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Differential forms: homework 1

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  • Prove the linearity of the exterior derivative, $d$.
  • Let $\Phi$ be the set of all closed intervals, $[a,b]$, in ${\bf R}$.

Suppose $f$ is a given function (fixed!), continuous. Define a function $F \colon \Phi \rightarrow {\bf R}$ by $F([a,b]) = \displaystyle\int_a^b f(x) dx$. "Turn" $F$ into a linear operator.

  • Hint 1: If $\Phi = {\bf R}^2$ show $F$ is not linear (consider $F[2a,2b] = 2F[a,b]$).
  • Hint 2: Enlarge $\Phi$ to some $\Phi^*$.

Comments: Adding apples to oranges

Comments: Linear algebra in elementary calculus

Discussion:

$[a,b] + [b,c] \sim [a,c]$ (definition)

Equivalence relation:

  • 1.) $[a,b] \sim [a,b]$
  • 2.) $[a,b] \sim [c,d]$ implies $[c,d] \sim [a,b]$ (Symmetry)
  • 3.) etc

To verify, rewrite $[a,b]+[b,c] \sim [a,d]+[d,c]$

Also this: $[a,a] + [a,c] = 0 + [a,c]$, since $[a,a]$ is a $0$-cell and $[a,c]$ is a $1$-cell.