Green's Theorem
Green’s theorem, area formulas, curl and divergence, simply connected and conservative fields
Recent posts
Posts sorted by last_modified_at (most recently updated first).
Surface Integrals and Flux
Parametric surfaces, normal vectors, surface area, scalar surface integrals, flux
Line Integrals
Scalar and vector line integrals, work, fundamental theorem, path independence and conservative fields
Vector Fields
Vector fields and gradient fields, conservative fields and potentials, divergence and curl, differential identities
Curves and Vector-Valued Functions
Vector-valued functions, parametric curves, velocity and tangent, arc length, curvature and acceleration decomposition
Multiple Integrals
Multiple integrals, Fubini’s theorem, change of variables, and the Jacobian
Multivariable Functions and Partial Derivatives
Partial derivatives, gradient, differentiability, multivariable chain rule, extrema
Minimal Polynomial
Cayley-Hamilton theorem and minimal polynomial
Quadratic Forms and Sylvester's Law of Inertia
Classification of real symmetric bilinear forms
Singular Value Decomposition
Orthogonal decomposition of arbitrary real matrices
The Fundamental Theorem of Calculus
The fundamental theorem, existence of antiderivatives, Leibniz rule, term-by-term integration of power series
Improper Integrals
Infinite and singular integrals, comparison test, absolute convergence
Integration
Antiderivatives, indefinite integrals, Riemann sums, definite integrals, and the mean value theorem
Taylor's Theorem
Taylor polynomials, Lagrange remainder, Maclaurin series, approximation and limits
Complex Spectral Theorem
unitary diagonalization of normal operators
Spectral Theorem
Orthogonal diagonalization of self-adjoint operators
Mean Value Theorem
Mean value theorem and its applications: monotonicity, extrema, convexity, L’Hopital’s rule, optimization
Complex Inner Product Spaces
Hermitian inner products over complex numbers
Differentiation
Termwise differentiation of power series, derivatives of elementary functions, product, quotient, and chain rules
Power Series
Power series, radius of convergence, elementary function expansions, and analytic functions
Infinite Series
Partial sums and convergence, geometric and p-series, convergence tests, absolute and conditional convergence
Limits of Sequences
Convergence, limit laws, standard limits, e, and monotone convergence
Continuous Functions
Definition of continuity and properties of continuous functions: extreme value and intermediate value theorems
Quotient Spaces
Quotient spaces formed by modding out a subspace
Limits of Functions
Defining limits via epsilon-delta and proving limit laws and the squeeze theorem
Richardson Varieties and Peterson Varieties
Richardson varieties as transversal intersections of opposite Schubert varieties, and Peterson varieties from regular nilpotents: definitions, dimensions, in...
Bruhat Decomposition
Cell decomposition of homogeneous spaces, parabolic subgroups, and Schubert varieties on Grassmannians
Differentiation and Derivatives
Definition of derivative, differentiability and continuity, derivative and higher-order derivatives
The Givental J-Function and the Mirror Theorem
Fundamental solutions of quantum differential equations and the I equals J theorem
Dubrovin Connection
A flat connection on a Frobenius manifold with a spectral parameter
Frobenius Manifolds
Frobenius manifolds and the WDVV equation
Overview of Mirror Symmetry
Historical background and the Hori-Vafa mirror
Fano Varieties
Reflexive polytopes and the corresponding Gorenstein Fano toric varieties
Torus-Invariant Divisors and Line Bundles
Torus-invariant divisors and line bundles arising from the rays of a fan
Definition of Toric Varieties
General toric varieties obtained by gluing affine toric varieties from a fan
Chow Groups
Chow groups and the cycle class map
Intersection Product
The intersection product on Chow groups
Kodaira Vanishing Theorem
The Kodaira vanishing theorem and its applications
The Riemann–Roch Theorem for Surfaces
Intersection theory on surfaces and its applications
The Riemann–Roch Theorem for Curves
The Riemann–Roch theorem for curves
Serre Duality
Serre duality theorem and its applications
Derived Categories
Construction of derived categories via chain complexes and quasi-isomorphisms
Spectral Sequences
Spectral sequences that approximate the cohomology of a filtered complex page by page
Cohomology of Projective Space
Bott’s formula and the cohomology of line bundles on projective space
Sheaf Cohomology
Sheaf cohomology and its applications
Canonical Line Bundle
Canonical bundle and canonical divisor
Linear Systems
Complete linear systems, base loci, and ampleness
Divisors
Weil divisors, Cartier divisors, and divisor class groups
Line Bundles and Vector Bundles
Line bundles, invertible sheaves, and the Picard group
Grassmannians
Grassmannians as parameter spaces of linear subspaces