Vector And Tensor Analysis ~UPD~

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A tensor is a generalization of matrix and vector. A representation of a tensor using matrix notation is called a matrix representation. A representation using vector notation is called a vector representation.

The information is given in the form of a 3D tensor, T, that is a number, with a shape of (N, 3) that means that the tensor has N rows and 3 columns. Each row is made of a number associated with a specific point or direction. The tensor can be a vector, a matrix or any other kind of data structure.

For example, the vector of a force F acting on a point or vector v can be defined as F=tv. The vector of a force that acts on a line can be defined as F=tv+tv', where v and v' are two points, that can be given in spherical coordinate system by r, φ and z. Since the two points v and v' are on the same line, the vector that acts on it must be the same, so we get F=trv+t(rφ+z).

So, tensors of rank 2 are useful to represent and process the information about the interactions between the directions in space. The number of tensors is equal to the number of dimensions that can be considered in an object, a system or a process.

Matrix math can be applied to tensors, in the same way as for vectors.2) Cramer’s rule is used for solving linear equations, Jacobian matrix is used for solving non-linear equations. So the general formula can be applied to solve a system of non-linear equations, as in the case of the biological neural network.

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