Akademik Dersler - İçerik

Explains the concept of Riemannian integral.
Defines measure zero from sets of content zero and distinguishes the difference between them by examples.
Proves the Fubini Theorem and calculates multiple integrals.
Express the change of variable theorem and its proof.
Defines the algebraic structures of Tensors, alternating tensors and wedge product.
Express concepts of k-tensor and alternating tensor.
Express the concept of k-tensors and alternating tensors.
Express the compatibility of pullback map and wedge product.
Explains fundamental properties of differential forms and their exterior derivatives.
Defines the concepts of vector fields and differential form.
Express the properties of the exterior derivative.
Explains the compatibility of pullback map, wedge product and the exterior derivative.
Integrates the forms on chains, explains the Stokes Theorem for the chains.
Defines singular cube, chain and boundary operator.
Integrates the forms on the chains.
Proves the Stokes Theorem for chains and express that it is a generalization of fundamental theorem of calculus.
Obtains Green, Stokes (for surfaces) and Divergence Theorems as special cases of general Stokes Theorem.
Defines and integrates forms on manifolds, explains the Stokes Theorem for manifolds.
Defines manifolds and manifolds-with-boundary in R^n via coordinate systems, and determines tangent spaces.
Explains forms, the exterior derivative and orientation on manifolds.
Define the integral over a compact oriented manifold.
Proves and applies Stokes’ Theorem for manifolds.