I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It cannot be a stationary value because if so then a smaller value on real number line exist, so it must be a moving value. Moving value towards $0$ so in most places we use its magnitude equal to zero but at the same time we also know that infinitesimal is not equal so in all those places were we use value of infinitesimal equal to $0$ we are making an infinitesimal error and are not $100\%$ accurate, maybe $99.9999\dots\%$ accurate, but no $100\%$! So please explain infinitesimal and its applications and methodology in context to the above paragraph or elsewise intuitively please.
[Math] the meaning of infinitesimal
calculusinfinitesimalsintuitionnonstandard-analysisterminology
Related Solutions
Zev, I honestly think Thurston's tongue was firmly implanted in his cheek when he wrote this. So the key point is that a connection on a vector bundle gives you (a) a means of differentiating sections (generalizing the covariant derivative for a Riemannian manifold as a connection on the tangent bundle) and (b) a notion of parallelism (generalizing the notion of parallel transport of tangent vectors).
As you suggested, the differential of $f\colon D\to\mathbb R$ gives you a $1$-form, hence a section of the cotangent bundle $T^*D$. With the standard symplectic structure on $T^*D$, Lagrangian sections (i.e., ones that pull back the symplectic $2$-form to $0$) are precisely closed $1$-forms. [This is tautological: If $q_i$ are coordinates on $D$, a $1$-form on $D$ is given by $\omega = \sum p_i\,dq_i$ for some functions $p_i$. By definition, $d\omega = \sum dp_i\wedge dq_i$, and this is (negative of) the pullback by the section $\omega$ of the standard symplectic form $\sum dq_i\wedge dp_i$ (with canonical coordinates $(q_i,p_i)$ on $T^*D$).]
Now, a connection form on a rank $k$ vector bundle $E\to M$ is a map $\nabla\colon \Gamma(E)\to\Gamma(E\otimes T^*M)$ (i.e., a map from sections to one-form valued sections) that satisfies the Leibniz rule $\nabla(gs) = dg\otimes s + g\nabla s$ for all sections $s$ and functions $g$. In general, one specifies this by covering $M$ with open sets $U$ over which $E$ is trivial and giving on each $U$ a $\mathfrak{gl}(k)$-valued $1$-form, i.e., a $k\times k$ matrix of $1$-forms; when we glue open sets these matrix-valued $1$-forms have to transform in a certain way in order to glue together to give a well-defined $\nabla$.
OK, so Thurston takes the trivial line bundle $D\times\mathbb R$. A connection is determined by taking the global section $1$ and specifying $\nabla 1$ to be a certain $1$-form on $D$. The standard flat connection will just take $\nabla 1 = 0$ and then $\nabla g = dg$. I'm now going to have to take some liberties with what Thurston says, and perhaps someone can point out what I'm missing. Assume now that our given function $f$ is nowhere $0$ on $D$. We can now define a connection by taking $\nabla 1 = -df/f$. Then the covariant derivative of the section given by the function $f=f\otimes 1$ [to which he refers as the graph of $f$] will be $\nabla(f\otimes 1) = df - f(df/f) = 0$, and so this section is parallel.
Slightly less tongue-in-cheek, parallelism is the generalization of constant (in a vector bundle, we cannot in general say elements of different fibers are equal), and covariant derivative $0$ is the generalization of $0$ derivative.
Well, in a sense, you're right. When they say that the limit of $f(x)$ at $x=a$ is $L$ it doesn't necessarily mean that $f(a)=L$. Actually, the nice idea behind limits is that you can talk about the limit of a function even if the function is not defined at that value. This is a very powerful idea that later enables us to talk about derivatives as you possibly know.
For example $\displaystyle \lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1$ but the value of $\displaystyle \frac{\sin(x)}{x}$ is not defined at $x=0$. If you graph it on wolframalpha, you'll see that this means 'as we approach $x=0$ the value of $\displaystyle \frac{\sin(x)}{x}$ approaches 1'. We never claim that these two are equal! We just claim that the value of $f(x) = \displaystyle \frac{\sin(x)}{x}$ can become arbitrarily close to $1$ provided that we let $x$ be close enough to $0$ .
When we say that the limit of $f(x)$ at $x=a$ is $L$, we are claiming that we can make $f(x)$ arbitrarily close to $L$ provided that we take $x$ close enough to $a$. That's all.
Best Answer
The real numbers $\mathbb{R}$ is an example of a field, a space where you can add, subtract, multiply and divide elements. In addition, $\mathbb{R}$ is an example of an ordered field, i.e. for any $a, b \in \mathbb{R}$ we have either $a < b$, $a = b$, or $a > b$. Note, there are some further conditions on the interaction between inequalities and the field operations.
A positive infinitesimal in an ordered field is an element $e > 0$ such that $e < \frac{1}{n}$ for all $n \in \mathbb{N}$. A negative infinitesimal is $e < 0$ such that $-e$ is a positive infinitesimal. An infinitesimal is either a positive infinitesimal, a negative infinitesimal, or zero.
In $\mathbb{R}$ there is only one infinitesimal, zero - this is precisely the Archimedean property of $\mathbb{R}$. So while people use the word infinitesimal to convey intuition, the real numbers don't have any non-zero infinitesimals, so their explanation is flawed.
In the early development of calculus by Newton and Leibniz, the concept of an infinitesimal was used extensively but never defined explicitly. The way this has been rectified through history is via the introduction of limits which still capture the intuition, but are in fact defined perfectly well.
It should be noted that other ordered fields do have non-zero infinitesimals. You might even try to find an ordered field which contains all the real numbers that you know and love, but also has non-zero infinitesimals. Such a thing exists! Abraham Robinson first showed such an ordered field exists in $1960$ using model theory, but it can actually be constructed using something called the ultrapower construction. This is called the field of hyperreal numbers and is denoted ${}^*\mathbb{R}$. With the hyperreals at hand, you can take all the ideas that Newton and Leibniz used and interpret them almost literally. Calculus done in this way is often called non-standard analysis.