The key and central motivation for studying Lie groups and their Lie algebras appears to be to characterize symmetry in Lagrangian mechanics through Noether's theorem, just start from there.

Notably local symmetries appear to map to forces, and local means "around the identity", notably: local symmetries of the Lagrangian imply conserved currents.

More precisely: local symmetries of the Lagrangian imply conserved currents.

TODO Ciro Santilli really wants to understand what all the fuss is about:

Oh, there is a low dimensional classification! Ciro is a sucker for classification theorems! en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras

The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of continuous problems are simpler than discrete ones.

Bibliography:

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: