Search

3 results

2 - Transformation and Tensors
Joel Franklin, Reed College, Oregon
Book:

An Introduction to Gravity

Published online:

04 May 2024

Print publication:

11 April 2024, pp 51-101
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

Starting from the definition of tensorial objects by their response to coordinate transformation, this chapter builds the flat space vector calculus machinery needed to understand the role of the metric and its associated geodesic curves in general. The emphasis here is on using tensors to build equations that are “generally covariant,” meaning that their content is independent of the coordinate system used to express them. Motivated by the transformation of gravitational energy sources, the gravitational field should be a second-rank tensor, and given the way in which that tensor must show up in a particle motion Lagrangian, it is natural to interpret that tensor as a metric.

Appendix B - Basic Analysis Review
Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
Book:

Classical Numerical Analysis

Published online:

29 September 2022

Print publication:

20 October 2022, pp 854-873
- Chapter
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation
Summary

This appendix collects a review of the calculus and analysis in one and several variables that the reader should be familiar with. Notions of convergence, continuity, differentiability and integrability are recalled here.

On the Regularity of the Multisublinear Maximal Functions
Feng Liu, Huoxiong Wu
Journal:

Canadian Mathematical Bulletin / Volume 58 / Issue 4 / 01 December 2015

Published online by Cambridge University Press:

20 November 2018, pp. 808-817

Print publication:

01 December 2015
- Article
- - You have access
- PDF
- Export citation
This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.