α = \frac{1}{r !} α_{i_{1} \dots i_{r}} d x^{i_{1}} \land \dots \land d x^{i_{r}},

d α = \frac{1}{r !} \frac{\partial α _{i_{1} \dots i_{r}}}{\partial x ^{j}} d x^{j} \land d x^{i_{1}} \land \dots \land d x^{i_{r}} .

Applying

d

again gives symmetric second partial derivatives contracted with the antisymmetric wedge

d x^{k} \land d x^{j}

, hence

d^{2} = 0

. Expanding coefficients and moving

d α

past a degree-

r

form gives the graded Leibniz rule

d (α \land β) = d α \land β + (- 1)^{r} α \land d β .

f

and smooth

F : X \to Y

, the chain rule gives

d (F^{*} f) = d (f \circ F) = F^{*} (df) .

Every differential form is locally a sum of products

f_{0} d f_{1} \land \dots \land d f_{r}

. Since pullback preserves products and wedge products, the function case and the graded Leibniz rule imply

d (F^{*} α) = F^{*} (d α)

for every form.

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.