synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
An ordinary differential equation is a differential equation involving derivatives of a function with respect to one argument only, i.e. the function is on a manifold only of dimension $d = 1$. This function can be vector valued, what is sometimes viewed as a system of possibly coupled equations; still all of them have the derivatives taken with respect to the same parameter. (Note that a higher-order differential equation can be turned into a system of first-order equations.)
A basic theorem concerns existence and uniqueness of local solutions to initial value problems. Let $X$ be a Banach space; given $(t_0, y_0) \in \mathbb{R} \times X$ and $a, r \gt 0$, put $Q \coloneqq [t_0 - a, t_0 + a] \times B_r(y_0)$m where $B_r(y_0)$ is the closed ball in $X$ of radius $r$ about $y_0$.
(Picard–Lindelöf) Suppose $f: Q \to X$ is a function satisfying the following conditions:
(Continuity in $t$): Given any $x \in B_r(y_0)$, the function $f(-,x)$ (that is $t \mapsto f(t, x)$) is continuous from $[t_0 - a, t_0 + a]$ to $X$.
(Lipschitz continuity in $y$): There is a Lipschitz constant $L$ such that
for all $(t, x) \in Q$;
(Boundedness): There is a constant $K$ such that $\sup_{(t, x) \in Q} \|f(t, x)\| \leq K$.
Then for any $c \leq \min(a, r/K)$, there exists exactly one? solution $y: [t_0 - c, t_0 + c] \to X$ to the initial value problem
We will define an infinite sequence of approximate solutions to the problem, prove that its limit exists, prove that this limit is an exact solution, and prove that this solution is unique.
The infinite sequence is given by Picard iteration: Starting with the given constant $y_0$, recursively define
that is, $y_{n+1}$ is that indefinite integral? of $f(-,y_n)$ that takes the correct initial value. (To define $y_1$, use abuse of notation? to interpret $y_0(t)$ as $y_0$; that is, think of $y_0$ as a constant function.) To prove that this integral exists, use the continuity conditions and an inductive proof? that each $y_n$ is continuous to show that we are integrating a continuous function.
Thinking of Picard iteration as an operator between Banach spaces of continuous functions, use Lipschitz continuity and boundedness to show that the Banach fixed point theorem? applies, so that the sequence $(y_1, y_2, \ldots)$ uniformly converges to a limit $y$.
Since uniform convergence of continuous functions behaves well with integration, the limiting instance of Picard iteration holds:
By differentiating with respect to $t$, and by evaluating at $t_0$, we confirm that $y$ is a solution of the initial value problem.
Given any putative solution $z$, apply Grönwall's inequality? to $z - y$ to prove that $z = y$.