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# 数学代写|有限元方法代写finite differences method代考|Summation convention

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## 数学代写|有限元代写Finite Element Method代考|Summation convention

It is useful to abbreviate a summation of terms by understanding that a repeated index means summation over all values of that index. Thus the summation
$$\mathbf{A}=\sum_{i=1}^3 A_i \mathbf{e}i$$ can be shortened to $$\mathbf{A}=A_i \mathbf{e}_i$$ The repeated index is a dummy index and thus can be replaced by any other symbol that has not already been used. Thus we can also write $$\mathbf{A}=A_i \mathbf{e}_i=A_m \mathbf{e}_m$$ and so on. The “dot product” $\hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j$ and “cross product” $\hat{\mathbf{e}}_i \times \hat{\mathbf{e}}_j$ of base vectors in a righthanded system are defined by \begin{aligned} & \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j \equiv \delta{i j}= \begin{cases}0, & \text { if } i \neq j \ 1, & \text { if } i=j\end{cases} \ & \hat{\mathbf{e}}i \times \hat{\mathbf{e}}_j \equiv \varepsilon{i j k} \hat{\mathbf{e}}_k \end{aligned}

where $\delta_{i j}$ is the Kronecker delta and $\varepsilon_{i j k}$ is the alternating symbol or permutation symbol
\varepsilon_{i j k}=\left{\begin{aligned} 1, & \text { if } i, j, k \text { are in cyclic order } \ & \text { and not repeated }(i \neq j \neq k), \ -1, & \text { if } i, j, k \text { are not in cyclic order } \ & \text { and not repeated }(i \neq j \neq k), \ 0, & \text { if any of } i, j, k \text { are repeated. } \end{aligned}\right.
Note that in Eq. (2.2.6), $k$ is a dummy index, while $i$ and $j$ are not. The latter are called free indices. A free index can be changed to some other index only when it is changed in every expression of the equation to the same index. Thus, we can write Eq. (2.2.6) as
$$\hat{\mathbf{e}}m \times \hat{\mathbf{e}}_j=\varepsilon{m j k} \hat{\mathbf{e}}k ; \quad \hat{\mathbf{e}}_m \times \hat{\mathbf{e}}_n=\varepsilon{m n k} \hat{\mathbf{e}}k ; \quad \hat{\mathbf{e}}_p \times \hat{\mathbf{e}}_q=\varepsilon{p q k} \hat{\mathbf{e}}_k$$

## 数学代写|有限元代写Finite Element Method代考|The del operator

Differentiation of vector functions with respect to the coordinates is common in science and engineering. Most of the operations involve the “del operator”, denoted by $\nabla$. In a rectangular Cartesian system it has the form
$$\nabla \equiv \hat{\mathbf{e}}_x \frac{\partial}{\partial x}+\hat{\mathbf{e}}_y \frac{\partial}{\partial y}+\hat{\mathbf{e}}_z \frac{\partial}{\partial z}$$
It is important to note that the del operator has some of the properties of a vector but it does not have them all, because it is an operator. The operation $\nabla \phi(\mathbf{x})$ is called the gradient of a scalar function $\phi$ whereas $\nabla \times \mathbf{A}(\mathbf{x})$ is called the curl of a vector function $\mathbf{A}$. The operator $\nabla^2 \equiv \nabla \cdot \nabla$ is called the Laplace operator. In a 3-D rectangular Cartesian coordinate system it has the form
$$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$
We have the following relations between the rectangular Cartesian coordinates $(x, y, z)$ and cylindrical coordinates $(r, \theta, z)$ (see Fig. 2.2.2):

$$x=r \cos \theta, y=r \sin \theta, z=z$$
The base vectors in the two coordinate systems are related by
$$\hat{\mathbf{e}}r=\cos \theta \hat{\mathbf{e}}_x+\sin \theta \hat{\mathbf{e}}_y, \quad \hat{\mathbf{e}}\theta=-\sin \theta \hat{\mathbf{e}}x+\cos \theta \hat{\mathbf{e}}_y, \quad \hat{\mathbf{e}}_z=\hat{\mathbf{e}}_z$$ Note that the base vectors of the cylindrical coordinate system are not constant; the direction of $\theta$ and $r$-coordinates change as we move around the cylindrical surface. Thus, we have $$\frac{\partial \hat{\mathbf{e}}_r}{\partial \theta}=-\sin \theta \hat{\mathbf{e}}_x+\cos \theta \hat{\mathbf{e}}_y=\hat{\mathbf{e}}\theta, \frac{\partial \hat{\mathbf{e}}\theta}{\partial \theta}=-\cos \theta \hat{\mathbf{e}}_x-\sin \theta \hat{\mathbf{e}}_y=-\hat{\mathbf{e}}_r$$ and all other derivatives of the base vectors are zero. The operators $\nabla$ and $\nabla^2$ in the cylindrical coordinate system are given by (see Reddy $[2,3]$ ) $$\boldsymbol{\nabla}=\hat{\mathrm{e}}_r \frac{\partial}{\partial r}+\frac{1}{r} \hat{\mathrm{e}}\theta \frac{\partial}{\partial \theta}+\hat{\mathrm{e}}_z \frac{\partial}{\partial z}, \quad \nabla^2=\frac{1}{r}\left[\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r} \frac{\partial^2}{\partial \theta^2}+r \frac{\partial^2}{\partial z^2}\right]$$

