 Since "f"*["X"] = deg "f" · ["Y"], we have

for any "m"-form "ω" on "Y".If formula_23is a bounded region, formula_24 smooth, formula_25 a regular value of formula_5 and
formula_27, then the degree formula_28 is defined
by the formula
where formula_30 is the Jacobi matrix of formula_5 in formula_32.This definition of the degree may be naturally extended for non-regular values formula_25 such that formula_34 where formula_35 is a point close to formula_25.