coonsider my situation.I wont be lie to you guys.