Total derivative

The total derivative of a function $𝑓 : 𝑋 \to 𝑌$ at a point $𝑥$ gives the best linear approximation of $𝑓$ in a neighbourhood of $𝑥$ . Let $𝑈 \subseteq ℝ 𝑁$ and $𝑉 \subseteq ℝ 𝑀$ be open, $𝑥 \in 𝑈$ , and $𝑓 : 𝑈 \to 𝑉$ be $𝐶 1$ differentiable. The total derivative $𝐷 𝑓 (𝑥) : ℝ 𝑁 \to ℝ 𝑀$ at $𝑥$ is the linear map defined by¹ #m/def/anal

𝐷 𝑓 (𝑥) (⃗ 𝐚) = l i m 𝑡 \to 0 𝑓 (𝑥 + 𝑡 ⃗ 𝐚) - 𝑓 (𝑥) 𝑡

For more general spaces, see Tangent map.

#state/tidy | #lang/en | #SemBr

1997. Topology from the differentiable viewpoint, p. 3 ↩

Total derivative

Footnotes