Gradient of a vector

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. 

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Clearly grad is a vector function whose (x, y, z) components are the first partial derivatives of . The gradient of a vector function is undefined. Consider an infinitesimal vector displacement such that... 

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. 

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 ... 

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

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],... 

C Trondheim Bubble Column Model Gradient of a vector... 

The gradient of a vector u must be understood as the square matrix -[O/J resulting from the matrix product of the 1-line matrix of the transposed differential vector with the 1-column matrix of the vector o ... 

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. 

The gradient of a vector function A, which we write as V A, is a matrix in which element Ap is the derivative of component i of the vector with respect to coordinate... 

The operator V ( del ) can operate on vector functions as well as on scalar functions. The gradient of a vector... 

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

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