The name is a bit obscure if you don't think in very generalized terms right out of the gate. It refers to a linear polynomial of multiple variables, which by definition must have the super simple form of:and then we just put the unknown $y$ and each derivative into that simple polynomial:except that now the $c_i$ are not just constants, but they can also depend on the argument $x$ (but not on $y$ or its derivatives).

Explicit solutions exist for the very specific cases of:

Order of the highest derivative that appears.

This idea comes up particularly in the phase space coordinate of Hamiltonian mechanics.

This basically adds one more ingredient to partial differential equations: a function that we can select.

And then the question becomes: if this function has such and such limitation, can we make the solution of the differential equation have such and such property?

It's quite fun from a mathematics point of view!

Control theory also takes into consideration possible discretization of the domain, which allows using numerical methods to solve partial differential equations, as well as digital, rather than analogue control methods.

Here's an example of the chain rule. Suppose we want to calculate:So we have:and so:Therefore the final result is:

Given a function $f$:we want to find the points $x$ of the domain of $f$ where the value of $f$ is smaller (for minima, or larger for maxima) than all other points in some neighbourhood of $x$.

In the case of Functionals, this problem is treated under the theory of the calculus of variations.

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D:and in 2D: