外微分可以统一矢量算符
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#1 外微分可以统一矢量算符
0- form f 是函数
df 是gradient
1- form是矢量
df 是 curl
2-form
df divergence
df 是gradient
1- form是矢量
df 是 curl
2-form
df divergence
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#10 Re: 外微分可以统一矢量算符
为什么叫exterior differential,还有exterior algebra,还真不知道。
这有个解释听起来是合理的:
According to one of the web search results1, the term “exterior” comes from the geometric property of the exterior product, which is the key operation in the exterior algebra. The exterior product of two vectors is non-zero only if the vectors are geometrically one to the exterior of the other, meaning that they do not lie in the same subspace. For example, if x, y, and z are vectors, then x ∧ y ∧ z = 0 if x lies in the subspace spanned by y and z. The exterior product can be interpreted as the area or volume of the parallelotope formed by the vectors, and it also determines the orientation of the subspace. The exterior algebra is the algebraic system that uses the exterior product to study geometric questions1: https://math.stackexchange.com/question ... -it-exteri
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#11 Re: 外微分可以统一矢量算符
有道理,外积是向外扩张。TheMatrix 写了: ↑1月 28, 2024, 6:43 pm 为什么叫exterior differential,还有exterior algebra,还真不知道。
这有个解释听起来是合理的:
According to one of the web search results1, the term “exterior” comes from the geometric property of the exterior product, which is the key operation in the exterior algebra. The exterior product of two vectors is non-zero only if the vectors are geometrically one to the exterior of the other, meaning that they do not lie in the same subspace. For example, if x, y, and z are vectors, then x ∧ y ∧ z = 0 if x lies in the subspace spanned by y and z. The exterior product can be interpreted as the area or volume of the parallelotope formed by the vectors, and it also determines the orientation of the subspace. The exterior algebra is the algebraic system that uses the exterior product to study geometric questions1: https://math.stackexchange.com/question ... -it-exteri
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#12 Re: 外微分可以统一矢量算符
TheMatrix 写了: ↑1月 28, 2024, 6:43 pm 为什么叫exterior differential,还有exterior algebra,还真不知道。
这有个解释听起来是合理的:
According to one of the web search results1, the term “exterior” comes from the geometric property of the exterior product, which is the key operation in the exterior algebra. The exterior product of two vectors is non-zero only if the vectors are geometrically one to the exterior of the other, meaning that they do not lie in the same subspace. For example, if x, y, and z are vectors, then x ∧ y ∧ z = 0 if x lies in the subspace spanned by y and z. The exterior product can be interpreted as the area or volume of the parallelotope formed by the vectors, and it also determines the orientation of the subspace. The exterior algebra is the algebraic system that uses the exterior product to study geometric questions1: https://math.stackexchange.com/question ... -it-exteri
我一直以为exterior product 是跟inner product 相对应...
正准备收杆,咬钩的大鱼忽然开口说话了:“脱钩!脱钩!”
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Amorphous 的博客 - 帖子: 4077
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#13 Re: 外微分可以统一矢量算符
正准备收杆,咬钩的大鱼忽然开口说话了:“脱钩!脱钩!”
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#15 Re: 外微分可以统一矢量算符
正准备收杆,咬钩的大鱼忽然开口说话了:“脱钩!脱钩!”
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#16 Re: 外微分可以统一矢量算符
正准备收杆,咬钩的大鱼忽然开口说话了:“脱钩!脱钩!”
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#17 Re: 外微分可以统一矢量算符
数学家喜欢这样的东西,又统一了啥啥的。
而我对这种高度抽象的东西表示不喜欢。
物理学家喜欢能表达出底层物理思维的数学方程式。
这样才能有物理直觉,才能知道哪儿可以假定是正确的,哪儿需要work了。
没有光子;也没有量子能级,量子跃迁,量子叠加,量子塌缩和量子纠缠。
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#20 Re: 外微分可以统一矢量算符
就是算符化代数化,exterior algebra应用于对微分算符的分类
泛化一点就是graded algebra
manifold上的differential form构成一个De-Rham algebra
泛化一点就是graded algebra
manifold上的differential form构成一个De-Rham algebra