I was reading few pages from Zorich's book about submanifolds and I spent some time trying to understand and I think that I succeeded but there are still questions which I cannot answer myself. So I would be very thankful if someone can help me!
First of all, author gives a definition of a $k$-dimensional smooth manifold.
Definition 1. We shall call a set $S\subset \mathbb{R}^n$ a $k$-dimensional smooth surface in $\mathbb{R}^n$ (or a $k$-dimensional
submanifold of $\mathbb{R}^n$) if for every point $x_0\in S$ there
exist a neighborhood $U(x_0)$ in $\mathbb{R}^n$ and a diffeomorphism
$\varphi:U(x_0)\to I^n$ of this neighborhood onto the standard
$n$-dimensional cube $I^n=\{t\in \mathbb{R}^n: |t^i|<1,\ i=1,\dots,n\}$
of the space $\mathbb{R}^n$ under which the image of the set $S\cap
U(x_0)$ is the portion of the $k$-dimensional plane in $\mathbb{R}^n$
defined by the relations $t^{k+1}=0,\dots, t^n=0$ lying inside $I^n$.
Then the provides the following
Example. Let $F^i(x^1,\dots,x^n)$ $(i=1,\dots,n-k)$ be a system of smooth functions of rank $n-k$. One can show that the relations $$
\begin{cases} F^1(x^1,\dots,x^k,x^{k+1},\dots,x^n)=0,\\
\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\\
F^{n-k}(x^1,\dots,x^k,x^{k+1},\dots,x^n)=0 \end{cases} \quad \quad \quad (1)$$ define a
$k$-dimensional submanifold $S$ in $\mathbb{R}^n$.
So far so good. Then he discusses the notion of tangent space which confused me a bit.
Definition 2. If a $k$-dimensional surface $S\subset \mathbb{R}^n,1\leq k\leq n,$ is defined parametrically in a
neighborhood of $x_0\in S$ by means of a smooth mapping
$(t^1,\dots,t^k)=t\mapsto x=(x^1,\dots,x^n)$ such that $x_0=x(0)$ and
the matrix $x'(0)$ has rank $k$, then the $k$-dimensional surface in
$\mathbb{R}^n$ defined parametrically by the matrix equality $(2)$ is called
the tangent plane or tangent space to the surface $S$ at $x_0\in S$.
$$x-x_0=x'(0)t \quad \quad(2)$$
In coordinate form the following system of equations corresponds to equation $(2)$:
$$\begin{cases}
x^1-x^1_0=\frac{\partial x^1}{\partial t^1}(0)t^1+\dots+ \frac{\partial x^1}{\partial t^k}(0)t^k,\\
\dots\dots\dots\dots\dots\dots\dots\dots\dots \\
x^n-x^n_0=\frac{\partial x^n}{\partial t^1}(0)t^1+\dots+ \frac{\partial x^n}{\partial t^k}(0)t^k.
\end{cases} \quad \quad \quad (3)$$
We shall denote the tangent space to the surface $S$ at $x\in S$ by $TS_x$.
Then author asks the following question: how to write the equation of the tangent space for submanifolds in the above example?
One can determine the form of the equation of the tangent plane to the
$k$-dimensional surface $S$ defined in $\mathbb{R}^n$ by the system $(1)$.
One can show that the equation for the tangent space $TS_{x_0}\subset
\mathbb{R}^n$ is $$F'_x(x_0)(x-x_0)=0. \quad \quad \quad (4)$$ In coordinate
representation equation (4) is equivalent to the system of equations
\begin{cases} \frac{\partial F^1}{\partial
x^1}(x_0)(x^1-x^1_0)+\dots+\frac{\partial F^1}{\partial
x^n}(x_0)(x^n-x^n_0)=0,\\
\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\\
\frac{\partial F^{n-k}}{\partial
x^1}(x_0)(x^1-x^1_0)+\dots+\frac{\partial F^{n-k}}{\partial
x^n}(x_0)(x^n-x^n_0)=0 \end{cases}
Then he adds one more paragraph which confused me completely.
