概述
Maple 11 引入了 Physics
函数包,经过多年的持续开发,Maple的计算物理功能已经非常强大。Maple
允许你研究和处理计算物理领域中的广泛问题,包括经典力学、量子力学、相对论理论。同时它也提供研究生水平的场论使用资源。
![](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_9.gif)
• Physics package提供计算物理中计算对象的表示和相关的操作,包括时空矩阵
,Kronecker
和 Levi-Civita
对称和反对称符号,Pauli
和 Dirac
矩阵,微分算子
,以及d'Alembertian时空坐标上的微分算子 ☐,n 维 Dirac 函数
,量子算子,交换子和交换子代数,等等...
• Physics package扩展了标准的计算领域,提供了关于反交换和交换变量和函数的操作,以及相关的乘积和幂次操作;时空的张量指数,
spinor和/或gauge类型,泛函微分,关于反交换变量的微分,张量表达式的微分和简化使用爱因斯坦求和约定。通过这种方式,用户可以利用Maple强大的计算引擎,相比传统使用纸笔计算的方式,更直观和方便。
• 作为计算领域的延伸,该函数包包含了一个Vectors子函数包,用于实现抽象向量微积分。该函数包提供非投影三维向量的表示,非投影倒三角微分算子、梯度、散度、旋度、拉普拉斯算子的惰性和活动表示对象,以及笛卡尔、柱面、球面向量基下的投影三维向量的代数表示(非矩阵)。然后可以使用无坐标向量公式完成计算任务,探索其中向量和向量操作的无坐标属性,使用与教科书中相同的符号输入和操作向量表达式。
• 计算中的所有约定可以通过一个简单灵活的交互式助手设置。为了完成该计算领域,需要建立约定区别交换、反交换和非换变量、三维向量、张量等不同的对象。当用户在Maple工作表中加载Physics函数包时,会调入默认的约定设置,用户也可以使用设置助手修改这些约定。
• 教科书式的数学符号:反交换和非交换变量显示为不同的颜色,非投影向量和单位向量分别显示为箭头和在顶部显示符号、向量微分算子(倒三角算子)和拉普拉斯算子分别显示为 ∇ 和 ∆、Bras〈ψ⎢和
Kets⎢ψ〉等显示为与教科书相同的格式。
• 为每一个
Physics 命令提供大量的示例和说明,
提供示例说明如何使用函数包中的命令解决解析几何、力学、电动力学、量子力学中的问题。
• 微分几何函数包提供了完整的计算工具处理高级广义相对论。Maple 15新增加了十七个新的命令。
• Maple 在求常微分方程和偏微分方程符号解领域处于世界领先地位,包括物理中的许多领域。Maple 15提供了新的算法进一步增强领先地位。
• 特殊函数,用于表示计算物理的解,也是 Maple 的一个强项,同样在 Maple 15中得到增强。一组新的特殊函数,Bell 多项式,已经被加入到Maple
15中。
物理计算示例
力学:拉格朗日单摆
问题
求平面单摆的拉格朗日方程,端点处的质量为
m,假设条件为:
a) 以恒定频率
沿圆周均匀移动移动。
b) 单摆相对于
在平面上水面振荡。
解
a) 问题的原理图如下:
拉格朗日方程定义为:
>
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![with(Physics[Vectors]); -1](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_13.gif) ![Setup(mathematicalnotation = true)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_14.gif) |
![[mathematicalnotation = true]](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_15.gif) |
(2.1.1) |
>
|
![L := `+`(T, `-`(U))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_16.gif) |
![Typesetting:-mprintslash([L := `+`(T, `-`(U))], [`+`(T, `-`(U))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_17.gif) |
(2.1.2) |
其中 T 和 U 分别是系统的动能和势能,这里主要由质量点 m 产生。势能 U 是重力势能。
>
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![U := `+`(`-`(`*`(m, `*`(g, `*`(y)))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_18.gif) |
![Typesetting:-mprintslash([U := `+`(`-`(`*`(m, `*`(g, `*`(y)))))], [`+`(`-`(`*`(m, `*`(g, `*`(y)))))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_19.gif) |
(2.1.3) |
其中 g 是 重力常数
,我们选择沿着
y 轴方向,因此重力为
。动能为:
>
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![T := `+`(`*`(`/`(1, 2), `*`(m, `*`(Typesetting:-delayDotProduct(v_, v_)))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_21.gif) |
![Typesetting:-mprintslash([T := `+`(`*`(`/`(1, 2), `*`(m, `*`(`^`(Physics:-Vectors:-Norm(v_), 2)))))], [`+`(`*`(`/`(1, 2), `*`(m, `*`(`^`(Physics:-Vectors:-Norm(v_), 2)))))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_22.gif) |
(2.1.4) |
为了计算速度,单摆质量点的位置向量
为:
>
|
![r_ := `+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_24.gif) |
![Typesetting:-mprintslash([r_ := `+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)))], [`+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_25.gif) |
(2.1.5) |
选择水平 x
轴和参考坐标系的原点(圆圈的中心位置),得到 x 和 y 的坐标:
>
|
![parametric_equations := [x = `+`(`*`(a, `*`(cos(`*`(omega, `*`(t))))), `*`(l, `*`(sin(phi(t))))), y = `+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t)))))), `*`(l, `*`(cos(phi(t)))))]](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_26.gif) |
![Typesetting:-mprintslash([parametric_equations := [x = `+`(`*`(a, `*`(cos(`*`(omega, `*`(t))))), `*`(l, `*`(sin(phi(t))))), y = `+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t)))))), `*`(l, `*`(cos(phi(t)))))...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_27.gif) |
