# 微积分作业代写calclulus代考| Vector fields

my-assignmentexpert™ 微积分calculus作业代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。my-assignmentexpert™， 最高质量的微积分calculus作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于economics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此微积分calculus作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

my-assignmentexpert™ 为您的留学生涯保驾护航 在经济学作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的微积分calculus代写服务。我们的专家在微积分calculus学 代写方面经验极为丰富，各种微积分calculus相关的作业也就用不着 说。

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分作业代写calclulus代考|Conservative fields

Although scalar potentials and conservative fields arise in many areas of physics, it is far from true that all vector fields are conservative. That is, it is not generally true that all vector fields can be derived from scalar fields. In the next section we will discover an important and appealing mathematical property of conservative fields. For the moment we focus on the questions of establishing whether a vector field is conservative and if so what its scalar potential is.

To answer these questions we look at the properties of the scalar potential itself. Firstly, we see that for it to be a scalar potential, $\phi: D \longrightarrow \mathbb{R}$ must be at least $C^{1}$. Consequently, being a $C^{1}$ function, the differential of $\phi$ can be derived:
\begin{aligned} \mathrm{d} \phi &=\frac{\partial \phi}{\partial x} \mathrm{~d} x+\frac{\partial \phi}{\partial y} \mathrm{~d} y+\frac{\partial \phi}{\partial z} \mathrm{~d} z \ &=f_{1} \mathrm{~d} x+f_{2} \mathrm{~d} y+f_{3} \mathrm{~d} z \end{aligned}
The replacement of $\boldsymbol{\nabla} \phi$ with $\boldsymbol{f}$ in the last equation is valid since $\boldsymbol{f}=\boldsymbol{\nabla} \phi$ by assumption. The right-hand side of this last equation is thus an

exact differential since it equals $\mathrm{d} \phi$. That is, $f$ is conservative if $f_{1} \mathrm{~d} x+$ $f_{2} \mathrm{~d} y+f_{3} \mathrm{~d} z$ is an exact differential. Moreover, if $\phi$ is a $C^{2}$ function, then (Definition 2.8, Page 83 ) we may conclude that
$$\frac{\partial^{2} \phi}{\partial x \partial y}=\frac{\partial^{2} \phi}{\partial y \partial x}, \quad \frac{\partial^{2} \phi}{\partial x \partial z}=\frac{\partial^{2} \phi}{\partial z \partial x}, \quad \frac{\partial^{2} \phi}{\partial y \partial z}=\frac{\partial^{2} \phi}{\partial z \partial y}$$
Again, making the substitution $\nabla \phi=\boldsymbol{f}$, we see that if $\phi$ is a potential to $f$, then the above equations are equivalent to:
$$\frac{\partial f_{1}}{\partial y}=\frac{\partial f_{2}}{\partial x}, \quad \frac{\partial f_{3}}{\partial x}=\frac{\partial f_{1}}{\partial z}, \quad \frac{\partial f_{3}}{\partial y}=\frac{\partial f_{2}}{\partial z} .$$
These are necessary conditions for $\boldsymbol{f}=\boldsymbol{\nabla} \phi$ to be true and thus for $\boldsymbol{f}$ to be a conservative field. That is, the components of a conservative field must satisfy these interrelations. (See also Pages $258-260$ and 288.)

## 微积分作业代写calclulus代考|Vector fields

A $3 \mathrm{D}$ vector-valued function of a $3 \mathrm{D}$ vector variable,
$$f: D_{f} \subseteq \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}, \boldsymbol{x} \longmapsto \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}),$$
has special significance in physics and engineering, and other applications in the real world. Hence, it is given a special name: a vector field. To be explicit, an arbitrary vector field has the form
\begin{aligned} \boldsymbol{f}(\boldsymbol{x}) &=\left(f_{1}(x, y, z), f_{2}(x, y, z), f_{3}(x, y, z)\right) \ &=f_{1}(x, y, z) \mathbf{i}+f_{2}(x, y, z) \mathbf{j}+f_{3}(x, y, z) \mathbf{k} \end{aligned}
where the $f_{1}, f_{2}$, and $f_{3}$ are scalar functions of the three variables $x, y, z$.
Note that the subseripts “1”, “2”, and “3”, do not here refer to partial derivatives, they refer to the components of our vector field.

Unless otherwise stated, we shall assume that the vector fields we work with have continuous partial derivatives of order $m \geq 2$. We will often refer to these as smooth and presume the component functions are $C^{2}$ or better.

## 微积分作业代写calclulus代考|Conservative fields

dφ=∂φX dX+∂φ d+∂φ d =F1 dX+F2 d+F3 d

∂2φX=∂2φX,∂2φX=∂2φX,∂2φ=∂2φ

F1∂=∂F2∂X,∂F3∂X=∂F1∂,∂F3∂=∂F2∂.

## 微积分作业代写calclulus代考|Vector fields

F:DF⊆R3⟶R3,X=F(X),

F(X)=(F1(X,,),F2(X,,),F3(X,,)) =F1(X,,)一世+F2(X,,)j+F3(X,,)