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♥𝙖𝙙𝙙 𝙢𝙚 𝙥𝙡𝙚𝙖𝙨𝙚♥
The cross product a × b {\displaystyle a\times b} a\times b and the Lie bracket operation [] {\displaystyle [a,b]} [a,b] both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket.
In other words, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function.
The concept of a Taylor series was formulated by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
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