Lectures related to Total Differential: Definition and Integrals

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Geometrical Aspects of Differential Operators

Explores differential operators, regular curves, norms, and injective functions, addressing questions on curves' properties, norms, simplicity, and injectivity.

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Fields from Potential: Derivation and Curvilinear Integral

Explores deriving fields from a potential, curvilinear integrals, and necessary conditions for domains.

Regular Surfaces: Painting with Normal Vectors

Explores regular surfaces and painting with normal vectors, integral calculations, symmetries, and surface orientation.

Magnetostatics: Magnetic Field and Force

Covers magnetic fields, Ampère's law, and magnetic dipoles with examples and illustrations.

Harmonic Forms and Riemann Surfaces

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Curvilinear Coordinates: Calculations and Examples

Covers curvilinear coordinates and area calculations using double integrals with examples of different curves.

Calculus Applications: Lengths and Surfaces of Revolution

Discusses the applications of calculus in calculating lengths and surfaces of revolution, emphasizing integral calculus and geometric interpretations.

Curve Integrals: Parameterizations and Riemann Sums

Explores curve integrals, emphasizing parameterizations, geometric curves, and Riemann sums.

Mathematical Complements: df or δf

Explores mathematical tools for differentials of functions of multiple variables and their practical applications in thermodynamics and real-life scenarios.

Regular Curves: Parametrization and Tangent Vectors

Explores regular curves, examples like segments and functions, and curvilinear integrals along regular curves.

Curvilinear Integrals

Covers curvilinear integrals in R^n, focusing on continuous functions and curves.

Smooth maps and differentials: Differentials

Explores smooth maps, differentials, composition properties, linearity, and extensions on manifolds.

Functions: Parametric, Integrals, Multi-variable

Covers parametric functions, integrals, and the origin of plasticity in metals.

Linear Approximation and Derivative Parametric

Covers linear approximation, parametric derivatives, and differentiability conditions on intervals.

Green-Riemann Formula: Curvilinear Integrals

Covers the Green-Riemann formula, connectedness by arcs, parametrization of curves, and open simply connected domains in the Oxy plane.

Derivative of Integral with Parameter Dependency

Explores the derivative of an integral with parameter dependency and its continuity.

Integral Calculus: Darboux Sums

Covers Darboux sums, properties, and the fundamental theorem of calculus.