Substantial derivative of a vector

The reason to make this point is to contrast the situation for the substantial derivative of a vector field, as illustrated in the next section for the velocity vector. 

From Eq. 2.61, and the scalar substantial-derivative operator (Eq. 2.43), it is clear by inspection that the substantial derivative of a vector is not equivalent to the substantial derivatives of the vector s scalar components. Although there is a certain resemblance, there are extra terms that appear,... 

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. 

The substantial derivative operator is stated as follows. However, be cautious when applying the substantial derivative operator to a vector, since, in general, the substantial derivative of a vector does not equal the substantial derivative of the scalar components of the vector. In the following, the velocity vector V is presumed to have components v, where i indicates the directions of the coordinates. 

In vector form, the substantial derivative of a vector is defined as... 

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may oe expressed as... 

Some more recent software uses the tensor LEED approximation of Rous and Pen-dry which can save a substantial amount of computer time [2.268-2.270]. In tensor LEED the amplitudes (0) of all escaping electron waves (spots) are first calculated conventionally as described above for a certain reference geometry. Then the derivatives of these amplitudes 5Ag/5ri with respect to small displacements of each atom i in this reference geometry are calculated. These derivatives are the constituents of the "tensor". The wave amplitude for a modified model geometry where atom i is displaced by the vector Aq is then approximately given by ... 

Here N is the extensive variable associated with the conservation law (e.g., the momentum vector P), p is the fluid s mass density, and t] is the intensive variable associated with N (e.g., the velocity vector V). The volume of the control volume is given as 6V. In a cartesian coordinate system (, y, z),SV = dxdydz. The operator D/Dt is called the substantial derivative. 

To give a concrete example of the general vector representation, the substantial-derivative operator can be expanded in cylindrical coordinates as... 

This chapter established three important concepts that are essential for the derivation of the conservation equations governing fluid flow. First, the Reynolds transport theorem was developed to relate a system to an Eulerian control volume. The substantial derivative that emerges from the Reynolds transport theorem can be thought of as a generalized time derivative that accommodates local fluid motion. For example, the fluid acceleration vector... 

Discuss the pro s and con s of writing and using the spatial components of the substantial-derivative operator as either one of the two equivalent notations for either a scalar of vector field ... 

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 equations in this section retain some compact notation, including the substantial derivative operator D/Dt, the divergence of the velocity vector V-V, and the Laplacian operator V2. The expansion of these operations into the various coordinate systems may be found in Appendix A. 

See Ref. [148] for a physically appealing derivation.) Equation (12) yields an efficient, accurate and formally exact scheme to perform quantum dynamics. It involves the action of a handed, sparse and Toeplitz matrix on a vector. The propagation scheme has been shown to accurately represent [147] all quantum dynamical features including zero-point effects, tunneling as well as over-barrier reflections and in this sense differs from standard semi-classical treatments. The approach also substantially differs from other formalisms such as centroid dynamics [97,98,168-171], where the Feynman path centroid is propagated in a classical-like... 

The left side corresponds to the accumulation of internal energy in a control volume that moves at the local fluid velocity at each point on its surface. If the substantial derivative is expanded using vector notation, then there are actually two terms on the left side. 

The transformation between Lagrangian and Eulerian variables is determined by the motion vector x = r(X, t). Considering the rate of change of a physical variable G at a fixed material point, X, leads to the material or substantial derivative... 

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