Interactive derivative geometry
I read the Jacobian as the thing that moves tiny vectors, the Wronskian as the area spanned by solution states, and the Hessian as the record of how slope itself changes.
The unifying ideaI group the three together because each packages derivatives into linear algebra so geometry can do part of the reasoning.
Jacobian
For F: ℝⁿ → ℝᵐ, the entry in row i and column j is ∂Fᵢ/∂xⱼ. Thus DF(p)h predicts the output change produced by a small input displacement h.
F(p + h) = F(p) + DF(p)h + o(‖h‖)When the matrix is square, det DF gives signed local volume scale. A zero determinant means at least one infinitesimal direction has collapsed.
Wronskian
For two scalar functions, W = y₁y₂′ − y₂y₁′ is the signed area spanned by their state vectors (y, y′). I teach that picture before the determinant formula, because it is the one that survives once the algebra is forgotten. Nonzero W at one point certifies independence.
The caveat I always flag
For arbitrary differentiable functions, W ≡ 0 need not imply dependence. For solutions of the same regular linear ODE, Abel's identity restores the all-or-nothing conclusion.
Hessian
For a scalar field f, Hᶠ contains every second partial derivative. In a unit direction u, the second directional derivative is uᵀHᶠu.
Classification
At a critical point: two positive eigenvalues mean a local minimum; two negative a maximum; mixed signs a saddle. A zero eigenvalue makes the quadratic test inconclusive.
The habits I drill