The affine equation $(4)$ is equivalent (given the point $x_0$) to the
vector equation $$F'_x(x_0)\cdot \xi=0, \quad \quad \quad (5)$$ in
which $\xi=x-x_0$. Hence the vector $\xi$ lies in the plane $TS_{x_0}$
tangent at $x_0\in S$ to the surface $S\subset \mathbb{R}^n$ defined
by the equation $F(x)=0$ if and only if it satisfies condition $(5)$.
Thus $TS_{x_0}$ can be regarded as the vector space consisting of the
vectors that satisfy $(5)$.
Here is my question which bothers me a lot: So what is $TS_{x_0}$?
First he defined it as $$TS_{x_0}:= \{x\in \mathbb{R}^n: F'_x(x_0)(x-x_0)=0\}.$$
Then he somehow says that $$TS_{x_0}:= \{\xi\in \mathbb{R}^n: F'_x(x_0)\cdot \xi=0\}.$$
But obviously they are different objects. Can anyone explain where is my misunderstanding please?
I would be deeply thankful for the help since I am struggling with this a few days. But please do not use anything advanced. I would be happy to see the explanation within given definitions which I provided above.
EDIT: All this is from Zorich's book (volume 1, Chapter 8, Topic 8.7.1 $k$-Dimensional surfaces in $\mathbb{R}^n$).
Best Answer
Zorich's notation is indeed ambiguous and may lead to confusion (as your question shows).
Actually he introduces two different concepts, but does not properly explain what the difference is. More sharply formulated, he fails to give precise definitions.
This concept first occurs on p.471 where he defines the tangent plane to the graph $S$ of a real-valued function defined on an open subset of $\mathbb R^2$. He calls $S$ a surface in $\mathbb R^3$. The tangent plane at $\xi_0 \in S$ is an affine plane in $\mathbb R^3$, but in general not a linear subspace of $\mathbb R^3$. Intuitively we can understand it as the affine plane which best approximates $S$ in a neighborhood of $\xi_0$.
In section 8.7.1 the above concept of "surface" is generalized to that of a $k$-dimensional smooth surface in $\mathbb R^n$ and in section 8.7.2 the $k$-dimensional tangent plane to a $k$-dimensional smooth surface in $\mathbb R^n$ is introduced. This is again an affine subspace of $\mathbb R^n$.
This concept occurs first on p.435:
This is a bit confusing. An object $T\mathbb R^m_x$ is defined as the set of vectors attached at the point $x \in \mathbb R^m$ (whatever the precise meaning may be), on the other hand $\mathbb R^m$ itself is called the tangent space to $\mathbb R^m$ at $x \in \mathbb R^m$.
A bit more information is given in section 8.7.2, but in my opinion it is still confusing. In Definition 2 we can read
This suggests that tangent plane and tangent space are synonyms, but they are not. Zorichs's use of the two phrases is inconsistent. He says that (8.152) defines the tangent space $TS_{x_0}$ which means that $$TS_{x_0} = \{ x \in \mathbb R^n \mid F'_x(x_0) \cdot (x -x _0) = 0 \} .$$
Later he says that the affine equation (8.152) is equivalent (given the point $x_0$) to the vector equation (8.154) in which $\xi = x - x_0$. This is correct, but then he writes
This is plainly false. The set $TS_{x_0}$ is an affine subspace (which in general does not contain the origin) whereas the vector space consisting of the vectors $\xi$ that satisfy (8.154) is $$\{ \xi \in \mathbb R^n \mid F'_x(x_0) \cdot \xi = 0 \} .$$ This is a linear subspace of $\mathbb R^n$ and the standard is to call it the tangent space to $S$ at $x_0 \in S$.
Both objects (tangent plane and tangent space) carry the same information, but the tangent plane $TS_{x_0}$ is obtained from the tangent space via a translation by $x_0$. Here is an example.
The tangent space at $x_0$ is the one-dimensional linear subspace (= line through the origin) parallel to $TS_{x_0}$.