(2.1.6) |
>
|
![r_ := .(r_, parametric_equations)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_28.gif) |
![Typesetting:-mprintslash([r_ := `+`(`*`(`+`(`*`(a, `*`(cos(`*`(omega, `*`(t))))), `*`(l, `*`(sin(phi(t))))), `*`(_i)), `*`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t)))))), `*`(l, `*`(cos(phi(t))))), `*`...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_29.gif) |
(2.1.7) |
>
|
![v_ := diff(r_, t)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_30.gif) |
![Typesetting:-mprintslash([v_ := `+`(`*`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t))), `*`(omega)))), `*`(l, `*`(cos(phi(t)), `*`(diff(phi(t), t))))), `*`(_i)), `*`(`+`(`-`(`*`(a, `*`(cos(`*`(omega, `*`(...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_31.gif) |
(2.1.8) |
>
|
![T](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_32.gif) |
![Typesetting:-mprintslash([`+`(`*`(`/`(1, 2), `*`(m, `*`(`+`(`*`(`^`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t))), `*`(omega)))), `*`(l, `*`(cos(phi(t)), `*`(diff(phi(t), t))))), 2)), `*`(`^`(`+`(`-`(`*`...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_33.gif) |
(2.1.9) |
该表达式含有三角函数的乘积,所以可以使用Maple中的三角简化技术简化方程:
>
|
![combine(T, trig)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_34.gif) |
![Typesetting:-mprintslash([`+`(`*`(`/`(1, 2), `*`(m, `*`(`^`(a, 2), `*`(`^`(omega, 2))))), `-`(`*`(m, `*`(a, `*`(omega, `*`(l, `*`(diff(phi(t), t), `*`(sin(`+`(`*`(omega, `*`(t)), `-`(phi(t))))))))))),...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_35.gif) |
(2.1.10) |
对于重力势能,表示为质量点
m 的参数方程的形式。得到:
>
|
![U := .(U, parametric_equations)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_36.gif) |
![Typesetting:-mprintslash([U := `+`(`-`(`*`(m, `*`(g, `*`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t)))))), `*`(l, `*`(cos(phi(t))))))))))], [`+`(`-`(`*`(m, `*`(g, `*`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_37.gif) |
(2.1.11) |
从而得到期望的拉格朗日方程:
>
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![L := combine(L, trig); 1](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_38.gif) ![](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_39.gif) |
![Typesetting:-mprintslash([L := `+`(`*`(`/`(1, 2), `*`(m, `*`(`^`(a, 2), `*`(`^`(omega, 2))))), `-`(`*`(m, `*`(a, `*`(omega, `*`(l, `*`(diff(phi(t), t), `*`(sin(`+`(`*`(omega, `*`(t)), `-`(phi(t)))))))...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_40.gif) |
(2.1.12) |
考虑到拉格朗日系统的定义在建立关于时间
t 的微分之上,因此我们可以消除其中的两项
和
,从而得到:
>
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![select(has, L, [`*`(`^`(omega, 2)), sin(`*`(omega, `*`(t)))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_43.gif) |
![`+`(`*`(`/`(1, 2), `*`(m, `*`(`^`(a, 2), `*`(`^`(omega, 2))))), `-`(`*`(m, `*`(g, `*`(a, `*`(sin(`*`(omega, `*`(t)))))))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_44.gif) |
(2.1.13) |
>
|
![L := `+`(L, `+`(`-`(`*`(`/`(1, 2), `*`(m, `*`(`^`(a, 2), `*`(`^`(omega, 2)))))), `*`(m, `*`(g, `*`(a, `*`(sin(`*`(omega, `*`(t)))))))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_45.gif) |
![Typesetting:-mprintslash([L := `+`(`-`(`*`(m, `*`(a, `*`(omega, `*`(l, `*`(diff(phi(t), t), `*`(sin(`+`(`*`(omega, `*`(t)), `-`(phi(t))))))))))), `*`(`/`(1, 2), `*`(m, `*`(`^`(l, 2), `*`(`^`(diff(phi(...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_46.gif) |
(2.1.14) |
__________________________________________________________
b) 步骤与 a
部分相同:
>
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![L := `+`(T, `-`(U))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_47.gif) |
