# 数学代写|多变量微积分代写multivariable calculus代考|MATH3334

## 数学代写|多变量微积分代写multivariable calculus代考|Differentials and error analysis

Definition $2.4$ of a differentiable function allows the following interpretation:
$$f(x+\Delta x)-f(x)=\operatorname{grad} f(x) \cdot \Delta x+|\Delta x| \rho(\Delta x)$$
or, more simply, that $\Delta f(\boldsymbol{x} ; \Delta \boldsymbol{x}) \approx \operatorname{grad} f(\boldsymbol{x}) \cdot \Delta \boldsymbol{x}$ for $|\Delta \boldsymbol{x}| \ll 1$.
This leads to the idea of constructing a new function of both
$$\boldsymbol{x}=\left(x_1, x_2, \ldots, x_n\right) \text { and } \mathbf{d} \boldsymbol{x}=\left(\mathrm{d} x_1, \mathrm{~d} x_2, \ldots, \mathrm{d} x_n\right)$$
That is, it is a function of $2 n$ variables.

The function $\mathrm{d} f$ has the following three features:
(1) It is an approximation to the change in $f, \Delta f$, coming from a change $\boldsymbol{x} \rightarrow \boldsymbol{x}+\mathrm{d} \boldsymbol{x}$;
(2) it is linear in $\mathrm{d} \boldsymbol{x}$; and
(3) it is a natural tool to use if considering overall error estimates when individual errors $(\mathrm{d} x)$ are known.
The last feature identifies the differential’s most useful application.
Suppose we have a quantity $f$ whose value depends on many parameters, say $x_1, \ldots, x_n$. Any errors incurred in measuring the $x_i$ result in an error in the quantity $f$.
An estimate of the maximum error in $f$ is thus given hy
$$|\mathrm{d} f(\Delta \boldsymbol{x})| \leq\left.\left|\frac{\partial f}{\partial x_1}\right|_x|| \Delta x_1|+| \frac{\partial f}{\partial x_2}\right|_x\left|\Delta x_2\right|+\cdots+\left|\frac{\partial f}{\partial x_n}\right|_x|| \Delta x_n \mid$$
by the triangle inequality (Section 1.B).
The right-hand side of Equation (3.2) gives the maximum possible error only if one knows the maximum uncertainty of the individual $x_i$. If one knows the exact values of the $\mathrm{d} x_i$ (or $\Delta x_i$ ) including their signs then we use the differential $\mathrm{d} f(\boldsymbol{x}, \mathrm{d} \boldsymbol{x})$ directly to give an approximate value to the change $\Delta f=f(\boldsymbol{x}+\Delta \boldsymbol{x})-f(\boldsymbol{x})$.

## 数学代写|多变量微积分代写multivariable calculus代考|Method of least squares

In the year 1801 the world of astronomy was excited by the discovery of a new minor planet, Ceres, whose (rough) position in the night sky had been noted a few times before it vanished from view. The young Carl Friedrich Gauss $[19,20]$ used a method – one he had worked out while still a student – to plot the orbit of the planet, and he was able to tell astronomers where in the sky to search for it. That method is the subject of this section.

In the first example that follows, we imagine the planet’s orbit in the night sky to be a straight line which has to be fitted in some optimal fashion to a set of discrete pairs of observations which are subject to errors. In the second example, the method is applied to discrete observations on a supposed planetary orbit. In the third example, the method is extended to continuous domains.

The field of statistics deals with “observations” (measurements) on variables that are known to be subject to random errors. When we observe two or more variables at once, it is often appropriate to ask what is the relationship between these two variables. This question is the basis of the study known as “regression analysis”, which is outside the scope of this book. But the core of regression analysis is an application of the differential calculus called the method of least squares, invented independently by Gauss.

# 多变量微积分代考

## 数学代写|多变量微积分代写multivariable calculus代考|Differentials and error analysis

$$f(x+\Delta x)-f(x)=\operatorname{grad} f(x) \cdot \Delta x+|\Delta x| \rho(\Delta x)$$

$$\boldsymbol{x}=\left(x_1, x_2, \ldots, x_n\right) \text { and } \mathbf{d} \boldsymbol{x}=\left(\mathrm{d} x_1, \mathrm{~d} x_2, \ldots, \mathrm{d} x_n\right)$$

(1) 它是对变化的近似 $f, \Delta f$ ，来自变化 $x \rightarrow x+\mathrm{d} x$;
(2) 它是线性的 $\mathrm{d} \boldsymbol{x}$;
(3) 如果在个别错误时考虑总体错误估计，它是一种自然使用的工具 $(\mathrm{d} x)$ 是已知的。 最后一个特征确定了差速器最有用的应用。

$$|\mathrm{d} f(\Delta \boldsymbol{x})| \leq\left.\left|\frac{\partial f}{\partial x_1}\right|_x|| \Delta x_1|+| \frac{\partial f}{\partial x_2}\right|_x\left|\Delta x_2\right|+\cdots+\left|\frac{\partial f}{\partial x_n}\right|_x|| \Delta x_n \mid$$

(或者 $\Delta x_i$ ) 包括他们的标志然后我们使用微分 $\mathrm{d} f(x, \mathrm{~d} \boldsymbol{x})$ 直接给变化一个近似值
$$\Delta f=f(\boldsymbol{x}+\Delta \boldsymbol{x})-f(\boldsymbol{x}) .$$

## 数学代写|多变量微积分代写multivariable calculus代考|Method of least squares

1801 年，天文学界因一颗新的小行星谷神星的发现而兴奋不已，它在夜空中的（粗略）位置在它从视野中消失之前曾被记录过几次。年轻的卡尔·弗里德里希·高斯[19,20]他使用了一种方法——他还是学生时就想出的方法——绘制了行星的轨道，他能够告诉天文学家在天空中的哪个位置寻找它。该方法是本节的主题。

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: