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Vector and covector fields

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Given a vector space $V$ over $R$, covectors are elements of its dual space $V^*$, i.e., linear functionals $$f:V\to R.$$

An illustration of a vector in $V={\bf R}^2$ and a covector in $V^*$.

Vector and covector.png

Here a vector is just a pair of numbers, while a covector is a correspondence of each unit vector with a number.

Suppose $M=R^n$ is the space of locations and $V=R^n$ is the space of directions (over $R$).

namevector fieldcovector field
interpretationvectors parametrized by location, velocitiescovectors parametrized by location
definition$\psi_1:M\to V$$\phi^1:M\to V^*$
exponents$\psi_1(a)\in [a,V_a]=[a,[V_a^*,R]]=[a\times V_a^*,R]$$\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 problemSolve 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
interpretationtrajectory of the particlework over any path