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-dimentional 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:
Griffiths Introduction to Electrodynamics is a textbook that provides a comprehensive introduction to the field of electrodynamics. It covers topics such as vector calculus, electrostatics, magnetostatics, electromagnetic waves, and more.
Stokes Theorem Corollary is a mathematical concept that is derived from Stokes Theorem, which relates the surface integral of a vector field to the line integral of its curl. The corollary states that the line integral of the tangential component of a vector field over a closed path is equal to the surface integral of the normal component of that vector field over the surface enclosed by the path.
Stokes Theorem Corollary has various applications in physics and engineering, particularly in the fields of electromagnetism and fluid mechanics. For example, it is used to calculate the circulation of a fluid flow around a closed loop or the electric flux through a closed surface.
Stokes Theorem Corollary and Green's Theorem are both special cases of the more general Stokes' Theorem. Green's Theorem relates the line integral of a two-dimensional vector field over a closed curve to the double integral of the curl of that vector field over the region enclosed by the curve. Stokes Theorem Corollary is the three-dimensional version of this theorem.
Stokes Theorem Corollary can be a difficult concept to grasp at first, but with proper understanding of vector calculus and practice, it can become more intuitive. It is important to have a solid understanding of the fundamental concepts and equations in order to apply it correctly in various applications.