Trimester 02 · Course detail

Multidimensional Analysis & Classical Fields

Analysis on ℝⁿ, Lebesgue measure and integration, differential forms, advanced linear and multilinear algebra, and classical electrodynamics

Credits: 15
Sequence: 2 of 8
Structure: 4 subjects
Status: Full syllabus

Curriculum focus

Analysis on ℝⁿ, Lebesgue measure and integration, differential forms, advanced linear and multilinear algebra, and classical electrodynamics

Evidence of mastery

Analytical field-theory study + research brief

Subject 01

Topology & Differential Calculus in ℝⁿ

The metric structure of finite-dimensional spaces and the differential calculus of vector-valued maps, carried to submanifolds and constrained extrema.

Norms, Compactness & Derivatives

Equivalence of Norms in Finite Dimensions

Prove that all norms on a finite-dimensional space are equivalent, and derive the Heine–Borel theorem together with the Bolzano–Weierstrass property for closed bounded subsets of ℝⁿ.

Fréchet vs. Gâteaux Derivatives

Compare the full (Fréchet) derivative with directional (Gâteaux) differentiability; compute the Jacobi matrix and prove that continuous partials suffice for class C¹.

Mean Value Inequality for Vector Maps

Establish the operator-norm mean value bound ‖f(x) − f(y)‖ ≤ sup‖Df‖·‖x − y‖ and extend differentiation to higher multilinear maps.

Higher Derivatives, Submanifolds & Constrained Extrema

Schwarz Symmetry & the Taylor Formula

Prove equality of mixed partials under continuity hypotheses, interpret the Hessian as a quadratic form, and state the second-order Taylor expansion in several variables.

Inverse & Implicit Function Theorems

Prove both theorems via the contraction mapping principle and derive their equivalence through block-manipulation of the Jacobi matrix.

Sard's Theorem

Show that the set of critical values of a sufficiently smooth map has measure zero, the cornerstone of transversality arguments.

Submanifolds of ℝⁿ

Characterize submanifolds as level sets of submersions; construct tangent and normal spaces and develop Lagrange multipliers together with the KKT conditions for inequality constraints.

Subject 02

Measure, Integration & Differential Forms

From Lebesgue's construction of the integral to the generalized Stokes' theorem, unifying volume, surface integrals, and exterior calculus.

Lebesgue Measure & Integration

Outer Measure & Carathéodory Condition

Construct Lebesgue outer measure, characterize measurable sets via the Carathéodory splitting condition, generate the Borel σ-algebra, and exhibit Vitali's non-measurable set.

Convergence Theorems

Prove the Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem, the workhorses of passing limits through integrals.

Fubini–Tonelli & Change of Variables

Iterate integrals by Fubini and Tonelli, connect Cavalieri's principle to slicing, and compute multidimensional integrals under the Jacobian change of variables.

Lᵖ Spaces & Geometric Measure

Lᵖ Spaces & Riesz–Fischer

Establish completeness of Lᵖ by the Riesz–Fischer theorem and study the convolution algebra together with mollifier approximations to the identity.

Surface Measure & Cauchy–Binet

Define k-dimensional surface measure via Gram determinants, prove the Cauchy–Binet formula, and examine the Schwarz lantern paradox showing surface-area subtleties.

Exterior Calculus & Stokes' Theorem

Exterior k-Forms & the Wedge Product

Build alternating multilinear forms, the wedge product algebra, and their behaviour under pullback.

The Exterior Derivative

Define d with the identity d² = 0, prove the Poincaré lemma on star-shaped domains, and assemble the de Rham complex.

Generalized Stokes' Theorem

Prove ∫_∂M ω = ∫_M dω on manifolds with boundary and recover the classical divergence, Green's, and Kelvin–Stokes theorems as special cases.

Subject 03

Advanced Linear Algebra & Multilinear Structures

Canonical forms, duality, tensors, and the spectral theory of operators — the algebraic engine behind the trimester's analysis and field theory.

Canonical Forms

Minimal Polynomial & Cayley–Hamilton

Relate the minimal polynomial to invariant subspaces and prove the Cayley–Hamilton theorem as the bridge between a matrix and its characteristic polynomial.

Jordan & Frobenius Forms

Decompose endomorphisms into Jordan canonical form over ℂ and rational (Frobenius) canonical form over general fields, with invariant-factor uniqueness.

Duality, Quotients & Tensors

Dual Spaces & Annihilators

Construct dual bases, characterize annihilators of subspaces, and compute duals of quotients.

Quotient Spaces & Tensor Products

Form quotient vector spaces and build the tensor product V ⊗ W via its universal property, with multilinear maps as its morphisms.

Quadratic & Spectral Theory

Quadratic Forms & Sylvester's Law

Classify real quadratic forms by Sylvester's law of inertia, read off signature, and connect to affine quadrics.

Spectral Theorem, SVD & Pseudoinverse

Prove the spectral theorem for Hermitian and normal operators, derive the singular value decomposition, and construct the Moore–Penrose pseudoinverse for least-squares problems.

Courant–Fischer & Graph Laplacians

Use the min-max theorem and Rayleigh quotients to bound eigenvalues, and apply the theory to graph Laplacians and spectral graph analysis.

Subject 04

Classical Fields & Maxwell Electrodynamics

Electrostatics through radiation treated as applied vector analysis and special relativity: potentials, gauges, stress-energy bookkeeping, and the covariant field tensor.

Field Operators & Potential Theory

Vector Field Operators

Master ∇, divergence, curl, and the Laplacian, and prove the Helmholtz decomposition theorem for fields specified by divergence and curl.

Poisson & Laplace Equations

Establish Dirichlet and Neumann uniqueness theorems, construct Green's functions for standard geometries, and develop the multipole expansion.

Magnetostatics & Gauge Structure

Biot–Savart & Ampère's Law

Derive the magnetostatic field from current distributions and connect Ampère's law to the vector potential A.

Coulomb vs. Lorenz Gauge

Impose the Coulomb and Lorenz gauge conditions on the potentials and examine the residual gauge freedom.

Maxwell Dynamics & Radiation

Full Maxwell Equations

Assemble the complete system including the displacement current and derive Poynting's theorem together with the Maxwell stress tensor as momentum conservation.

Retarded Potentials & Liénard–Wiechert

Solve the wave equations for moving charges through retarded potentials, obtain the Liénard–Wiechert fields, and compute Larmor dipole radiation.

Media & Covariant Formulation

Dispersive Media

Model dielectric tensors in dispersive media and prove the Kramers–Kronig causality relations linking absorption and dispersion.

Covariant Electrodynamics

Package the fields into the field-strength tensor F^μν and its dual F̃^μν, and rewrite the Maxwell system covariantly with the four-current J^μ.