![Typesetting:-mprintslash([L := `+`(`*`(`/`(1, 2), `*`(m, `*`(`+`(`*`(`^`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t))), `*`(omega)))), `*`(l, `*`(cos(phi(t)), `*`(diff(phi(t), t))))), 2)), `*`(`^`(`+`(`-...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_48.gif)
![Typesetting:-mprintslash([L := `+`(`*`(`/`(1, 2), `*`(m, `*`(`+`(`*`(`^`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t))), `*`(omega)))), `*`(l, `*`(cos(phi(t)), `*`(diff(phi(t), t))))), 2)), `*`(`^`(`+`(`-...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_49.gif) |
(2.1.15) |
>
|
![U := `+`(`-`(`*`(m, `*`(g, `*`(y)))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_50.gif) |
![Typesetting:-mprintslash([U := `+`(`-`(`*`(m, `*`(g, `*`(y)))))], [`+`(`-`(`*`(m, `*`(g, `*`(y)))))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_51.gif) |
(2.1.16) |
>
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![T := `+`(`*`(`/`(1, 2), `*`(m, `*`(Typesetting:-delayDotProduct(v_, v_)))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_52.gif) |
![Typesetting:-mprintslash([T := `+`(`*`(`/`(1, 2), `*`(m, `*`(`+`(`*`(`^`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t))), `*`(omega)))), `*`(l, `*`(cos(phi(t)), `*`(diff(phi(t), t))))), 2)), `*`(`^`(`+`(`-...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_53.gif) |
(2.1.17) |
>
|
![r_ := `+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_54.gif) |
![Typesetting:-mprintslash([r_ := `+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)))], [`+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_55.gif) |
(2.1.18) |
现在,相对于 a 部分,唯一的不同是表达式使用
y 坐标,得到:
>
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![y = `+`(1, `*`(l, `*`(cos(phi(t)))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_56.gif) |
![y = `+`(1, `*`(l, `*`(cos(phi(t)))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_57.gif) |
(2.1.19) |
这种情况下的参数化方程是:
>
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![parametric_equations := [x = `+`(`*`(a, `*`(cos(`*`(omega, `*`(t))))), `*`(l, `*`(sin(phi(t))))), y = `+`(1, `*`(l, `*`(cos(phi(t)))))]](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_58.gif) |
![Typesetting:-mprintslash([parametric_equations := [x = `+`(`*`(a, `*`(cos(`*`(omega, `*`(t))))), `*`(l, `*`(sin(phi(t))))), y = `+`(1, `*`(l, `*`(cos(phi(t)))))]], [[x = `+`(`*`(a, `*`(cos(`*`(omega, ...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_59.gif) |
(2.1.20) |
>
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![r_ := .(r_, parametric_equations)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_60.gif) |
![Typesetting:-mprintslash([r_ := `+`(`*`(`+`(`*`(a, `*`(cos(`*`(omega, `*`(t))))), `*`(l, `*`(sin(phi(t))))), `*`(_i)), `*`(`+`(1, `*`(l, `*`(cos(phi(t))))), `*`(_j)))], [`+`(`*`(`+`(`*`(a, `*`(cos(`*`...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_61.gif) |
(2.1.21) |
>
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![v_ := diff(r_, t)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_62.gif) |
![Typesetting:-mprintslash([v_ := `+`(`*`(`+`(`-`(`*`(a, `*`(sin(`*`(omega, `*`(t))), `*`(omega)))), `*`(l, `*`(cos(phi(t)), `*`(diff(phi(t), t))))), `*`(_i)), `-`(`*`(_j, `*`(l, `*`(sin(phi(t)), `*`(di...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_63.gif) |
(2.1.22) |
对于重力势能,表示为质量点
m 的参数化方程形式,得到:
>
|
![U := .(U, parametric_equations)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_64.gif) |
![Typesetting:-mprintslash([U := `+`(`-`(`*`(m, `*`(g, `*`(`+`(1, `*`(l, `*`(cos(phi(t))))))))))], [`+`(`-`(`*`(m, `*`(g, `*`(`+`(1, `*`(l, `*`(cos(phi(t))))))))))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_65.gif) |
