Submanifold Helicity

Abstract

In this thesis we will focus on the topic of helicity. Helicity gives us a way to measure the coiling of flow lines in a vector field, computed by the formula [see formula in PDF file] . To start our exploration of helicity we begin with the topic of differential forms and their correspondence with vector fields in R 3 . Then we use the correspondence of forms and vector fields to show that Maxwell’s Equations can be reduced to two equations of differential forms. We continue our exploration of mathematics in physics with the Biot-Savart operator; this operator calculates the associated magnetic field on a domain Ω given an electric current on Ω. The Biot-Savart operator gives us a way to compute vector field helicity. We will begin our exploration of helicity with its standard definition above, and where it comes of use in both mathematics and physics. Next we return to differential forms to show how the correspondence to vector fields allows us to define the helicity of forms. Then using helicity of forms we can expand on the three-dimensional idea of helicity to both submanifolds and higher dimension ambient spaces.