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Define: grad f = ∇f the vector field given by: x7→ n X i=1 ∂f ∂xi ∂ ∂xi and div X the function x7→ n X i=1 ∂Xi ∂xi

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HOMEWORK 8 MATH-GA 2350.001 DIFFERENTIAL GEOMETRY I (due by November, 28, 2016)

1. Let U be an open in E = Rn. Let f : U → R be a smooth function, X = Pn

i=1Xi∂x∂

i ∈ X(U) be a vector field on U. Define: grad f = ∇f the vector field given by:

x7→

n

X

i=1

∂f

∂xi

∂

∂xi

and div X the function

x7→

n

X

i=1

∂Xi

∂xi. Show that

(a) LX(dx1∧. . .∧dxn) = (div X)dx1∧. . .∧dxn; (b) div(f X) =f divX +hgrad f, Xi.

2. Let A be a ring. Consider the following commutative diagram, where the maps are the maps ofA-modules:

B //

C //

D //

E //

F

B0 //C0 //D0 //E0 //F0

Show that it first, second, forth and fifth vertical maps are isomorphisms, then the third one is an isomorphism as well.

1

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