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Concept of Tensors
Published in Bhaben Chandra Kalita, Tensor Calculus and Applications, 2019
where (p, q, r, …, u) or (a, b, c,…, s) is the k number of indices. Any mixed tensor of any order can similarly be defined.
Tensors
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
i.e., using the outer product of n tangent vectors and m vectors from the dual basis. Such tensors are said to be of type (n, m). If n = 0 the tensors are covariant and if m = 0 they are contravariant, in both cases referring to the way the tensor components transform under coordinate changes. A tensor that is neither covariant nor contravariant is a mixed tensor and its components transform according to the position of the different indices, i.e., upper indices transform contravariantly, while lower indices transform covariantly () Tb′1…b′ma′1…a′n=Tb1…bma1…an(∏k=1n∂y′a′k∂yak)(∏l=1n∂ybl∂y′b′k).
Applications of periodic unfolding on manifolds
Published in Applicable Analysis, 2018
We overcome this difficulty with the result of step 3: there it is shown that the expression , corresponding to B with a lowered index k, is symmetric. Since B is contravariant in k, is covariant in k and thus, due to the symmetry, also in i. Therefore, B has to be covariant in i as well, and B is finally a well-defined mixed tensor corresponding to a linear map acting on vector fields.
A novel variational method for 3D viscous flow in flow channel of turbomachines based on differential geometry
Published in Applicable Analysis, 2020
Guoliang Ju, Jingzhi Li, Kaitai Li
Covariant component and contravariant component of the metric tensor on a surface ℑ, Christoffel symbol and covariant tensor and mixed tensor of the curvature are defined, respectively, as The area element of a surface ℑ is , where and Without the loss of generality, we assume that the disk revolves around the z-axis with angular velocity and are orthonormal bases of a cylindrical coordinate system on the disk. Denote N as the number of blades and , then rotating one blade can reach the next (Figures 1 and 2). Thus the channel is determined by the boundary , where are inlet and outlet, are top surface (shroud) and bottom surface (hub), and are positive pressure surface and negative pressure surface.