Geometrical meaning of Curl(Gradient(T))=0

  • #1
Titan97
Gold Member
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What is the geometrical meaning of ##\nabla\times\nabla T=0##?

The gradient of T(x,y,z) gives the direction of maximum increase of T.
The Curl gives information about how much T curls around a given point.

So the equation says "gradient of T at a point P does not Curl around P.
To know about how much T curls around a particular point, I need to know about the direction of T on other points around the required point.
 

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  • #2
Geofleur
Science Advisor
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Suppose that ## T = T(x,y) ##. Then the equations ## T(x,y) = const ## will describe curves of constant ## T ## in the ##x##-##y## plane. At any point on a given curve, ## \nabla T ## will then be perpendicular to that curve. Imagine the whole vector field of the ## \nabla T ##'s everywhere. Within a small neighborhood around any given point, these vectors will not have a "circulatory" pattern. That is, if the ## \nabla T ## vectors represented fluid velocities, then around any point, a small paddle wheel placed in the fluid would not rotate. That is the geometric meaning of ## \nabla \times \nabla T = 0 ##.
 
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  • #3
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The gradient of a magnitude is the flux through and infinitesimal closed 3D area divided by this area. I don´t thik infinitesimals as a serious mathematical concept, but you can translate to differentiation formal way.
 
  • #4
I find it helpful to think about conservative fields here. A force field grad T is a conservative field because it is derived from a scalar potential T. Now any circulatory path through a conservative field results in no change in potential, since it starts and ends at the same point. Hence curl of the force field must be zero, intuitively. You could prove that using Stokes Theorem: 0 = integral of gradT.dr along a circulatory path = surface integral of curl (gradT) .dS. Can only happen for all possible closed paths if curl(gradT) =0
 

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