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## 数学代写|变分法代写Calculus of Variations代考|The Cartesian Second Order Tensors

The definition of a vector based on the coordinate transformation is extended, the definition of a tensor can be obtained.

Suppose that a quantity $\boldsymbol{T}$ is a ordered population composed of three components $T_j$, if they from a rectangular coordinate system $O x_1 x_2 x_3$ according to the following transformation rule
$$T_i^{\prime}=\alpha_{i j} T_j$$
are transformed to three components $T_i^{\prime}$ in another rectangular coordinate system $O x_1^{\prime} x_2^{\prime} x_3^{\prime}$, then the $\boldsymbol{T}$ is called the Cartesian first order tensor, it is called the first order tensor for short. This shows that a vector is the Cartesian first order tensor. A first order tensor needs a freedom index to express.

Similarly, suppose that a quantity $\boldsymbol{T}$ is a ordered population composed of nine components $T_{l m}$, if they from a rectangular coordinate system $O x_1 x_2 x_3$ according to the following transformation rule
$$T_{i j}^{\prime}=\alpha_{i l} \alpha_{j m} T_{l m}$$
are transformed to three components $T_i^{\prime}$ in another rectangular coordinate system $O x_1^{\prime} x_2^{\prime} x_3^{\prime}$, then the $\boldsymbol{T}$ is called the Cartesian second order tensor, it is called the second order tensor or tensor of second order for short. $T_{l m}$ and $T_{i j}^{\prime}$ are called the component of the Cartesian second order tensor. The above Kronecker symbol is a second order tensor. A second order tensor needs two freedom indexes to express. A second order tensor is usually expressed in the following several ways
$$\boldsymbol{T}=\left{T_{i j}\right}=T_{i j}=\left[\begin{array}{lll} T_{11} & T_{12} & T_{13} \ T_{21} & T_{22} & T_{23} \ T_{31} & T_{32} & T_{33} \end{array}\right]$$
where, the tensor and its component uses the same symbol $T_{i j}$, taking note that $T_{i j}$ represents different meaning in use.

The tensor is an invariant, namely, it has nothing to do with the choice of coordinate system, but its components change with the choice of coordinate system. The scalar and vector can be merged into the tensor, the scalar is a zero order tensor, the vector is a first order tensor, the stress is a second order tensor, and a third order, fourth order or higher order tensor. For a three-dimensional space, the $n$th order tensor has the $3^n$ components. In general, for a $m$-dimensional space, the $n$th order tensor has the $m^n$ components. It is thus clear that the tensor is the more general description of invariant.

## 数学代写|变分法代写Calculus of Variations代考|Algebraic Operations of Cartesian Tensors

Let $A_{i j}=\alpha_{l i} \alpha_{m j} A_{l m}^{\prime}$ and $B_{i j}=\alpha_{l i} \alpha_{m j} B_{l m}^{\prime}$ be two same order (Here is a second order) Cartesian tensors, do addition or subtraction of every component of the first order tensor and the corresponding component of the second order tensor, the result is made a new tensor with the same structure. For instance
$$C_{i j}=A_{i j} \pm B_{i j}=\alpha_{l i} \alpha_{m j}\left(A_{l m}^{\prime} \pm B_{l m}^{\prime}\right)=\alpha_{l i} \alpha_{m j} C_{l m}^{\prime \prime}$$
Because Eq. (1.7.17) satisfies the definition of a second order tensor, so $C_{i j}$ is a second order tensor. It is thus clear that the sum (or difference) of the two same order Cartesian tensors is still the same order tensors, their components equal the sum (or difference) of the components of the two same order Cartesian tensors.
(2) Multiplicative operation
Arbitrary tensors of the same space can be continually multiplied, instead of requiring them to have the same structure, but the order can not be disorderly, can also not have the same index, this kind of operation is called the outer product, exterior product, external product or outer multiplication of a tensor. For instance, the components of the two tensors are $A_{i j}$ and $B_{l m n}$, the cross product is a fifth order tensor, its composition is
$$C_{i j l m n}=A_{i j} B_{l m n}$$

# 变分法代考

## 数学代写|变分法代写Calculus of Variations代考|The Cartesian Second Order Tensors

$$T_i^{\prime}=\alpha_{i j} T_j$$

$$T_{i j}^{\prime}=\alpha_{i l} \alpha_{j m} T_{l m}$$

$\backslash 1$ eft 的分隔符缺失或无法识别

## 数学代写|变分法代写Calculus of Variations代考|Algebraic Operations of Cartesian Tensors

$$C_{i j}=A_{i j} \pm B_{i j}=\alpha_{l i} \alpha_{m j}\left(A_{l m}^{\prime} \pm B_{l m}^{\prime}\right)=\alpha_{l i} \alpha_{m j} C_{l m}^{\prime \prime}$$

(2) 乘法运算

$$C_{i j l m n}=A_{i j} B_{l m n}$$

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