The discussion revolves around the application of Stokes' Theorem as presented in Griffiths' Introduction to Electrodynamics, specifically addressing a question related to a line integral and its evaluation in the context of a closed rectangle shape.
The conversation is ongoing, with participants providing insights into the definitions and implications of closed surfaces and boundaries. Some participants challenge the assumptions made about the nature of the rectangle and its relevance to Stokes' Theorem.
There is a noted confusion regarding the dimensionality of objects and their boundaries, particularly in relation to the definitions used in physics and mathematics. The discussion reflects a mix of interpretations about the corollary's application in physical contexts.
It is an integration over a closed rectangle shapeOrodruin said:
The rectangle is not a closed surface. The definition of a closed surface is that it has no boundary. The rectangle clearly has a boundary - the four straight lines that form its border.cemtu said:
sir there is nothing without boundaries and borders, so how exactly this second corollary of stokes theorem has a proper use in physics?Orodruin said:
This is wrong. For example, the two-dimensional sphere has no boundary curve.cemtu said:
sir, spheres are three-dimensional objects.Orodruin said:
You are thinking of a ball or the fact that the sphere is embedded in three-dimensional space. The sphere is the two-dimensional boundary of a three-dimensional ball. The boundary of an n-dimensional object is (n-1)-dimensional. In order to know where you are on a sphere you need two coordinates (eg, longitude and latitude), this makes the sphere two-dimensional.cemtu said: