微分形式

${\displaystyle \xi \otimes \eta }$

${\displaystyle \sum a_{ij}\,\mathrm {d} x_{i}\,\mathrm {d} x_{j}}$

リーマン計量は多様体上の各点での接ベクトルの大きさを定めるものであり、局所的に線素の「長さ」を定めていることになる。ガウスが曲面論で示したように、このような局所的な情報から、多様体全体の形や大きさをかなりの程度知ることができる。

${\displaystyle \sum a_{ij}\,\mathrm {d} x_{i}\wedge \mathrm {d} x_{j}}$

${\displaystyle \mathrm {d} x_{i}\wedge \mathrm {d} x_{j}=-\mathrm {d} x_{j}\wedge \mathrm {d} x_{i}}$

${\displaystyle f_{1}(p)\,{\mathrm {d} x_{1}}^{p}+f_{2}(p)\,{\mathrm {d} x_{2}}^{p}+\dotsb +f_{n}(p)\,{\mathrm {d} x_{n}}^{p}}$

${\displaystyle \xi =\sum f_{i_{1},\dotsc ,i_{k}}\mathrm {d} x_{i_{1}}\wedge \dotsb \wedge \mathrm {d} x_{i_{k}}}$

${\displaystyle \xi _{p}=\sum f_{i_{1},\dotsc ,i_{k}}(p)\,{\mathrm {d} x_{i_{1}}}^{p}\wedge \dotsb \wedge {\mathrm {d} x_{i_{k}}}^{p}}$

${\displaystyle \xi \colon \mathrm {T} _{p}^{k}(M)\to \mathbf {R} }$

${\displaystyle \xi _{P}=\phi _{1}\wedge \dotsb \wedge \phi _{P}}$

${\displaystyle \xi _{P}{\bigl (}X_{1},X_{2},\dotsc ,X_{k}{\bigr )}={\frac {1}{k!}}\det {\begin{pmatrix}\phi _{1}(X_{1})&\phi _{1}(X_{2})&\cdots &\phi _{1}(X_{k})\\\phi _{2}(X_{1})&\phi _{2}(X_{2})&\cdots &\phi _{2}(X_{k})\\\vdots &\vdots &\ddots &\vdots \\\phi _{k}(X_{1})&\phi _{k}(X_{2})&\cdots &\phi _{k}(X_{k})\end{pmatrix}}}$

正確には、この写像は k によって定義域や値域が異なる写像であり d^k のように k を明示して区別すべきであるが、特に気にせず、どれも d で表すことが多い。

${\displaystyle \mathrm {d} f={\frac {\partial f}{\partial x_{1}}}\,\mathrm {d} x_{1}+{\frac {\partial f}{\partial x_{2}}}\,\mathrm {d} x_{2}+\dotsb +{\frac {\partial f}{\partial x_{n}}}\,\mathrm {d} x_{n}}$

${\displaystyle \xi =\sum f_{i_{1},\dotsc ,i_{k}}\mathrm {d} x_{i_{1}}\wedge \dotsb \wedge \mathrm {d} x_{i_{k}}}$

${\displaystyle \mathrm {d} \xi =\sum \mathrm {d} f_{i_{1},\dotsc ,i_{k}}\wedge \mathrm {d} x_{i_{1}}\wedge \dotsb \wedge \mathrm {d} x_{i_{k}}}$

${\displaystyle \mathrm {d} (\mathrm {d} \xi )=0}$

${\displaystyle \mathrm {d} x_{a}\wedge \mathrm {d} x_{b}=-\mathrm {d} x_{b}\wedge \mathrm {d} x_{a}}$ （交代性）

${\displaystyle \mathrm {d} x_{a}\wedge \mathrm {d} x_{a}=0}$

${\displaystyle \mathrm {d} x_{i_{1}}\wedge \dotsb \wedge \mathrm {d} x_{i_{k}}=\operatorname {sgn} (\sigma )\,\mathrm {d} x_{\sigma (i_{1})}\wedge \dotsb \wedge \mathrm {d} x_{\sigma (i_{k})}}$

i₁, …, i_k を並べ替えたときに、それが奇置換なら符号は負になるということである。

${\displaystyle \mathrm {d} x_{1}\wedge \mathrm {d} x_{3}\wedge \mathrm {d} x_{4}\wedge \mathrm {d} x_{1}=0}$

${\displaystyle f\,\mathrm {d} x_{a}\wedge \mathrm {d} x_{b}=(f\,\mathrm {d} x_{a})\wedge \mathrm {d} x_{b}=\mathrm {d} x_{a}\wedge (f\,\mathrm {d} x_{b})}$

${\displaystyle \xi \wedge \eta =(-1)^{kl}\eta \wedge \xi }$

${\displaystyle \xi \wedge \xi =-\xi \wedge \xi }$

${\displaystyle \xi \wedge \xi =0}$

(f₁ dx₁ + f₂ dx₂) ∧ (f₃ dx₃ + f₄ dx₄) = f₁ f₃ dx₁ ∧ dx₃ + f₁ f₄ dx₁ ∧ dx₄ + f₂ f₃ dx₂ ∧ dx₃ + f₂ f₄ dx₂ ∧ dx₄

${\displaystyle {\begin{aligned}y_{1}&=y_{1}(x_{1},x_{2})\\y_{2}&=y_{2}(x_{1},x_{2})\end{aligned}}}$

${\displaystyle {\begin{aligned}\mathrm {d} y_{1}\wedge \mathrm {d} y_{2}&=\left({\frac {\partial y_{1}}{\partial x_{1}}}\,\mathrm {d} x_{1}+{\frac {\partial y_{1}}{\partial x_{2}}}\,\mathrm {d} x_{2}\right)\wedge \left({\frac {\partial y_{2}}{\partial x_{1}}}\,\mathrm {d} x_{1}+{\frac {\partial y_{2}}{\partial x_{2}}}\,\mathrm {d} x_{2}\right)\\&={\frac {\partial y_{1}}{\partial x_{1}}}{\frac {\partial y_{2}}{\partial x_{1}}}\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{1}+{\frac {\partial y_{1}}{\partial x_{1}}}{\frac {\partial y_{2}}{\partial x_{2}}}\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\\&\qquad {}+{\frac {\partial y_{1}}{\partial x_{2}}}{\frac {\partial y_{2}}{\partial x_{1}}}\,\mathrm {d} x_{2}\wedge \mathrm {d} x_{1}+{\frac {\partial y_{1}}{\partial x_{2}}}{\frac {\partial y_{2}}{\partial x_{2}}}\,\mathrm {d} x_{2}\wedge \mathrm {d} x_{2}\\&={\frac {\partial y_{1}}{\partial x_{1}}}{\frac {\partial y_{2}}{\partial x_{2}}}\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}+{\frac {\partial y_{1}}{\partial x_{2}}}{\frac {\partial y_{2}}{\partial x_{1}}}\,\mathrm {d} x_{2}\wedge \mathrm {d} x_{1}\\&=\left({\frac {\partial y_{1}}{\partial x_{1}}}{\frac {\partial y_{2}}{\partial x_{2}}}-{\frac {\partial y_{1}}{\partial x_{2}}}{\frac {\partial y_{2}}{\partial x_{1}}}\right)\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\\&={\frac {\partial (y_{1},y_{2})}{\partial (x_{1},x_{2})}}\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\end{aligned}}}$

${\displaystyle \iint f(y_{1},y_{2})\,\mathrm {d} y_{1}\mathrm {d} y_{2}=\iint f{\bigl (}y_{1}(x_{1},x_{2}),y_{2}(x_{1},x_{2}){\bigr )}\left|{\frac {\partial (y_{1},y_{2})}{\partial (x_{1},x_{2})}}\right|\mathrm {d} x_{1}\mathrm {d} x_{2}}$

${\displaystyle y_{m}=y_{m}(x_{1},x_{2},\dotsc ,x_{n})\quad 1\leq m\leq n}$

${\displaystyle \mathrm {d} y_{i_{1}}\wedge \dotsb \wedge \mathrm {d} y_{i_{k}}=\sum _{j_{1}<\dotsb <j_{k}}{\frac {\partial {\bigl (}y_{i_{1}},y_{i_{2}},\dotsc ,y_{i_{k}}{\bigr )}}{\partial {\bigl (}x_{j_{1}},x_{j_{2}},\dotsc ,x_{j_{k}}{\bigr )}}}\mathrm {d} x_{j_{1}}\wedge \dotsb \wedge \mathrm {d} x_{j_{k}}}$

∑ が付くのは k < n の時を含めた一般の式だからである。

${\displaystyle \xi =f\,\mathrm {d} x_{1}\wedge \dotsb \wedge \mathrm {d} x_{k}}$

${\displaystyle \int _{D}\xi =\int _{D}f\,\mathrm {d} x_{1}\dotsm \mathrm {d} x_{k}}$

通常は積分 ${\textstyle \int f(x)\,\mathrm {d} x}$ において、∫ と dx は一対の記号であり、別々に用いることはできないが、微分形式としての意味を与えたことによって dx は一つの記号として意味を持ったことになる。

${\displaystyle {\bigl (}\phi _{1}\circ {\phi _{2}}^{-1}{\bigr )}\colon \phi _{2}(U_{1}\cap U_{2})\to \phi _{1}(U_{1}\cap U_{2})}$

${\displaystyle {\begin{aligned}\phi _{1}&=(x_{1},\dotsc ,x_{n})\\\phi _{2}&=(y_{1},\dotsc ,y_{n})\end{aligned}}}$

${\displaystyle f\colon M\to N}$

${\displaystyle {\begin{aligned}&f_{p}^{*}\colon \mathrm {T} _{q}^{*}(N)\to \mathrm {T} _{p}^{*}(M)\\&f^{*}(\xi _{q})=\xi _{q}\circ \mathrm {d} f_{p}\end{aligned}}}$

M と N は次元が異なってもよい。

${\displaystyle \int _{M}\xi =\sum _{k}\int _{U_{k}}f_{k}\xi _{k}}$

多様体が局所的に Rⁿ と見なせることから、局所的に計算した積分を足し合わせようという定義であり、開被覆で重なっている部分については 1 の分割により、重なっているそれぞれの座標近傍系に積分を割り振って計算しようということである。

${\displaystyle \mathrm {d} \xi =0}$

${\displaystyle \xi =\mathrm {d} \omega }$