Curl of the gradient

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... 

Consider first a general gauge transformation of the vector potential. For any scalar function / (r), the curl of the gradient vanishes identically ... 

Two other possibilities for successive operation of the del operator are the curl of the gradient and the gradient of the divergence. The curl of the gradient of any differentiable scalar function always vanishes. 

A useful relation that can be easily proved in any set of coordinates is that the curl of the gradient of scalar field is identically zero ... 

Note that as discussed in previous sections, under static conditions, these two antennas give no fields. In going between two static conditions, one can have the same fields at intermediate times, but a change in the electric impulse, this being related to a change in the Lorenz vector potential or to a nonzero time integral of the gradient of the Lorenz scalar potential. However, with no fields, the vector potential has zero curl, which in a QED sense is not measurable. 

Recall that the operation of V on a scalar quantity is the gradient, which is a vector. For example, if V is operated on a scalar pressure field P, then V P is the pressure gradient vector field, which can have different values in the three spatial directions. The operation of V on a vector field can either be the divergence or the curl of the vector field. The former is obtained by the dot product (also called the scalar product) as V v or div 1 , where the result is a scalar whereas, the latter is obtained by the cross product (also called the vector product) V / v. or curl v, and the result is a vector field. 

The machinery of geometric algebra makes it possible to differentiate and integrate functions of vector variables in a coordinate-free manner. The conventionally separated concepts of the gradient, divergence, and curl are... 

Evaluation of the Curl of the Divergence of the Velocity Gradient. Begin by expressing the velocity gradient tensor using summation notation ... 

The vector potential is not uniquely defined since the gradient of any scalar function may be added (the curl of a gradient is always zero). For an external magnetic field, it is conventional to write it as in eq. (10.65). 

Note that the physically equivalent vector potentials also may contain a vector field component A (the same in both cases), which is not a gradient (of any function). For example, it contains a vortexlike field, which is not equivalent to any gradient field. The curl of such a field is nonzero, while the curl of a gradient of any function does equal zero. Thus, the A itself may contain an unknown admixture of the gradient of a function. Hence, any experimental observation is determined solely by the non-gradient component of the field. For example, the magnetic field H2 for the vector potential A2 is... 

In developing the first item, the fact that, according to Gauss equation H8.2, the divergence of the electric field is proportional to the charge concentration has been taken into account, so its gradient is zero because the charge concentration is supposed to be uniform or equal to zero. For the second item, the absorption factor has been taken as a constant. With these developments, the curl of the potential density becomes... 

The proof of all these relations is straightforward and thus omitted here. Eq. (A.17) states that any rotational vector field has no sources, and Eq. (A.18) summarizes the fact that the curl of any gradient field is zero. 

The vector + d L has a vanishing curl and can, therefore, be represented as the gradient of some scalar o... 

Let us express Eqs. (58)-(61) in terms of potentials, rather than fields. Toward that end, let us invoke a general result from field theory (see, e.g., Kellogg [88], p. 76) Any vector field F(r,w), sufficiently differentiable, is the sum of a gradient and a curl. Then, fields P, N are given by... 

Because the term in parentheses above has a curl of zero, it can be written as the gradient of some scalar function, U, which is named the scmlar electromagnetic... 

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