### 数学代写|微积分代写Calculus代写|MATH141

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|微积分代写Calculus代写|Divergence Theorem

Combining Eq. (1.16) for the average value of the divergence over a finite volume $V$ with the definition of a volume average that
$$\langle\nabla \cdot \mathbf{E}\rangle_V=\frac{1}{V} \int_V \nabla \cdot \mathbf{E} d^3 r,$$
we get
$$\int_V^r \nabla \cdot \mathbf{E} d^3 r=\oint_{J_S} \mathbf{d} \mathbf{A} \cdot \mathbf{E} .$$
In this form, it is called the divergence theorem.
The definition of the divergence given by Eq. (1.17) can be used to evaluate the divergence of the position vector. We apply the definition to a sphere of radius $R$, getting ${ }^2$
\begin{aligned} \nabla \cdot \mathbf{r} & =\lim {V \rightarrow 0} \frac{1}{V} \oint_S \mathbf{d A}^{\prime} \cdot \mathbf{r}^{\prime}=\lim {R \rightarrow 0} \frac{1}{V} \oint R^3 d \Omega^{\prime} \ & =\lim {R \rightarrow 0} \frac{4 \pi R^3}{(4 / 3) \pi R^3}=3 . \end{aligned} As we did with the gradient, we now show what the divergence would look like in Cartesian coordinates. Figure $1.3$ shows an infinitesimal volume (a parallelepipid in Cartesian coordinates) of dimensions $\Delta x \times \Delta y \times \Delta z$, that will shrink to zero at the point $x, y, z$. The surface integral in the definition of $\operatorname{div} \mathbf{E}$ is over the six faces of the parallelepipid, I-VI, so the integral can be written as $$\lim {V \rightarrow 0} \frac{1}{V} \oint_S \mathbf{d} \mathbf{A} \cdot \mathbf{r}=I+I I+I I I+I V+V+V I,$$
where $I$ indicates the integral over face $\mathrm{I}$, and similarly for the other faces.

## 数学代写|微积分代写Calculus代写|Curl

Next, we look at how a vector $\mathbf{E}$ can vary across its direction, and we give a physical definition of the curl of a vector field. Figure $1.4$ shows a vector field having such a variation, with the density of lines being proportional to the strength of the field. If this were a velocity field, such as the current of water in a stream, this variation could be measured experimentally by placing a paddle wheel in the stream as shown in the figure. Then the rotation of the paddle wheel would be a measure of the variation of the vector field. This can be done without getting wet by calculating a line integral around a typical closed curve $C$, as shown on the figure.

The line integral can be used to define an average value of the variation (called curl) over a surface $S$ bounded by the curve $C$. The average curl is defined by
$$\langle\hat{\mathbf{n}} \cdot \operatorname{curl} \mathbf{E}\rangle_S=\frac{1}{S} \oint_C \mathrm{dr}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right),$$
where $\hat{\mathbf{n}}$ is the unit vector normal to the surface $S$ at any point. Note that, by this detinition, the vector average $\langle\hat{\mathbf{n}} \cdot \mathbf{c u r l} \mathbf{E}\rangle_S$ does not depend on the shape of the surface $S$, but only on the bounding path $C$ and the area of the surface. Since the variation will be different in different directions, it is the average value of the normal component of curl that is defined by Eq. (1.25).

The positive sign for the direction of $\hat{\mathbf{n}}$ is taken by convention to be the boreal direction. That is, if the integral around the contour $C$ is taken in the direction of the rotation of the earth, then the north pole is in the positive direction as shown on Fig. 1.5a.

This is also stated as the right hand rule: If the integral around the contour $C$ is taken in the direction that the four fingers of the right hand curl as they tend to close, then the right thumb points in the positive direction for $\hat{\mathbf{n}}$, as shown in Fig. 1.5b. This will be our general sign convention relating the direction of integration around a closed curve and the positive direction of the normal vector to any surface bounded by the curve.

I’he value of curl $\mathbf{E}$ at a point $\mathbf{r}$ can be defined by starting with a smooth surface through the point and taking the limit as the curve bounding the surface shrinks about the point, and the enclosed surface shrinks to zero area. This gives the definition of curl at a point:
$$[\operatorname{curl} \mathbf{E}(\mathbf{r})]n=\lim {S \rightarrow 0} \frac{1}{S} \oint_C \mathbf{d r}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right) .$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|Divergence Theorem

$$\langle\nabla \cdot \mathbf{E}\rangle_V=\frac{1}{V} \int_V \nabla \cdot \mathbf{E} d^3 r$$

$$\int_V^r \nabla \cdot \mathbf{E} d^3 r=\oint_{J_S} \mathbf{d} \mathbf{A} \cdot \mathbf{E} .$$

$$\nabla \cdot \mathbf{r}=\lim V \rightarrow 0 \frac{1}{V} \oint_S \mathbf{d} \mathbf{A}^{\prime} \cdot \mathbf{r}^{\prime}=\lim R \rightarrow 0 \frac{1}{V} \oint R^3 d \Omega^{\prime} \quad=\lim R \rightarrow 0 \frac{4 \pi R^3}{(4 / 3) \pi R^3}=3$$

$$\lim V \rightarrow 0 \frac{1}{V} \oint_S \mathbf{d A} \cdot \mathbf{r}=I+I I+I I I+I V+V+V I$$

## 数学代写|微积分代写Calculus代写|Curl

$$\langle\hat{\mathbf{n}} \cdot \operatorname{curl} \mathbf{E}\rangle_S=\frac{1}{S} \oint_C \mathrm{dr}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right),$$

（1） 定义的是卷曲法向分量的平均值。(1.25)。

$$[\operatorname{curl} \mathbf{E}(\mathbf{r})] n=\lim S \rightarrow 0 \frac{1}{S} \oint_C \mathbf{d r}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right)$$

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## MATLAB代写

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