(2.1.23) |
从而得到拉格朗日方程:
>
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![L](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_66.gif) |
![Typesetting:-mprintslash([`+`(`*`(`/`(1, 2), `*`(m, `*`(`^`(a, 2), `*`(`^`(omega, 2))))), `-`(`*`(m, `*`(a, `*`(omega, `*`(l, `*`(diff(phi(t), t), `*`(sin(`+`(`*`(omega, `*`(t)), `-`(phi(t))))))))))),...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_67.gif) |
(2.1.24) |
>
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![L := combine(L, trig)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_68.gif) |
![Typesetting:-mprintslash([L := `+`(`*`(`/`(1, 2), `*`(m, `*`(`^`(a, 2), `*`(`^`(omega, 2))))), `-`(`*`(m, `*`(a, `*`(omega, `*`(l, `*`(diff(phi(t), t), `*`(sin(`+`(`*`(omega, `*`(t)), `-`(phi(t)))))))...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_69.gif) |
(2.1.25) |
获取 L 中的不可微分项:
>
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![select(has, L, [`*`(`^`(omega, 2)), cos(`+`(`*`(2, `*`(omega, `*`(t)))))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_70.gif) |
![`+`(`*`(`/`(1, 2), `*`(m, `*`(`^`(a, 2), `*`(`^`(omega, 2))))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_71.gif) |
(2.1.26) |
从而得到拉格朗日方程:
>
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![L := `+`(L, `-`(`*`(`/`(1, 2), `*`(m, `*`(`^`(a, 2), `*`(`^`(omega, 2)))))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_72.gif) |
![Typesetting:-mprintslash([L := `+`(`-`(`*`(m, `*`(a, `*`(omega, `*`(l, `*`(diff(phi(t), t), `*`(sin(`+`(`*`(omega, `*`(t)), `-`(phi(t))))))))))), `*`(`/`(1, 2), `*`(m, `*`(`^`(l, 2), `*`(`^`(diff(phi(...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_73.gif) |
(2.1.27) |
电动力学:旋转带电圆盘的磁场
问题
圆盘的半径为 a,均匀带电电荷的表面密度是
以恒定角速度
围绕轴线旋转,其中
是圆柱坐标(极角)。计算圆盘轴上的磁场。
解
磁场
的表达式,依赖于电荷的电流
:
这里
是空间中任一点的位置向量,
是存在电流的任一点的位置向量,在这种情况下圆盘直径是 a,
是面积单元。
表示积分域,上面的表达式是一个曲面积分。
>
|
![restart; -1; with(Physics[Vectors]); -1](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_85.gif) ![Setup(mathematicalnotation = true)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_86.gif) |
![[mathematicalnotation = true]](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_87.gif) |
(2.2.1) |
的表达式可以输入为圆柱坐标系下的二重积分 (
); 该坐标系下圆盘的面积表示为
, 其中
变换范围是 0 到 a,
的范围是 0 到
。
>
|
![H_ := Int(Int(`/`(`*`(`&x`(J_, `+`(r_, `-`(R_))), `*`(rho)), `*`(c, `*`(`^`(Norm(`+`(r_, `-`(R_))), 3)))), rho = 0 .. a), phi = 0 .. `+`(`*`(2, `*`(Pi))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_94.gif) |
![Typesetting:-mprintslash([H_ := Int(Int(`/`(`*`(Physics:-Vectors:-`&x`(J_, `+`(r_, `-`(R_))), `*`(rho)), `*`(c, `*`(`^`(Physics:-Vectors:-Norm(`+`(r_, `-`(R_))), 3)))), rho = 0 .. a), phi = 0 .. `+`(`...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_95.gif) |
(2.2.2) |
我们选择与前一个问题相同的参考系统,原点在圆盘的中心,z 轴的方向垂直于圆盘。z
轴上一点的位置向量是:
>
|
![r_ := `*`(z, `*`(_k))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_96.gif) |
![Typesetting:-mprintslash([r_ := `*`(z, `*`(_k))], [`*`(z, `*`(_k))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_97.gif) |
(2.2.3) |
圆盘上一点的位置向量是:
>
|
![R_ := `*`(rho, `*`(_rho))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_98.gif) |
![Typesetting:-mprintslash([R_ := `*`(rho, `*`(_rho))], [`*`(rho, `*`(_rho))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_99.gif) |
(2.2.4) |
根据定义,一点
上的电流
等于电荷密度乘以电荷速度,也就是:
>
|
![J_ := `*`(sigma, `*`(V_))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_102.gif) |
![Typesetting:-mprintslash([J_ := `*`(sigma, `*`(V_))], [`*`(sigma, `*`(V_))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_103.gif) |
