Posted on Categories:Structural Mechanics, 物理代写, 结构力学

# 物理代写|结构力学代写Structural Mechanics代考|CIVE514 Tensor invariants

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 物理代写|结构力学代写Structural Mechanics代考|Tensor invariants

Tensor invariants. In subsequent chapters we will have occasions to wonder whether there are properties of the tensor components that do not depend upon the choice of basis. These properties will be called tensor invariants. The identities of Eqn. (43) will be useful in proving the invariance of these properties. The argument will go something like this: Let $f\left(T_{i j}\right)$ be a function of the components of the tensor $\mathbf{T}$. Under a change of basis, we can write this function in the form $f\left(Q_{i k} Q_{j i} T_{k 1}\right)$. If the function has the property that
$$f\left(Q_{i k} Q_{j l} T_{k l}\right)=f\left(T_{i j}\right)$$
then the function $f$ is a tensor invariant. Since it does not depend upon the coordinate system, we can say that it is an intrinsic function of the tensor $T$, and write $f(\mathrm{~T})$. Three fundamental tensor invariants are given by
$$f_{1}(\mathbf{T}) \equiv T_{i i} \quad f_{2}(\mathbf{T}) \equiv T_{i j} T_{j i} \quad f_{3}(\mathbf{T}) \equiv T_{i j} T_{j k} T_{k i}$$

## 物理代写|结构力学代写Structural Mechanics代考|Eigenvalues and eigenvectors of symmetric tensors

Eigenvalues and eigenvectors of symmetric tensors. A tensor has properties independent of any basis used to characterize its components. As we have just seen, the components themselves have mysterious properties called invariants that are independent of the basis that defines them. It seems reasonable to expect that we might be able to find a representation of a tensor that is canonical. Indeed, this canonical form is the spectral representation of the tensor that can be built from its eigenvalues and eigenvectors. In this section we shall build the mathematics behind the spectral representation of tensors.

Recall that the action of a tensor is to stretch and rotate a vector. Let us consider a symmetric tensor $\mathbf{T}$ acting on a unit vector $\mathbf{n}^{\dagger}$ If the action of the tensor is simply to stretch the vector but not to rotate it then we can express it as
$$\mathbf{T n}=\mu \mathbf{n}$$
where $\mu$ is the amount of the stretch. This equation, by itself, begs the question of existence of such a vector $\mathbf{n}$. Is there any vector that has the special property that action by $\mathrm{T}$ is identical to multiplication by a scalar? Is it possible that more than one vector has this property?

Equation (47) is called an eigenvalue problem. Eigenvalue problems show up all over the place in mathematical physics and engineering. The tensor in three dimensional space is a great context in which to explore the eigenvalue problem because the computations are quite manageable (as opposed to, say, solving the vibration eigenvalue problem of structural dynamics on a structure with a million degrees of freedom).

## 物理代写|结构力学代写Structural Mechanics代考| Tensor invariants

（43）的晅等式侍有助于证明这些性质的不变性。争论将变成䢒样: 让 $f\left(T_{i j}\right)$ 是张量分量的函数 $\mathbf{T}$. 在基数栾化的情况下，我们可 以将此函数编写为 $f\left(Q_{i k} Q_{j i} T_{k 1}\right)$. 如果函数具有以下属性:
$$f\left(Q_{i k} Q_{j l} T_{k l}\right)=f\left(T_{i j}\right)$$

$$f_{1}(\mathbf{T}) \equiv T_{i i} \quad f_{2}(\mathbf{T}) \equiv T_{i j} T_{j i} \quad f_{3}(\mathbf{T}) \equiv T_{i j} T_{j k} T_{k i}$$

## 物理代写|结构力学代写Structural Mechanics代考| Eigenvalues and eigenvectors of symmetric tensors

$$\mathbf{T n}=\mu \mathbf{n}$$

## 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。