“differential form”的概念、定义、翻译、参考文献-科学参考

differential form

Mathematics

A differential form of degree 0 on ℝ³, or a 0-form, is a differentiable real function f(x,y,z). A 1-form is written

$f_{1} d x + f_{2} d y + f_{3} d z$

where f₁,f₂,f₃ are differentiable real functions of x,y,z. The exterior derivative d maps 0-forms to 1-forms via the chain rule

$d f = \frac{\partial f}{\partial x} d x + \frac{\partial f}{\partial y} d y + \frac{\partial f}{\partial z} d z .$

Note how the terms here are the same as those that appear in the gradient of f. More generally, the exterior derivative takes k-forms to (k + 1)-forms, but the product used is the exterior product, so that, for example, dx dy = –dy dx and dx² = 0. In the case of applying d to a 1-form we get

$d (F_{1} d x + F_{2} d y + F_{3} d z) = (\frac{\partial F_{1}}{\partial x} d x + \frac{\partial F_{1}}{\partial y} d y + \frac{\partial F_{1}}{\partial z} d z) d x + \dots$

$= (\frac{\partial F_{3}}{\partial y} - \frac{\partial F_{2}}{\partial z}) d y d z + (\frac{\partial F_{1}}{\partial z} - \frac{\partial F_{3}}{\partial x}) d z d x + (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d x d y,$

once simplified. Note how the terms here are the same as those that appear in the curl of (F₁,F₂,F₃). A similar calculation shows that d acts like divergence when mapping 2-forms to 3-forms.

In general, d² = d∘d = 0 generalizing curl(grad) = 0 and div(curl) = 0. Differential forms generalize to ℝⁿ for all n and to manifolds where they can be expressed in terms of local coordinates. See Stokes’ Theorem (generalized form).

单词	differential form
释义	differential form Mathematics A differential form of degree 0 on ℝ³, or a 0-form, is a differentiable real function f(x,y,z). A 1-form is written $f_{1} d x + f_{2} d y + f_{3} d z$ where f₁,f₂,f₃ are differentiable real functions of x,y,z. The exterior derivative d maps 0-forms to 1-forms via the chain rule $d f = \frac{\partial f}{\partial x} d x + \frac{\partial f}{\partial y} d y + \frac{\partial f}{\partial z} d z .$ Note how the terms here are the same as those that appear in the gradient of f. More generally, the exterior derivative takes k-forms to (k + 1)-forms, but the product used is the exterior product, so that, for example, dx dy = –dy dx and dx² = 0. In the case of applying d to a 1-form we get $d (F_{1} d x + F_{2} d y + F_{3} d z) = (\frac{\partial F_{1}}{\partial x} d x + \frac{\partial F_{1}}{\partial y} d y + \frac{\partial F_{1}}{\partial z} d z) d x + \dots$ $= (\frac{\partial F_{3}}{\partial y} - \frac{\partial F_{2}}{\partial z}) d y d z + (\frac{\partial F_{1}}{\partial z} - \frac{\partial F_{3}}{\partial x}) d z d x + (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d x d y,$ once simplified. Note how the terms here are the same as those that appear in the curl of (F₁,F₂,F₃). A similar calculation shows that d acts like divergence when mapping 2-forms to 3-forms. In general, d² = d∘d = 0 generalizing curl(grad) = 0 and div(curl) = 0. Differential forms generalize to ℝⁿ for all n and to manifolds where they can be expressed in terms of local coordinates. See Stokes’ Theorem (generalized form).
随便看	Mubarak, Hosni (1928–2020) mucigel mucilage mucin mu circuit mucopolysaccharide mucoprotein mucosa mucosal-associated lymphoid tissue mucous membrane mucus MUD mud cake mudcracks mud drape mud, drilling mud filtrate mud flat mudflat mud flow mudflow mud log mudlump mud mound mudrock