## 数学代写|有限元代写Finite Element Method代考|Summation convention

$$\mathbf{A}=\sum_{i=1}^3 A_i \mathbf{e}i$$可以缩写为$$\mathbf{A}=A_i \mathbf{e}_i$$重复索引是一个虚拟索引，因此可以用任何其他尚未使用过的符号代替。因此，我们也可以写$$\mathbf{A}=A_i \mathbf{e}_i=A_m \mathbf{e}_m$$等等。右手坐标系中基向量的“点积”$\hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j$和“叉积”$\hat{\mathbf{e}}_i \times \hat{\mathbf{e}}_j$定义为 \begin{aligned} & \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j \equiv \delta{i j}= \begin{cases}0, & \text { if } i \neq j \ 1, & \text { if } i=j\end{cases} \ & \hat{\mathbf{e}}i \times \hat{\mathbf{e}}_j \equiv \varepsilon{i j k} \hat{\mathbf{e}}_k \end{aligned}

\varepsilon_{i j k}=\left{\begin{aligned} 1, & \text { if } i, j, k \text { are in cyclic order } \ & \text { and not repeated }(i \neq j \neq k), \ -1, & \text { if } i, j, k \text { are not in cyclic order } \ & \text { and not repeated }(i \neq j \neq k), \ 0, & \text { if any of } i, j, k \text { are repeated. } \end{aligned}\right.

$$\hat{\mathbf{e}}m \times \hat{\mathbf{e}}_j=\varepsilon{m j k} \hat{\mathbf{e}}k ; \quad \hat{\mathbf{e}}_m \times \hat{\mathbf{e}}_n=\varepsilon{m n k} \hat{\mathbf{e}}k ; \quad \hat{\mathbf{e}}_p \times \hat{\mathbf{e}}_q=\varepsilon{p q k} \hat{\mathbf{e}}_k$$

## 数学代写|有限元代写Finite Element Method代考|The del operator

$$\nabla \equiv \hat{\mathbf{e}}_x \frac{\partial}{\partial x}+\hat{\mathbf{e}}_y \frac{\partial}{\partial y}+\hat{\mathbf{e}}_z \frac{\partial}{\partial z}$$

$$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

$$x=r \cos \theta, y=r \sin \theta, z=z$$

$$\hat{\mathbf{e}}r=\cos \theta \hat{\mathbf{e}}_x+\sin \theta \hat{\mathbf{e}}_y, \quad \hat{\mathbf{e}}\theta=-\sin \theta \hat{\mathbf{e}}x+\cos \theta \hat{\mathbf{e}}_y, \quad \hat{\mathbf{e}}_z=\hat{\mathbf{e}}_z$$注意圆柱坐标系的基向量不是恒定的;当我们绕圆柱面移动时，$\theta$和$r$坐标的方向会改变。因此，我们有$$\frac{\partial \hat{\mathbf{e}}_r}{\partial \theta}=-\sin \theta \hat{\mathbf{e}}_x+\cos \theta \hat{\mathbf{e}}_y=\hat{\mathbf{e}}\theta, \frac{\partial \hat{\mathbf{e}}\theta}{\partial \theta}=-\cos \theta \hat{\mathbf{e}}_x-\sin \theta \hat{\mathbf{e}}_y=-\hat{\mathbf{e}}_r$$所有基向量的导数都是零。柱坐标系中的算子$\nabla$和$\nabla^2$由(参见Reddy $[2,3]$)给出。 $$\boldsymbol{\nabla}=\hat{\mathrm{e}}_r \frac{\partial}{\partial r}+\frac{1}{r} \hat{\mathrm{e}}\theta \frac{\partial}{\partial \theta}+\hat{\mathrm{e}}_z \frac{\partial}{\partial z}, \quad \nabla^2=\frac{1}{r}\left[\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r} \frac{\partial^2}{\partial \theta^2}+r \frac{\partial^2}{\partial z^2}\right]$$

## MATLAB代写

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