What is a second order tensor equation, the elemental unit of a second order tensor

Tensor is one of the fundamental concepts in geometry and algebra.

Algebraically speaking, it's a generalization of vectors. We know that vectors can be seen as one-dimensional "**" (i.e., the components are arranged in a sequential row), and matrices are two-dimensional "**" (the components are arranged in vertical and horizontal positions), so the n-order tensor is the so-called n-dimensional "**". The strict definition of a tensor is described using a linear map.

Geometrically, it is a true geometric quantity, that is, it is something that does not change with the coordinate transformation of the frame of reference. Vectors also have this property.

Sometimes, people directly represent tensors in a coordinate system by several numbers (called components), and between the components in different coordinate systems, certain transformation rules should be satisfied (see covariance law, inverse change law), such as matrices, multivariate linear forms, etc. Some physical quantities such as stress, strain of an elastomer, and the energy momentum of a moving object are expressed by tensors. In the development of differential geometry, Gauss, b

Riemann, Christofel and others introduced the concept of tensors in the 19th century, and then introduced them by GRich and his student TLevitsita developed tensor analysis, a

Einstein made extensive use of tensors in his general theory of relativity.

Scalars can be seen as tensors of order 0, and vectors can be seen as tensors of order 1.

There are many special forms of tensors, such as symmetric tensors, antisymmetric tensors, and so on.

The second order is the second order.

Second orderTensorsThe unit of the element is a vector.

For a point in space, given a location, there is a value of pressure. Then start doing the gradient field of pressure. Scalar.

The gradient field of is a vector, i.e., a tensor of order 1.

For a location in space, there is a value for a vector. But what makes more sense is that for a position in space, for an arbitrary nucleus, roll a unit vector.

You can find the change in the pressure along the direction of the unit vector in the field of this point, that is, the value of the pressure gradient point multiplied by the unit vector.

Example of open burning. A tensor can be expressed as a sequence of values, defined by a domain of vector values.

and a range of scalar values.

Functional representation of . The vectors in these defined fields are natural numbers.

and these numbers are called indicators. For example, a third-order tensor can have dimensions and 7. Here, the indicators range from <1,1,1,> to <2,5,7>.

The tensor can have one value > the indicator <1,1,1, another value in the indicator <1,1,2, >, and so on for a total of 70 values.

The above content reference: Encyclopedia of Hundred Corrections Surplus - Tensors.

Pure product, linear transformation, linear mapping, inner product, bilinear type, etc., are all tensors.

In fact, tensors from a modern point of view.

It is defined in terms of multiple linear mappings, but of course there are many equivalent definitions, which can be found in the wiki tensor entry. The most obvious thing to notice right away is that the second-order tensors are all n*n. But the most essential difference is:

Tensors are geometric, and matrices are algebraic. A tensor is geometric, which means that it is represented by different bases, and it is certainly not the same when written, but they all represent the same tensor, and a definite geometric object is not different because of the different representations. It also shows that tensors must follow certain rules when transforming.

For example, an ellipse is geometric, and x2 a2+y2 b2=1 is algebraic; Another example is the three-dimensional European space.

A point in is geometric, but it is represented by three coordinates. Tensors and matrix operations are similar in that they are inherently corresponding. <>