For a set A ⊆ R^{n}(or a banach space(?)) we define
Df:A→Hom(R^{n},R^{m});
x ↦ (Df)(x),
(Df)(x):R^{n}→R^{m}; h ↦ (Df)(x)(h)
= f'(x)·h with f'(x)∈R^{m×n}
(which is the functional matrix)
df/dx = f' = ∂f/∂x = (∂f_{i}/∂x_{j})_{i,j} = | ( | ∇f_{1} | ) | = | [ | ∂f_{1}/∂x_{1}, | ∂f_{1}/∂x_{2}, , | …, | ∂f_{1}/∂x_{n} | ] |
∇f_{2} | ∂f_{2}/∂x_{1}, | ∂f_{2}/∂x_{2}, | …, | ∂f_{2}/∂x_{n} | ||||||
⋮ | ⋮ | … | ⋮ | |||||||
∇f_{m} | ∂f_{m}/∂x_{1}, | ∂f_{m}/∂x_{2}, | …, | ∂f_{m}/∂x_{n} |