Curl of a Vector Field

A differential element of volume is conventionally written as dx. Sometimes the notation dV is used for S to indicate the surface area enclosing the volume V. 

The divergence of the gradient of a scalar field occurs in several fundamental equations of electromagnetism, wave theory, and quantum mechanics. In Cartesian coordinates. 

FIGURE 11.13 Schematic representations of vector fields with divergence (left) and curl (right). 

FIGURE 11.14 Circulation of a vector field A(r) about an element of area Acr. In the limit Acr 0, this gives the z component of the curl V X A. 

we divide both sides by Act = AxAy and take the limits Lx, Ly 0. Since 

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The curl of a vector field u, written curl u, is given in Cartesian coordinates by... 

The curl of a vector field, representing the velocity of a fluid, is related to the phenomenon of rotation of the fluid. Consider V a vector field that represents the velocity field of a fluid particle, and let a point located at (x, y, z). The particles situated in the vicinity of this point tend to rotate around the axis formed by the... 

Taking the curl of both sides of the third and fourth Maxwell equations and using the identity that relates the curl of the curl of a vector field to its divergence and its laplacian, Eq. (G.21), we find... 

The differential operators encountered often in the description of the physical properties of solids are the gradient of a scalar field V,4)(r), the divergence of a vector field Vr F(r), the curl of a vector field x F(r), and the laplacian of a scalar field V vector field is simply the vector addition of the laplacian of its components, V F = + V F z). These operators in three dimensions are... 

Therefore, on the 0(3) level, the magnetic part of the complete free field is defined as a sum of a curl of a vector potential and a vacuum magnetization inherent in the structure of the B cyclic theorem. On the U(l) level, there is no B(3) field by hypothesis. 

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 second step consists of finding the curl of the potential density from its expression H8.7 as a function of the electric field. The curl of the curl of the electric field and the curl of a vector product are developed according to known formulas of vector calculus (Woan 2003) ... 

A direct consequence of Maxwell s equations is that the electric and magnetic fields can be expressed in terms of the scalar and vector potentials , A, which now include both spatial and temporal dependence. From the second Maxwell equation, Eq. (A.22), we conclude that we can express the magnetic field as the curl of the vector potential A(r, t), since the divergence of the curl of a vector potential vanishes identically (see Appendix G) ... 

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... 

Now, if a Beltrami field is simultaneously complex lamellar, (1) combined with (3), then curl v is both perpendicular and parallel to v. This can happen only if curl v is zero (that is, the field v is curl-less, or lamellar). Hence a vector field that is simultaneously a complex lamellar and a Beltrami field is necessarily lamellar. If the divergence of (5) is taken, we obtain... 

The curl u, or V x u, or rot u, or "curly function" describes how "curly" a vector field (array) of vectors u is. Verify that in a Cartesian system the pseudovector ... 

Helmholtz analysis. So, a vector field with the desired curl is evaluated and projected in the space of divergence-free vectors. This concept leads to the next assertion ... 

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