How did they get that the vorticity = 2##\omega##?

AI Thread Summary
The discussion centers on understanding the relationship between vorticity and angular velocity in meteorology, particularly as presented in the book "Atmospheric Science" by Wallace and Hobbs. The user is struggling to grasp the mathematical definition of vorticity, specifically the equation ∇ × u = 2ω, given their familiarity with related concepts like angular velocity and differentiation. They seek clarification on how the natural coordinate system defined in the book relates to the calculation of vorticity. The user expresses confusion about the application of these concepts in the context of fluid dynamics and meteorology. Overall, the thread highlights the complexities of applying theoretical definitions to practical scenarios in atmospheric science.
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Homework Statement
Deriving an expression for vorticity.
Relevant Equations
Vorticity =##2 \omega##
I'm learning Meteorology, and using the book Atmospheric Science by Wallace and Hobbs. We're discussing the kinematics of the winds (fluids). I shall post some images to say what I don't understand. This is how they define their natural coordinate system
At any point on the surface one can define a pair of axes of a sys- tem of natural coordinates (s, n), where s is arc length directed downstream along the local streamline, and n is distance directed normal to the streamline and toward the left,
Screenshot 2022-05-25 at 5.48.06 PM.png

Screenshot 2022-05-25 at 5.51.01 PM.png
Screenshot 2022-05-25 at 5.51.31 PM.png
I know and understand the concepts of angular velocity, shear, curvature, differentiation, and partial differentiation but for some reason, which is latent to me, I cannot understand anything that the book has done. Will you please explain it to me?
 
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I am not sure whether I understand the book but mathematical definition of vorticity says
\nabla \times \mathbf{u}=2\mathbf {\omega}
where
u_x=- \omega y,\ u_y=\omega x,\,\ u_z=0
 
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