Curl, definition

The Ampere law is next used with a covariant definition of the curl operator V flx = Vx+i A x (351)... 

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 ... 

This result is a consequence of the dependence of the definition of Fp% of the spin curvature tensor, Kpk, according to Eq. (49) (as well as qpk = 0). This is because the spin curvature tensor Kpk is the four-dimensional curl of a 4-vector in configuration space ... 

Common examples of pseudo-vectors that will be relevant later include the angular velocity vector f2, the torque T, the vorticity vector co (or the curl of any true vector), and the cross product of two vectors. The inner scaler product of a vector and a pseudo-tensor or a pseudo-vector and a regular tensor will both produce a pseudo-vector. It will also be useful to extend the notion of a pseudo-vector to scalers that are formed as the product of a vector and a pseudo-vector. The third-order, alternating tensor e is a pseudo-tensor of third order as may be verified by reviewing its definition... 

Ectorhizosphere The soil layer surrounding roots and affected by the activity of roots themselves, and the microorganisms. The thickness of this soil portion usually ranges from one to a few millimetres. The ectorhizosphere was initially defined by Hiltner (1904), who referred to it simply as the rhizosphere. The same definition and term were reported by Curl and Truelove (1986). 

Chip-forms, Chip Breakability and Chip Control, Fig. 13 Definition of twist angle, which would indicate ratio of upcurl to side curl in a 3D chip form... 

Carbon 60 (C60, Buckyball) is this third form of carbon, discovered in 1985 by Richard Smalley, Harold Kroto, and Robert Curl for which they won the 1996 Nobel Prize in chemistry. It is named as Buckministerfuller to honor the architect of the geodesic dome, Buckminster Puller, because the dome s shell resembles the fullerenes hollow-core construction. Fullerene structure of carbon is face-centered cubic having carbon molecules at the corners and at the center of the faces and belonging to the fullerene family. In the world of symmetry it is definitely a new form of pattern created by the existing symmetry operations. 

The second angle of importance in machining is inclination. This is the angle that the cutting edge makes with the reference plane, which, by definition, is per-pendiciilar to the direction of tool travel. Inclination determines the direction of chip curling. When the inclination is zero, chip fiow is virtually in the plane of true rake. 

Thus, if the mass conservation requirement div q = 0 holds, and it does for incompressible flows where q is the Darcy velocity, the identity guarantees that we can represent q = curl V, which reduces to our planar in that limit. The use of a three-dimensional T, however, does not render path tracing any easier than dealing with three velocity functions (in two-dimensional problems, a single streamfunction suffices). Thus, we will not pursue any further discussion. But the idea of streamlines or pathlines as tangents locally parallel to the velocity vector is still attractive, and kinematically, we would expect a definition along the lines of dx/dt u, dy/dt v, and dz/dt w, where denotes proportionality. Let us consider an interface located anywhere within a flow, that is, any surface marked by red dye, and describe it by the locus of points f(x,y,z,t) = 0 (4-103)... 

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] ... 

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