(2.2.5) |
最后,圆盘上一点
上的速度
可以通过计算
相对于时间
t 的导数得到,同时我们需要考虑单位向量
随时间的变化因素,这是因为它依赖于角度
,圆盘是旋转的。
对
的导数计算有两种不同的方法。一种方法是改变
从圆柱坐标系到笛卡尔坐标系的投影转换,明确
对
的依赖关系。
>
|
![ChangeBasis(R_, 1)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_113.gif) |
![`+`(`*`(rho, `*`(cos(phi), `*`(_i))), `*`(rho, `*`(sin(phi), `*`(_j))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_114.gif) |
(2.2.6) |
现在让
依赖于 时间 t, 然后求微分。
>
|
![subs(phi = phi(t), `+`(`*`(rho, `*`(cos(phi), `*`(_i))), `*`(rho, `*`(sin(phi), `*`(_j)))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_116.gif) |
![`+`(`*`(rho, `*`(cos(phi(t)), `*`(_i))), `*`(rho, `*`(sin(phi(t)), `*`(_j))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_117.gif) |
(2.2.7) |
>
|
![diff(`+`(`*`(rho, `*`(cos(phi(t)), `*`(_i))), `*`(rho, `*`(sin(phi(t)), `*`(_j)))), t)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_118.gif) |
![Typesetting:-mprintslash([`+`(`-`(`*`(rho, `*`(sin(phi(t)), `*`(diff(phi(t), t), `*`(_i))))), `*`(rho, `*`(cos(phi(t)), `*`(diff(phi(t), t), `*`(_j)))))], [`+`(`-`(`*`(rho, `*`(sin(phi(t)), `*`(diff(p...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_119.gif) |
(2.2.8) |
让
, 然后移除
对 t 的显式依赖关系,得到
的表达式。
>
|
![factor(subs([diff(phi(t), t) = omega, phi(t) = phi], `+`(`-`(`*`(rho, `*`(sin(phi(t)), `*`(diff(phi(t), t), `*`(_i))))), `*`(rho, `*`(cos(phi(t)), `*`(diff(phi(t), t), `*`(_j)))))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_123.gif) |
![`*`(rho, `*`(omega, `*`(`+`(`-`(`*`(sin(phi), `*`(_i))), `*`(cos(phi), `*`(_j))))))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_124.gif) |
(2.2.9) |
或者使用更简单的方法,知道
因此
通过
依赖于时间,用户可以计算
= ![](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_130.gif)
。为此目的,使用 VectorDiff 命令,自动考虑
依赖于
。
>
|
![V_ := `*`(omega, `*`(VectorDiff(R_, phi)))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_134.gif) |
![Typesetting:-mprintslash([V_ := `*`(omega, `*`(rho, `*`(_phi)))], [`*`(omega, `*`(rho, `*`(_phi)))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_135.gif) |
(2.2.10) |
此时,我们已经定义在选定坐标系上的所有量,表示为恒定角速度
,圆盘半径为 a。磁场的表达式如下:
>
|
![H_](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_137.gif) |
![Typesetting:-mprintslash([Int(Int(`/`(`*`(`+`(`*`(sigma, `*`(omega, `*`(rho, `*`(z, `*`(_rho))))), `*`(sigma, `*`(omega, `*`(`^`(rho, 2), `*`(_k))))), `*`(rho)), `*`(c, `*`(`^`(`+`(`*`(`^`(rho, 2)), `...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_138.gif) |
(2.2.11) |
但是为了完成积分,我们仍需要将
表示为积分变量
的函数。出于该目的,需要将
和
改变为笛卡尔基下的形式。
>
|
![R_ := ChangeBasis(R_, 1)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_143.gif) |
![Typesetting:-mprintslash([R_ := `+`(`*`(rho, `*`(cos(phi), `*`(_i))), `*`(rho, `*`(sin(phi), `*`(_j))))], [`+`(`*`(rho, `*`(cos(phi), `*`(_i))), `*`(rho, `*`(sin(phi), `*`(_j))))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_144.gif) |
(2.2.12) |
>
|
![V_ := ChangeBasis(V_, 1)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_145.gif) |
![Typesetting:-mprintslash([V_ := `+`(`-`(`*`(omega, `*`(rho, `*`(sin(phi), `*`(_i))))), `*`(omega, `*`(rho, `*`(cos(phi), `*`(_j)))))], [`+`(`-`(`*`(omega, `*`(rho, `*`(sin(phi), `*`(_i))))), `*`(omega...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_146.gif) |
(2.2.13) |
改变后的
如下:
>
|
![H_](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_148.gif) |
![Typesetting:-mprintslash([Int(Int(`/`(`*`(`+`(`*`(sigma, `*`(omega, `*`(rho, `*`(cos(phi), `*`(z, `*`(_i)))))), `*`(sigma, `*`(omega, `*`(rho, `*`(sin(phi), `*`(z, `*`(_j)))))), `*`(sigma, `*`(omega, ...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_149.gif) |
(2.2.14) |
现在可以完成积分计算,得到磁场
的值。
>
|
