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# 物理代写|结构力学代写Structural Mechanics代考|CE57000 The spectral decomposition

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## 物理代写|结构力学代写Structural Mechanics代考|The spectral decomposition

The spectral decomposition. If the eigenvalues and eigenvectors are known, we can express the original tensor in terms of those objects in the following manner
$$\mathbf{T}=\sum_{i=1}^{3} \mu_{i} \mathbf{n}{i} \otimes \mathbf{n}{i}$$
Note that we need to suspend the summation convention because of the number of times that the index $i$ appears in the expression. This form of expression of the tensor $\mathbf{T}$ is called the spectral decomposition of the tensor. How do we know that the tensor $\mathbf{T}$ is equivalent to its spectral decomposition? As we indicated earlier, the operation of a second-order tensor is completely defined by its operation on three independent vectors. Let us assume that the eigenvectors $\left{\mathbf{n}{1}, \mathbf{n}{2}, \mathbf{n}{3}\right}$ are orthogonal (which means that any eigenvectors associated with repeated eigenvalues were orthogonalized). Let us examine how the tensor and its spectral decomposition operate on $\mathbf{n}{j}$
$$\mathbf{T n}{j}=\sum{i=1}^{3} \mu_{i}\left[\mathbf{n}{i} \otimes \mathbf{n}{i}\right] \mathbf{n}{j}=\sum{i=1}^{3} \mu_{i}\left(\mathbf{n}{j} \cdot \mathbf{n}{i}\right) \mathbf{n}{i}=\sum{i=1}^{3} \mu_{i} \delta_{i j} \mathbf{n}{i}=\mu{j} \mathbf{n}_{j}$$
Thus, we have concluded that both tensors operate the same way on the three eigenvectors. Therefore, the spectral representation must be equivalent to the original tensor. A corollary of the preceding construction is that any two tensors with exactly the same eigenvalues and eigenvectors are equivalent.

## 物理代写|结构力学代写Structural Mechanics代考|Vector and Tensor Calculus

A field is a function of position defined on a particular region. In our study of mechanics we shall have need of scalar, vector, and tensor fields, in which the output of the function is a scalar, vector, or tensor, respectively. For problems defined on a region of three-dimensional space, the input is the position vector x. A function defined on a three-dimensional domain, then, is a function of three independent variables (the components $x_{1}, x_{2}$, and $x_{3}$ of the position vector $\mathbf{x}$ ). In certain specialized theories (e.g., beam theory, plate theory, and plane stress) position will be described by one or two independent variables.

A field theory is a physical theory built within the framework of fields. The primary advantage of using field theories to describe physical phenomena is that the tools of differential and integral calculus are available to carry out the analysis. For example, we can appeal to concepts like infinitesimal neighborhoods and limits. And we can compute rates of change by differentiation and accumulations and averages by integration.

Figure 15 shows the simplest possible manifestation of a field: a scalar function of a scalar variable, $g(x)$. A scalar field can, of course, be represented as a graph with $x$ as the abscissa and $g(x)$ as the ordinate. For each value of position $x$ the function produces as output $g(x)$. The derivative of the function is defined through the limiting process as

$$\frac{d g}{d x} \equiv \lim _{\Delta x \rightarrow 0}\left(\frac{g(x+\Delta x)-g(x)}{\Delta x}\right) \equiv g^{\prime}(x)$$
The derivative has the familiar geometrical interpretation of the slope of the curve at a point and gives the rate of change of $g$ with respect to change in position $x$. Many of the graphical constructs that serve so well for scalar functions of scalar variables do not generalize well to vector and tensor fields. However, the concept of the derivative as the limit of the ratio of flux, $g(x+\Delta x)-g(x)$ in the present case, to size of the region, $\Delta x$ in the present case, will generalize for all cases.

## 物理代写|结构力学代写Structural Mechanics代考| The spectral decomposition

$$\mathbf{T}=\sum_{i=1}^{3} \mu_{i} \mathbf{n} i \otimes \mathbf{n} i$$

$$\mathbf{T} \mathbf{n} j=\sum i=1^{3} \mu_{i}[\mathbf{n} i \otimes \mathbf{n} i] \mathbf{n} j=\sum i=1^{3} \mu_{i}(\mathbf{n} j \cdot \mathbf{n} i) \mathbf{n} i=\sum i=1^{3} \mu_{i} \delta_{i j} \mathbf{n} i=\mu j \mathbf{n}{j}$$ 因此，我们得出结论，两个张量在三个特征向量上以相同的方式工作。因此，光谱表示必须等价于原始张量。上述构造的推论是， 任何两个具有完全相同特征值和特征向量的张量都是等价的。

## 物理代写结构力学代写Structural Mechanics代考| Vector and Tensor Calculus

$$\frac{d g}{d x} \equiv \lim _{\Delta x \rightarrow 0}\left(\frac{g(x+\Delta x)-g(x)}{\Delta x}\right) \equiv g^{\prime}(x)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。