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2. Curl (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Curl_(mathematics)

   The above formula says that the curl of a vector field at a point is the infinitesimal volume density of this "circulation vector" around the point. To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the volume integral of the curl of a vector field to the above surface integral taken ...

3. Vector calculus identities - Wikipedia

   en.wikipedia.org/wiki/Vector_calculus_identities

   C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist.

4. Stokes' theorem - Wikipedia

   en.wikipedia.org/wiki/Stokes'_theorem

   The line integral of a vector field over a loop is equal to the surface integral of its curl over the enclosed surface. Stokes' theorem is a special case of the generalized Stokes theorem . [ 5 ] [ 6 ] In particular, a vector field on R 3 {\displaystyle \mathbb {R} ^{3}} can be considered as a 1-form in which case its curl is its exterior ...

5. Generalized Stokes theorem - Wikipedia

   en.wikipedia.org/wiki/Generalized_Stokes_theorem

   In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or , and the divergence theorem is the case of a volume in . [2] Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus.

6. Euler equations (fluid dynamics) - Wikipedia

   en.wikipedia.org/wiki/Euler_equations_(fluid...

   The ideal gas law can be recast into the formula: ... The vector calculus identity of the cross product of a curl holds: ... In 1D the volume reduces to an ...

7. Stream function - Wikipedia

   en.wikipedia.org/wiki/Stream_function

   The volume flux through the test surface connecting the points and . The total volumetric flux through the ... satisfies the curl-divergence equation ...