<img border="0" alt="H_ := `assuming`([value(H_)], [`<`(0, a), ` |
![Typesetting:-mprintslash([H_ := `+`(`-`(`/`(`*`(2, `*`(omega, `*`(sigma, `*`(_k, `*`(Pi, `*`(`+`(`*`(2, `*`(`^`(`+`(`*`(`^`(z, 2)), `*`(`^`(a, 2))), `/`(1, 2)), `*`(z))), `-`(`*`(`^`(a, 2))), `-`(`*`(...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_152.gif) |
(2.2.15) |
量子力学:角动量:
和
1.考虑量子动力学中的角动量算子
,
,
, 和
。我们需要验证
的 对易(Commutator) , 使得
中的任意 components 为 0 (例如见 Chapter VI of
Cohen-Tannoudji)。出于此目的,
的三维向量量子算子可通过 Vectors 函数包构建 ( vectorpostfix identifier 是 '_'), 以及
![](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_162.gif)
,
以及
和
和它们的元素可以设置为量子算子。
想要设置
和
为量子算子,只需要设置
和
。
>
|
![Setup(quantumoperators = {L, L_, p, p_, r_, x, y, z})](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_174.gif) |
![[quantumoperators = {L, L_, p, p_, r_, x, y, z}]](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_175.gif) |
(2.3.1) |
因此对于
以及
自身表示为向量算子
和
的形式。
>
|
![LL := Typesetting:-delayDotProduct(L_, L_)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_180.gif) |
![Typesetting:-mprintslash([LL := `*`(`^`(Physics:-Vectors:-Norm(L_), 2))], [Physics:-`^`(Physics:-Vectors:-Norm(L_), 2)])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_181.gif) |
(2.3.2) |
>
|
![L_ := `&x`(r_, p_)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_182.gif) |
![Typesetting:-mprintslash([L_ := Physics:-Vectors:-`&x`(r_, p_)], [Physics:-Vectors:-`&x`(r_, p_)])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_183.gif) |
(2.3.3) |
其中,
>
|
![r_ := `+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)), `*`(z, `*`(_k)))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_184.gif) |
![Typesetting:-mprintslash([r_ := `+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)), `*`(z, `*`(_k)))], [`+`(`*`(x, `*`(_i)), `*`(y, `*`(_j)), `*`(z, `*`(_k)))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_185.gif) |
(2.3.4) |
>
|
![p_ := `+`(`*`(p[x], `*`(_i)), `*`(p[y], `*`(_j)), `*`(p[z], `*`(_k)))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_186.gif) |
![Typesetting:-mprintslash([p_ := `+`(`*`(p[x], `*`(_i)), `*`(p[y], `*`(_j)), `*`(p[z], `*`(_k)))], [`+`(`*`(p[x], `*`(_i)), `*`(p[y], `*`(_j)), `*`(p[z], `*`(_k)))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_187.gif) |
(2.3.5) |
的对易规则是关于
和
元素对易规则的子序列。这些规则可以通过使用 Setup 命令设置。这里需要输入许多交换子,一个方便的替代方法是使用索引(张量)符号(见下面的问题2)或者创建一个 Matrix 的索引过程。例如:
![Typesetting:-mprintslash([algebra := proc (i, j) options operator, arrow; %Commutator(Physics:-Vectors:-Component(r_, i), Physics:-Vectors:-Component(p_, j)) = Physics:-`*`(I, Physics:-KroneckerDelta[...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_196.gif) |
(2.3.6) |
现在可以使用 Matrix 构造器生成交换子,整个矩阵可以传递给 Setup 。
>
|
![Matrix(3, 3, algebra)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_197.gif) |
>
|
![Setup(%)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_204.gif) |
中的元素是:
>
|
![L[x] := Component(L_, 1)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_209.gif) |
![Typesetting:-mprintslash([L[x] := `+`(Physics:-`*`(y, p[z]), `-`(Physics:-`*`(z, p[y])))], [`+`(Physics:-`*`(y, p[z]), `-`(Physics:-`*`(z, p[y])))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_210.gif) |
(2.3.9) |
>
|
![L[y] := Component(L_, 2)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_211.gif) |
![Typesetting:-mprintslash([L[y] := `+`(Physics:-`*`(z, p[x]), `-`(Physics:-`*`(x, p[z])))], [`+`(Physics:-`*`(z, p[x]), `-`(Physics:-`*`(x, p[z])))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_212.gif) |
(2.3.10) |
>
|
![L[z] := Component(L_, 3)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_213.gif) |
