Study of analysis 'in the large' on (possibly infinite-dimensional) smooth manifolds — where local chart-by-chart calculus must be glued consistently and global invariants emerge. Core themes: (i) Morse theory (Morse 1925, Bott 1954,…
global-analysis
Morse theory: critical-point indices, handle-attachment, Morse inequalities
A smooth f : M → R on a compact manifold is Morse iff every critical point (df = 0) is non-degenerate (Hessian invertible). At a…
Banach manifold: charts into Banach spaces + Fréchet-derivative calculus
An infinite-dimensional generalisation of a smooth manifold: charts take values in a fixed Banach space E (rather than R^n), with smooth…
Jet bundle J^k(M,N): truncated Taylor fibre + prolongation functor
Ehresmann's 1951 jet-bundle construction. For M, N smooth manifolds the k-jet j^k f(x) of a smooth map f : M → N at x ∈ M is the…
Paraboloid f=x²−y²: grad(0)=0; Hessian diag(2,−2); det −4; Morse index 1
Sympy-exact Morse-critical-point witness for the canonical saddle f(x, y) = x² − y² on R². Step 1 — gradient: ∇f(x, y) = (2x, −2y),…
Height on S²: perfect Morse, c=(1,0,1), Σ(−1)^i c_i = χ(S²) = 2
Sympy-exact arithmetic witness of the Morse inequalities equality case on S². The height function f(x, y, z) = z restricted to the round…
2-jet of x³: j²(0)=(0,0,0); j²(1)=(1,3,6); j²(2)=(8,12,12)
Sympy-exact verification of the 2-jet prolongation operator for the canonical cubic f(x) = x³. Derivatives: f'(x) = 3x², f''(x) = 6x. …