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Vector and covector fields
From Mathematics Is A Science
Revision as of 16:40, 18 September 2013 by imported>WikiSysop
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 $V^*$.
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 $R$).
name | vector field | covector field |
interpretation | vectors parametrized by location, velocities | covectors 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 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 |