ODEs dump...

name	vector field	covector field
interpretation	vectors parametrized by location, velocities	covectors parametrized by location
definition	$ψ_1:M\to V$	$ϕ_1:M\to V^∗$
exponents	$ψ_1(a)\in [a,V_a]=[a,[V^∗_a,R]]=[a\times V^∗_a,R]$	$\phi_1(a)\in [a,V^∗_a]=[a,[V_a,R]]=[a\times V_a,R]$
dual definition	$\psi_1:M\times V^∗\to R$	$\phi_1:M\times V\to R$
calculus problem	Solve the ODE: find a curve $p_1$ in $M$ such that $p'_1(t)=\psi_1(p_1(t)),p_1(0)=a\in M$	Evaluate the integral for any curve $p_1$ in $M:q^1(p_1)=\int _{p_1}\Phi^1 =\int _0^1\Phi^1(p_1(t),p'_1(t))dt$
solution	$p_1\in C_1(M)$, a chain	$q_1\in C^1(M)$, a cochain
interpretation	trajectory of the particle	work over any path