Griffiths Introduction to Electrodynamics (Stokes Theorem Corollary)

[cemtu]

Homework Statement

Electrodynamics, Stokes Theorem

Relevant Equations

No Equations needed

Although Stokes Theorem says that the line integral of a closed surface equals to zero why do we get a non-zero value out of this question 1.11 (and figure 1.33) in the Griffits Introduction to Eletrodynamics Book?

 

[Orodruin]

It is not an integration over a closed surface.

 

[cemtu]

It is not an integration over a closed surface.

It is an integration over a closed rectangle shape

 

[Orodruin]

It is an integration over a closed rectangle shape

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]

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.

sir there is nothing without boundaries and borders, so how exactly this second corollary of stokes theorem has a proper use in physics?

 

[Orodruin]

sir there is nothing without boundaries and borders, so how exactly this second corollary of stokes theorem has a proper use in physics?

This is wrong. For example, the two-dimensional sphere has no boundary curve.

 

[cemtu]

This is wrong. For example, the two-dimensional sphere has no boundary curve.

sir, spheres are three-dimensional objects.

 

[Orodruin]

sir, spheres are three-dimensional objects.

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.