Method 1: T_{a}(R^{n}) = {(a, x) | x ∈ R^{n}} ϕ(ax) = x − a canonical basis {ϕ^{−1}(e_{i}) | i = 1, ..., n} Method 2: Let x(t) : R → R^{n}, a C^{1} curve such that x(0) = a x(t) ∼ y(t) if x′_{i}(t) = y′_{i}(t) for t ∈ (−ϵ, ϵ) Let f([x(t)]) = x′(0) = (x′_{1}(0), ..., x′_{n}(0)) Let T_{a}(R^{n}) = {[x(t)] | x ∈ C^{1}, x(0) = a} [x(t)] + [y(t)] = f^{−1}(x′(0) + y′(0)) α[x(t)] = f^{−1}(αx′(0))

Let f : R^{n} → R and let v ∈ R^{n} such that ||v|| = 1 The directional derivative of f at a in the direction of v is D_{v}f(a) = lim_{h→0} \frac{f(a+hv)−f(a)}{h} = D[f(a + tv)]_{0} = Df_{a}v = Df_{a} · v = (\frac{∂f}{∂x_{1}} , ..., \frac{∂f}{∂x_{n}})|_{a} · v =