Physical interpretation of total derivative Can I get some help interpreting the following?

"Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy."

This is from the derivation of the form of the kinetic energy from https://en.wikipedia.org/wiki/Kinetic_energy#Derivation
This interpretation of a total derivative is new to me (and may cure my perpetual confusion about the topic when talking about varying actions/Euler-Lagrange/etc).
 A: Statistical mechanics
In the context of statistical mechanics one distinguishes quantities that are state functions and those which are not. State functions are functions only of the thermodynamic variables characterizing the system, e.g., of pressure, temperature and chemical potential $P,T,\mu$. That is, the function always have the same value for the same set of arguments $P,T,\mu$.
Some quantities may depend not only on the state of the system, but on external variables or history: the most glaring example is the quantity of heat which may be different, depending on the process by which the change between two states is caused. The differentials of these quantities in respect to only the system variables are obviously not full differentials, since some variables are missing.
Mathematics
In strictly mathematical sense, the total differential of a function of several variables is
$$df(x_1, x_2, ... , x_n) = \sum_{i=1}^n\frac{\partial f}{\partial x_i}dx_i,$$
whereas each term $\frac{\partial f}{\partial x_i}dx_i$ in this expansion
is called a partial differential. Existence of a differential implies uniqueness of the function, which means that the difference of function values between two points is the same, regardless of the path we use for integrating between these points:
$$f(\mathbf{x}_2) - f(\mathbf{x}_1) = \int_{\mathbf{x}_1}^{\mathbf{x}_2}df(\mathbf{x}).$$
This is however not true for a sum of several partial differentials, the integral of which will depend on path. E.g., in case of only two variables:
$$\int_{x_1, y_1}^{x_2,y_2}\frac{\partial f(x,y)}{\partial x}dx$$
obviously depends on the intermediate values of $y$.
Remark: differential $\neq$ derivative
In this context it is useful to stress the difference between the increment $\Delta x$, the differential $dx$, and the derivative $\frac{\partial f}{\partial x}$.
In case of a function of single variable the increment and the differential are essentially the same thing, $\Delta x = dx$, whereas the derivative is just a proportionality coefficient between the differential of a function and the differential of its argument, which is why one conventionally writes $\frac{d f}{d x}$ instead of $\frac{\partial f}{\partial x}$. In the case of a function of multiple variables these are three different things. (Confusingly, Wikipedia contains multiple articles for differential, total differential and total derivative, which are not very consistent.)
