“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).
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