# Derivative of functional on path space

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Let $M$ be a differential manifold and $\Omega = \Omega(M;p,q)$ be a path space on $M$ (viz, the space of piecewise smooth paths from $p$ to $q$ in $M$.
Let $W$ be an element of the tangent space at a path $\omega \in \Omega$ and let $F$ be a functional from $\Omega$ to $\R$. Then the derivative of $F$ at the point $\omega \in \Omega$ with respect to the tangent direction $W$ is defined as follows:
Let $\alpha$ be any variation whose variation vector field is $W$. The derivative is then:
$\frac{d}{du}|_{u=0} F(\alpha(u))$