![Typesetting:-mprintslash([L[z] := `+`(Physics:-`*`(x, p[y]), `-`(Physics:-`*`(y, p[x])))], [`+`(Physics:-`*`(x, p[y]), `-`(Physics:-`*`(y, p[x])))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_214.gif) |
(2.3.11) |
使用 expansion 展开交换子:
>
|
![%Commutator(LL, L[x]); 1](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_215.gif) |
![Typesetting:-mprintslash([Typesetting:-msub(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-msup(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mrow(Typese...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_216.gif) |
(2.3.12) |
>
|
![value(%)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_217.gif) |
>
|
![zero := expand(%)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_221.gif) |
![Typesetting:-mprintslash([zero := `+`(`-`(`*`(`+`(I), `*`(Physics:-`*`(z, p[y], y, p[y])))), `*`(I, `*`(Physics:-`*`(p[z], `*`(`^`(y, 2)), p[y]))), `-`(`*`(`+`(I), `*`(Physics:-`*`(y, p[z], p[y], y)))...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_222.gif) |
(2.3.14) |
为了验证上面的表达式确实等于0,需要考虑下面的交换子规则:
>
|
![Setup(alg)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_223.gif) |
使用 Simplify :
>
|
![Simplify(zero)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_228.gif) |
![0](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_229.gif) |
(2.3.16) |
>
|
![%Commutator(LL, Ly); 1](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_230.gif) |
![Typesetting:-mprintslash([Typesetting:-msub(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-msup(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mrow(Typese...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_231.gif) |
(2.3.17) |
>
|
![Simplify(%)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_232.gif) |
![0](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_233.gif) |
(2.3.18) |
>
|
![Simplify(Commutator(LL, Lz))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_234.gif) |
![0](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_235.gif) |
(2.3.19) |
______________________________________________________________
2. 使用张量符号表示量子算子元件
,
显示为
(请参考Chap VI
in Cohen-Tannoudji练习部分)。
设置时空张量为
3、欧几里德三维空间,因此“时空”张量实际上是三维空间张量。为了使用教科书式的符号,使用 lowercaselatin 张量索引(见帮助
Setup )。
>
|
![Setup(dimension = 3, signature = `+`, spacetimeindices = lowercaselatin, quiet)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_238.gif) |
![[dimension = 3, signature = `+`, spacetimeindices = lowercaselatin]](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_239.gif) |
(2.3.20) |
使用张量符号设置r和p的
Commutator 规则,使用 Simplify 命令应用爱因斯坦求和约定求积, Define r 和 p 为三维欧几里德空间的张量。
>
|
![Define(r, p)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_240.gif) |
![`Defined objects with tensor properties`](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_241.gif) |
|
![{p, r, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_242.gif) |
(2.3.21) |
现在可以使用张量符号设置相关的交换子规则;消除前面关于量子算子的设置和代数规则(通过使用 Setup 中的 redo
参数项消除前面的定义,这里的例子不是必须的,但某些情况下需要)。
>
|
![Setup(redo, quantumoperators = '{L, L_, p, r}', algebrarules = {%Commutator(p[i], p[j]) = 0, %Commutator(r[i], p[j]) = `*`(I, `*`(kd_[i, j])), %Commutator(r[i], r[j]) = 0})](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_243.gif)
![Setup(redo, quantumoperators = '{L, L_, p, r}', algebrarules = {%Commutator(p[i], p[j]) = 0, %Commutator(r[i], p[j]) = `*`(I, `*`(kd_[i, j])), %Commutator(r[i], r[j]) = 0})](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_244.gif) |
![Typesetting:-mprintslash([[algebrarules = {Typesetting:-msub(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_245.gif) |
(2.3.22) |
验证这些代数规则如何工作:
>
|
