Curl equations definition

Taking the vector curl of the right-hand side causes the first and last terms to drop out, since the curl of the gradient vanishes. However, for variable density, the left-hand side expands to long, complex, and not-too-useful expression (see Section A.14). Therefore let us restrict attention to incompressible flows, namely constant density. The curl of the incompressible Navier-Stokes equation, incorporating the definition of vorticity u = VxV, yields... 

In this case the pressure is eliminated altogether, since by vector identity, the curl of the gradient of a scalar field vanishes. From the definition of vorticity, Eq. 2.103, a simple diffusion equation emerges for the vorticity... 

The bracket in the second equation in (5) denotes a cyclic sum, and we use units (henceforth in this article) with c = 1. Combining the definition of /p as the four-dimensional curl of a 4 vector, as in Eq. (4), Maxwell s equations in terms of the vector potential are ... 

Transport processes in electrochemical systems should be analyzed with vector analysis, a part of calculus. The main definitions of vector analysis are terms like scalar, vector, gradient, divergence, and curl values. The reader is encouraged to refresh his or her memory for definitions of these values. There are two key equations that are fundamental to transport processes in electrochemical systans. The first describes the flux vector, of the ith species [1] ... 

This is simply a consequence of the prudent choice of coordinates. In addition to the z-components, we need the other x-components of H and E, respectively, in order to apply the boundary conditions. They can be found from the z-components with the help of Maxwell s equations and the definition of the curl operator. For the TE and TM waves the tangential components in the x-direction, using the generalized form, are... 

According to the first Maxwell equation the curl of the electrical field is zero in the absence of time dependent magnetic fields. This means that, in this case, no turbulent electrical fields occur. It is then and only then that we can define a scalar electrical potential, which we require. This is visible from the definition of curl E... 

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