10. Calculus identities#

Here some calculus identities are listed and proved, often using Cartesian coordinates and index notation. While these identities are independent on the choice of the coordinates - as calculus is - a generic set of coordinates can be used to prove them, and Cartesian coordinates are the most convenint choice.

(10.1)#Δv=(v)××v .
Proof.

Uusing the identity

εijkεilm=δjlδkmδjmδkl

and index notation for Cartesian coordinates,

(××v)i=e^i(××v)==εijkj(εklmlvm)==εkijεklmjlvm==(δilδjmδimδjl)jlvm==ijvjjjvi==e^i((v)Δv)==((v)Δv)i .