![%Commutator(r[m], p[n])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_246.gif) |
![Typesetting:-mprintslash([Typesetting:-msub(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_247.gif) |
(2.3.23) |
>
|
![value(%)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_248.gif) |
![`*`(I, `*`(Physics:-KroneckerDelta[m, n]))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_249.gif) |
(2.3.24) |
>
|
![Commutator(r[m], r[n])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_250.gif) |
![0](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_251.gif) |
(2.3.25) |
>
|
![Commutator(p[m], p[n])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_252.gif) |
![0](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_253.gif) |
(2.3.26) |
现在输入
, 以及
和
表示的
。对 LeviCivita 伪张量使用默认的缩写 ep_ 。
>
|
![rule := %Commutator(L[i], L[j]) = `*`(I, `*`('ep_[i, j, k]', `*`(L[k])))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_258.gif) |
![Typesetting:-mprintslash([rule := Typesetting:-msub(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_259.gif) |
(2.3.27) |
>
|
![L[i] := `*`(ep_[i, m, n], `*`(r[m], `*`(p[n])))](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_260.gif) |
![Typesetting:-mprintslash([L[i] := `*`(Physics:-`*`(r[m], p[n]), `*`(Physics:-LeviCivita[i, `~m`, `~n`]))], [`*`(Physics:-`*`(r[m], p[n]), `*`(Physics:-LeviCivita[i, `~m`, `~n`]))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_261.gif) |
(2.3.28) |
>
|
![L[j] := subs(i = j, L[i])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_262.gif) |
![Typesetting:-mprintslash([L[j] := `*`(Physics:-`*`(r[m], p[n]), `*`(Physics:-LeviCivita[j, `~m`, `~n`]))], [`*`(Physics:-`*`(r[m], p[n]), `*`(Physics:-LeviCivita[j, `~m`, `~n`]))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_263.gif) |
(2.3.29) |
>
|
![L[k] := subs(i = k, L[i])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_264.gif) |
![Typesetting:-mprintslash([L[k] := `*`(Physics:-`*`(r[m], p[n]), `*`(Physics:-LeviCivita[k, `~m`, `~n`]))], [`*`(Physics:-`*`(r[m], p[n]), `*`(Physics:-LeviCivita[k, `~m`, `~n`]))])](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_265.gif) |
(2.3.30) |
因此,
可以给定为:
>
|
![rule](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_267.gif) |
![Typesetting:-mprintslash([Typesetting:-msub(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_268.gif) |
(2.3.31) |
用户可以展开这个规则,得到实际值,然后使用 Simplify 命令,
>
|
![expand(rule)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_269.gif) |
![Typesetting:-mprintslash([`+`(`-`(`*`(Physics:-LeviCivita[i, `~m`, `~n`], `*`(Physics:-LeviCivita[j, `~a`, `~b`], `*`(`+`(`-`(`*`(`+`(I), `*`(Physics:-KroneckerDelta[b, m], `*`(Physics:-`*`(r[a], p[n]...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_270.gif) |
(2.3.32) |
>
|
![Simplify(%)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_271.gif) |
![Typesetting:-mprintslash([`+`(`*`(I, `*`(Physics:-`*`(r[i], p[j]))), `-`(`*`(`+`(I), `*`(Physics:-`*`(r[j], p[i]))))) = `+`(`*`(I, `*`(Physics:-`*`(r[i], p[j]))), `-`(`*`(`+`(I), `*`(Physics:-`*`(r[j]...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_272.gif) |
(2.3.33) |
或者使用 Simplify 规则,而不是首先展开。
>
|
![Simplify(rule)](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_273.gif) |
![Typesetting:-mprintslash([`+`(`*`(I, `*`(Physics:-`*`(r[i], p[j]))), `-`(`*`(`+`(I), `*`(Physics:-`*`(r[j], p[i]))))) = `+`(`*`(I, `*`(Physics:-`*`(r[i], p[j]))), `-`(`*`(`+`(I), `*`(Physics:-`*`(r[j]...](http://www.maplesoft.com.cn/products/maple/physics/images/physics-package-in-maple_274.gif) |
(2.3.34) |
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