# Liouville's Theorem (Differential Equations)

## Theorem

Let $\map \Phi t$ be a solution to the matrix differential equation:

$X' = \map A t X$

with $\map A t$ continuous on the interval $I$ such that $t_0 \in I$.

Then:

$\det \map \Phi t = e^{\int_{t_0}^t \tr \map A s \rd s} \det \map \Phi {t_0}$

## Source of Name

This entry was named for Joseph Liouville.