Meaning of Curl from stokes' theorem

In summary, Divergence and Curl can be explained using Gauss' and Stokes' theorems, respectively. While Gauss' theorem explains net outward flux per unit volume, Stokes' theorem relates total circulation of a vector field along a closed path to the component of Curl perpendicular to a small square. However, it is important to note that the absence of shear does not necessarily mean the absence of Curl, as there may still be viscosity present.
  • #1
Titan97
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Divergence can be defined as the net outward flux per unit volume and can be explained using Gauss' theorem. (I read this in Feynman lectures Vol. 2)
IMG_20151023_184133_507.JPG


In the next page, He derives Stokes' theorem using small squares.
IMG_20151023_183636_615.JPG

The left side of equation represents the total circulation of a vector field along a closed path S.
The right side contains the component of Curl perpendicular to ##\Delta a##
This only gives meaning to a particular component of Curl. How can I explain Curl using stokes' theorem just like how divergence is explained?
 
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  • #2
http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/4.-triple-integrals-and-surface-integrals-in-3-space/part-c-line-integrals-and-stokes-theorem/session-90-curl-in-3d/MIT18_02SC_MNotes_v4.3.pdf and find this longer one
another link
 
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  • #3
So the total shear in a region R is related to the circulation of the field about the boundary of R.
 
  • #4
Yes, but be careful: no shear doesn't have to mean no curl. There is the viscosity somewhere in between and that can be zero (in theory).
 
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Related to Meaning of Curl from stokes' theorem

1. What is the meaning of curl in Stokes' theorem?

The curl in Stokes' theorem refers to the measure of the rotation or circulation of a vector field around a closed curve in a three-dimensional space.

2. How is curl related to divergence in Stokes' theorem?

Stokes' theorem states that the surface integral of the curl of a vector field over a closed surface is equal to the line integral of the vector field along the boundary of the surface. This relationship between the curl and the line integral is similar to the relationship between the divergence and the volume integral in Gauss's law.

3. What is the significance of the curl in electromagnetic theory?

In electromagnetic theory, the curl of the electric field is related to the rate of change of the magnetic field, while the curl of the magnetic field is related to the rate of change of the electric field. This relationship is described by Maxwell's equations and is essential in understanding the behavior of electromagnetic waves.

4. Can the curl of a vector field be zero?

Yes, the curl of a vector field can be zero if the vector field is irrotational, meaning it has no rotation or circulation. This can occur when the vector field is conservative, which means it can be expressed as the gradient of a scalar field.

5. How is the curl calculated in three dimensions?

In three dimensions, the curl of a vector field is calculated using the cross product of the del operator (∇) with the vector field. This results in a vector quantity with components representing the rate of change of the vector field in each direction. The magnitude and direction of this vector give the magnitude and direction of the curl.

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