### 数学代写|微积分代写Calculus代写|MATH1051

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Vector Differential Operators

A scalar field (that is, a scalar function of the position vector $\mathbf{r}$ ) is conveniently pictured by means of surfaces (in three dimensions) or lines (in two dimensions) along which its magnitude is constant. Depending on the physical application, these constant magnitude surfaces or lines could be called equipotentials, isobars, isotherms, or whatever applies in the given situation.

A common example, shown in Fig. 1.1, is a topographic map of a hillside.

The lines of equal altitude shown on the map are equipotentials of the gravitational field. No work is done in moving along an equipotential, and the direction of steepest slope is everywhere perpendicular to the equipotential. That perpendicular direction is defined as the direction of the gradient of the potential. In our topographic example, this direction is the steepest direction up the hill.

Experimentally, this would be opposite to the direction a ball would roll if placed at rest on the hillside.

The magnitude of the gradient is defined to be the rate of change of the potential with respect to distance in the direction of maximum increase. This provides a mathematical definition of the gradient as
$$\operatorname{grad} \phi=\hat{\mathbf{n}} \frac{d \phi}{|\mathbf{d r}|} .$$
In Eq. (1.1), the unit vector $\hat{\mathbf{n}}(=\mathbf{n} /|\mathbf{n}|)$ is in the direction of maximum increase of $\phi$, and $\mathbf{d r}$ is taken in that direction of maximum increase.

For infinitesimal displacements, an equipotential surface can be approximated by its tangent plane (or tangent line in two dimensions), so the change in a scalar field in an infinitesimal displacement $\mathbf{d r}$ will vary as the cosine of the angle between the direction of maximum gradient and dr. Then the differential change of $\phi$ in any direction is given by
$$d \phi(\mathbf{r})=\mathbf{d r} \cdot \mathbf{g r a d} \phi .$$

## 数学代写|微积分代写Calculus代写|Divergence

A vector field can have two different types of variation. It can vary along its direction, for instance like the velocity field, v, of a stream as the slope gets steeper. The vector field can also vary across its direction, as when the velocity is faster in the middle of the stream than near the edges. How can these two variations be measured?

The rate of increase of a vector field along its direction is called the divergence of the vector field. A simple example for a vector field $\mathbf{E}$ is shown in Fig. 1.2. A measure of the strength of the field is the density of lines of force in the figure, with the increase in the field indicated by increasing lines of force.
We construct a mathematical volume $V$ enclosed by a surface $S$, as shown in the figure. The increase in $\mathbf{E}$ can be seen in the figure as more lines of $\mathbf{E}$ leaving the volume than entering it (‘diverging’ from the volume).

A quantitative measure of the excess of lines leaving the volume is given by the integral $\oint_S \mathbf{d A} \cdot \mathbf{E} .{ }^1$ This integral can be used to define an average divergence (written as ‘div’) of the lines of the vector field. That is
$$\langle\operatorname{div} \mathbf{E}\rangle_V=\frac{1}{V} \oint_S \mathbf{d A}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right),$$
where the notation $\langle\operatorname{div} \mathbf{E}\rangle_V$ denotes the average of $\operatorname{div} \mathbf{E}$ over the volume $V$.
The value of $\operatorname{div} \mathbf{E}$ at a point $\mathbf{r}$ can be defined by shrinking the integral about the point, so
$$\operatorname{div} \mathbf{E}(\mathbf{r})=\lim _{V \rightarrow 0} \frac{1}{V} \oint_S \mathbf{d} \mathbf{A}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right)$$ gives the divergence of the vector field at the point $\mathbf{r}$ (if the limit exists), and is a measure of its rate of increase along the direction of the vector field. We take Eq. (1.17) as the definition of the divergence operator.

We will show on the next page that div $\mathbf{E}$ can be written as $\boldsymbol{\nabla} \cdot \mathbf{E}$, corresponding to the dot product of the vector differential operator $\boldsymbol{\nabla}$ with a vector E. We start using that notation now, so that the following equations will be in the usual notation for the divergence.

# 微积分代考

## 数学代写|微积分代写Calculus代写|Vector Differential Operators

$$\operatorname{grad} \phi=\hat{\mathbf{n}} \frac{d \phi}{|\mathbf{d r}|} .$$

$$d \phi(\mathbf{r})=\mathbf{d r} \cdot \operatorname{grad} \phi$$

## 数学代写|微积分代写Calculus代写|Divergence

$$\langle\operatorname{div} \mathbf{E}\rangle_V=\frac{1}{V} \oint_S \mathbf{d} \mathbf{A}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right)$$

$$\operatorname{div} \mathbf{E}(\mathbf{r})=\lim _{V \rightarrow 0} \frac{1}{V} \oint_S \mathbf{d} \mathbf{A}^{\prime} \cdot \mathbf{E}\left(\mathbf{r}^{\prime}\right)$$

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## MATLAB代写

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