Introduction to differential forms and applications

• Grant number: 21/03338-9
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): October 01, 2021
• Effective date (End): September 30, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Sérgio Leandro Nascimento Neves
• Grantee: Marlon Henrique Vieira Galafo
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil

Abstract

In this project we shall make an introduction to differential forms, beginning from a review of the main concepts of linear algebra that will be necessary later, we shall revisit the main contents of the calculus courses with the language of differential forms. We will make an introduction to the differential forms defined in R2 and R3, considering its natural generalization to Rn and also differential forms defined on curves and surfaces. In the end, we plan to get an unified theory for the vector calculus in R3 with the theorems of Green, Gauss and Stokes related through the concept of exterior differentiation. It is an opportunity for the student to learn important concepts that are not covered in undergraduate courses like, for example, duality, multilinear algebra, tensors, exterior algebra, tangent vectors as derivations and to learn an unified theory for the integral calculus in R2 and R3 with the language of differential forms. (AU)