Differential algebra

A differential field F is a field F 0 (rational functions over the rationals Q for example) together with a derivation map u  → ∂ u . (Here ∂ u is a new function. Sometimes the notation u  ′ is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear : partial (u + v) = partial u + partial v and satisfies the Product rule : partial(ucdot v)=partial ucdot v+ucdotpartial v,.