Gradient of a vector field

If V(r, x) were a known function, this linear expansion could be used to determine how the velocity varies for short intervals of time and in any arbitrary short spatial direction dx. In a Taylor-series expansion of a scalar field, it is often conventional to post-multiply by the dx. Since the gradient of a scalar field is a vector and because the inner product of two vectors is commutative, the order of the product is unimportant. However, because of the tensor structure of the gradient of a vector field, the pre-multiply is essential. 

Because simple translation of the entire solid is not of interest, this class of motion is eliminated to give a parameter related only to local deformations of the solid this parameter is the displacement gradient, V . The gradient of a vector field Vu is a second-rank tensor, specified by a 3 by 3 matrix. The elements of this displacement gradient matrix are given by (Vu),y = dujdxj, also denoted Uij in which i denotes the i" displacement element and j denotes a derivative with respect to the y spatial coordinate, i.e. [1],... 

The derivative of a scalar a with respect to a vector is a vector. The gradient of a vector field v is a tensor of rank two... 

The gradient of a vector field v x) can be calculated in a similar way. It states how the vector field changes in each spatial direction and is thus a tensor field of second order. The rule to calculate this gradient can be most easily written in component notation ... 

V operates on the quantity that follows it. The notation V T means to perform the V operation on T. How the V operator is applied depends on whether it is operating on a scalar or on a vector function. For a scalar function, Equation (17.14) dehnes V as the following operation take the partial derivative of the function with respect to each Cartesian coordinate and multiply it by the unit vector in the direction of the coordinate, then sum the products. The result is the gradient vector. Now let s find the gradient of a vector field. 

In this text we shall define the gradient of a vector field v by [161, 273]... 

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... 

In the derivation of the Navier-Stokes equations, it is necessary to consider the gradient of the vector velocity field. The gradient of a vector produces a second-order tensor. 

The functional gradient of F (or gradient of a vector function) can be defined by Gp, and the inner product with a velocity field v ... 

When contraction is performed once (summation over repeated indices), the divergence is obtained instead of the gradient. The divergence of a vector field v is a scalar... 

The gradient of a scalar field A(x, y,z) is a vector field, each point of which gives the magnitude and direction of the largest change in the scalar field ... 

Equation (14) is known as the curl-condition and derives from the elementary fact that the vector field Fi2(Q) should be curl-free if expressible as the gradient of a scalar field according to Eq. (13), since one has ... 

Gradient of a scalar The gradient of a scalar field p is a vector defined by Vp, or Qp/dxi. 

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... 

Scalar, vector, and tensor fields. The gradient of a scalar field 0(x) is denoted by V0 and is the veetor defined by ... 

The divergence div u and gradient Vm of a vector field u(x) are respectively a scalar and a second-order tensor field, defined by ... 

Nearly two years ago, studying electrodynamics in curved space-time I found1 that Maxwell s equations impose on space-time a restriction which can be formulated by saying that a certain vector q determined by the curvature field must be the gradient of a scalar function, or... 

So far, Santos has been able to express the relation between a set of coefficients af, aj J 6 / describing a vector field and the overall curvature of the stream lines of this vector field. Based on the curvature field, they constructed the measure E of the curvature distribution in the simulation box. Provided that the homogeneous curvature field of curvature c0 is the one that minimizes E, the problem of packing has been recast as a minimization problem. However, the lack of information about the gradient of the error function to be minimized does not facilitate the search. Fortunately, appropriate computer simulation schemes for similar minimization problems have been proposed in the literature [105-109]. 

Assume that a differentiable scalar field depends on certain independent variables as S = S(xi, x2, X3). The gradient of the scalar field produces a vector, described below in different coordinate systems. 

The quantity A appears in these equations and is the vector potential of electromagnetic theory. In a very elementary discussion of the static electric field we are introduced to the theory of Coulomb. It is demonstrated that the electric field can be written as the gradient of a scalar potential E = —Vc)>, constant term to this potential leaves the electric field invariant. Where you choose to set the potential to zero is purely arbitrary. In order to describe a time-varying electric field a time dependent vector potential must be introduced A. If one takes any scalar function % and uses it in the substitutions... 

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. 

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