Unit Vector Derivatives

The second term in Eq. 2.33 requires taking the divergence of a tensor. This operation, V pVV which produces a vector, is expanded in several coordinate systems in Section A.ll. In noncartesian coordinate systems, since the unit-vector derivatives do not all vanish, the divergence of a tensor produces some unexpected terms. 

The extra terms in the bottom row are a result of nonvanishing unit-vector derivatives. The tensor products of unit vectors (e.g., ezer) are called unit dyads. In matrix form, where the unit vectors (unit dyads) are implied but usually not shown, the velocity-gradient tensor is written as... 

The extra terms appear because in noncartesian coordinate systems the unit-vector derivatives do not all vanish. Only in cartesian coordinates are the components of the substantial derivative of a vector equal to the substantial derivative of the scalar components of the vector. The acceleration in the r direction is seen to involve w2, the circumferential velocity. This term represents the centrifugal acceleration associated with a fluid packet as it moves in an arc defined by the 9 coordinate. There is also a G acceleration caused by a radial velocity. In qualitative terms, one can visualize this term as being related to the circumferential acceleration (spinning rate) that a dancer or skater experiences as she brings her arms closer to her body. 

Figure 2.20 illustrates the unit vectors er and in a cylindrical coordinate system. Based on a geometric interpretation in the limit A0 - 0, develop expressions for der/d8 and deg/dd. Explain why the remaining seven unit-vector derivatives vanish (e.g., der/dr= 0). 

Carry out all the operations to evaluate the divergence of the stress tensor, V T. Be careful to consider that some unit-vector derivatives do not vanish. Check the results with those provided in the Appendix. 

We first derive the so-called continuity equation, which is a direct consequence of the conservation of mass. If p is the density, or mass per unit volume, then the total mass of a fluid contained in F is equal to M = fj p dF. Letting dS — fi dS be an element of the surface, with n a unit vector perpendicular to the surface, the mass flow per unit time through the surface element is pv dS. The total fluid flow out of the volume F is then given by... 

Here the unit vector n and radius vector R have opposite directions. The volume V is surrounded by the surface S as well as a spherical surface with infinitely large radius. In deriving this equation we assume that the potential U p) is a harmonic function, and the Green s function is chosen in such a way that allows us to neglect the second integral over the surface when its radius tends to an infinity. The integrand in Equation (1.117) contains both the potential and its derivative on the spherical surface S. In order to carry out our task we have to find a Green s function in the volume V that is equal to zero at each point of the boundary surface ... 

Here e is a unit vector normal to the surface at the POI defined by e = pH xpv/ pH xpv, and subscripts of p denote the partial derivatives. Thus the mean, H, and Gaussian, K, curvatures are expressed as... 

Figure 3.15. Results after applying three different derivative preprocessing tools to a sample vector, (a) A sample vector with noise and an offset of one unit. ) The derivative calculated by simple difference, (c) The derivative calculated using a running mean difference with a window width of 15. d) The derivative calculated using the Gorry method with a window width of 15.

In cartesian coordinates, the vector-tensor operator can be readily seen by inspection. In other coordinate systems, however, terms like the ones in the third row of the equation above result physically from the fact that control-surface areas vary, and mathematically from the fact that the derivatives of the unit vectors do not all vanish. We have recovered the expression in the previous section, which was developed entirely from vector-tensor manipulations ... 

A word of caution is needed in going from Eq. 6.47 to Eq. 6.48. Even though there is only one vorticity component for this situation, cog, Eq. 6.47 is still a vector equation. As such, care must be exercised with the substantial-derivative and Laplacian operators, since they involve nonvanishing unit-vector detivatives. The Laplacian of the vector oj produces... 

The l.h.s. of Eq. (2.38) is the /th element of the vector g(4+I). On the r.h.s. of Eq. (2.38), since the partial derivative of q with respect to its / th coordinate is simply the unit vector in the /th coordinate direction, the various matrix multiplications simply produce the / th element of the multiplied vectors. Because mixed partial derivative values are independent of the order of differentiation, the Hessian matrix is Hermitian, and we may simplify Eq. (2.38) as... 

The other arises from the rotation of the unit vectors i,j, and k with angular velocity m. Since the time derivatives of the unit vectors are the vector products of [Pg.9]

Although eq. (19) has been derived for R((j> z), all rotations R(4> n) through the same angle are in the same class (and therefore have the same character) irrespective of the orientation of the unit vector n. Therefore, eq. (19) holds for a rotation through 4> about any rotation axis. A formal proof that... 

This relation is of central importance for the dispersion of phonons and po-laritons. Special cases can immediately be derived from it. The case of the cubic diatomic crystal treated above follows from ej = e2 = e3 = e. Because s is a unit vector, s + s + s = 1. Therefore from Eq. (II.22a) follows... 

An alternative and simpler approach to deriving the result in equation (4.12) is to express the polarizability tensor as a general expansion in the two orthogonal unit vectors, u and p, embedded on the principal axes shown in Figure 4.4. Evidently, using Einstein notation, the polarizability can be written as... 

The rotary diffusion (Fokker-Planck) equation for the distribution function W(e,t) of the unit vector of the particle magnetic moment was derived by Brown [47]. As shown in other studies [48,54], it may be reduced to a compact form... 

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