跳到主要内容

3 Implicit Function and Inverse Mapping Theorem

3.1 Implicit Function Theorem​

Theorem 3.1 (Implicit Function Theorem). Let F:W→RnF: W \to \mathbb{R}^n be a Ck\mathscr{C}^k differentiable mapping on an open set W⊆Rm×RnW \subseteq \mathbb{R}^m \times \mathbb{R}^n. Suppose (x0,y0)∈W(\mathbf{x}_0, \mathbf{y}_0) \in W satisfies F(x0,y0)=0F(\mathbf{x}_0, \mathbf{y}_0) = 0 and the partial derivative ∂yF(x0,y0)\partial_{\mathbf{y}} F(\mathbf{x}_0, \mathbf{y}_0) is invertible.

Then there exist a neighborhood U⊆RmU \subseteq \mathbb{R}^m of x0\mathbf{x}_0, a neighborhood V⊆RnV \subseteq \mathbb{R}^n of y0\mathbf{y}_0, and a Ck\mathscr{C}^k mapping f:U→Rnf: U \to \mathbb{R}^n such that:

(1) For all x∈U\mathbf{x} \in U,

F(x,f(x))=0,\begin{align*} F(\mathbf{x}, f(\mathbf{x})) = 0, \end{align*}

and

(2) If (x,y)∈U×V(\mathbf{x}, \mathbf{y}) \in U \times V satisfies F(x,y)=0F(\mathbf{x}, \mathbf{y}) = 0, then

y=f(x).\begin{align*} \mathbf{y} = f(\mathbf{x}). \end{align*}

3.2 Inverse Mapping Theorem​

Theorem 3.2 (Inverse Mapping Theorem). Let f:U→Rnf: U \to \mathbb{R}^n be Ck\mathscr{C}^k differentiable on an open set UU. If df(x0)\mathrm{d}f(\mathbf{x}_0) is invertible, then there exist a neighborhood VV of x0\mathbf{x}_0 and a neighborhood WW of f(x0)f(\mathbf{x}_0) such that

f:V→W\begin{align*} f: V \to W \end{align*}

has a Ck\mathscr{C}^k inverse mapping.

Definition. A mapping f:U→Vf: U \to V is called a Ck\mathscr{C}^k diffeomorphism if ff is a Ck\mathscr{C}^k mapping and there exists a Ck\mathscr{C}^k inverse mapping g:V→Ug: V \to U, such that

g(f(x))=x,f(g(y))=y,∀x∈U,y∈V.\begin{align*} g(f(\mathbf{x})) = \mathbf{x}, \quad f(g(\mathbf{y})) = \mathbf{y}, \quad \forall \mathbf{x} \in U, \mathbf{y} \in V. \end{align*}

Theorem 3.3. Let U⊆RnU \subseteq \mathbb{R}^n be an open set, and let f:U→Rnf: U \to \mathbb{R}^n be a Ck\mathscr{C}^k mapping. Then

V=f(U) is an open set and f:U→V is a Ck diffeomorphism\begin{align*} V = f(U) \text{ is an open set and } f: U \to V \text{ is a } \mathscr{C}^k \text{ diffeomorphism} \end{align*}

if and only if

f is injective, and for every x∈U,df(x) is an invertible linear mapping.\begin{align*} f \text{ is injective, and for every } \mathbf{x} \in U, \mathrm{d}f(\mathbf{x}) \text{ is an invertible linear mapping.} \end{align*}

3.4 Surfaces, Tangent Planes and Normal Vectors​

Surface​

Let 1≤m≤n−11 \leq m \leq n-1. We say that Σ⊂Rn\Sigma \subset \mathbb{R}^n is an mm-dimensional Ck\mathscr{C}^k surface, if for any x0∈Σ\mathbf{x}_0 \in \Sigma, Σ\Sigma is the graph of an mm-variable Ck\mathscr{C}^k mapping in a neighborhood of x0\mathbf{x}_0. That is, there exists a neighborhood UU of x0\mathbf{x}_0, a permutation σ(1),σ(2),…,σ(n)\sigma(1), \sigma(2), \dots, \sigma(n) of 1,2,…,n1, 2, \dots, n, and a Ck\mathscr{C}^k mapping gg such that for any x=(x1,x2,…,xn)∈U\mathbf{x} = (x^1, x^2, \dots, x^n) \in U,

x=(x1,x2,…,xn)∈Σ  ⟺  (xσ(m+1),…,xσ(n))=g(xσ(1),…,xσ(m)).\begin{align*} \mathbf{x} = (x^1, x^2, \dots, x^n) \in \Sigma \iff (x^{\sigma(m+1)}, \dots, x^{\sigma(n)}) = g(x^{\sigma(1)}, \dots, x^{\sigma(m)}). \end{align*}

When m=n−1m = n-1, Σ\Sigma is called a Ck\mathscr{C}^k hypersurface.

When m=1m = 1, Σ\Sigma is a Ck\mathscr{C}^k curve.

Theorem 3.4.

  1. If aa is a regular value of a Ck\mathscr{C}^k mapping F:U→RnF: U \to \mathbb{R}^n (where U⊆RmU \subseteq \mathbb{R}^m is a region), meaning that for any x∈U\mathbf{x} \in U satisfying F(x)=aF(\mathbf{x}) = a, dF(x)\mathrm{d}F(\mathbf{x}) has full row rank, then the level set of FF
Σ={x∈U∣F(x)=a}\begin{align*} \Sigma = \{\mathbf{x} \in U \mid F(\mathbf{x}) = a\} \end{align*}

is an (m−n)(m - n)-dimensional Ck\mathscr{C}^k surface in Rm\mathbb{R}^m.

  1. Let U⊆RmU \subseteq \mathbb{R}^m be an open set. A Ck\mathscr{C}^k injective mapping f:U→Rnf: U \to \mathbb{R}^n (x=f(u1,…,um)\mathbf{x} = f(u^1, \dots, u^m)) satisfies for any u∈U\mathbf{u} \in U, rank(df(u))=m\text{rank}(\mathrm{d}f(\mathbf{u})) = m, i.e., df(u)\mathrm{d}f(\mathbf{u}) has full column rank. Then
f(U)⊆Rn\begin{align*} f(U) \subseteq \mathbb{R}^n \end{align*}

is an mm-dimensional Ck\mathscr{C}^k surface in Rn\mathbb{R}^n. ff is called a Ck\mathscr{C}^k parametric representation of this surface.

Tangent Space​

Let Σ\Sigma be a Ck\mathscr{C}^k surface in Rn\mathbb{R}^n, and x0∈Σ\mathbf{x}_0 \in \Sigma. Let γ:(a,b)→Rn\gamma: (a, b) \to \mathbb{R}^n be a C1\mathscr{C}^1 curve on Σ\Sigma passing through x0\mathbf{x}_0, meaning that for any t∈(a,b)t \in (a, b), γ(t)∈Σ\gamma(t) \in \Sigma, and γ(t0)=x0\gamma(t_0) = \mathbf{x}_0. In this case, γ′(t0)∈Rn\gamma'(t_0) \in \mathbb{R}^n is called a tangent vector of Σ\Sigma at x0\mathbf{x}_0. Let Tx0ΣT_{\mathbf{x}_0}\Sigma denote the set consisting of all tangent vectors of Σ\Sigma at x0\mathbf{x}_0, which is called the tangent space of Σ\Sigma at x0\mathbf{x}_0.

The tangent plane of the surface Σ\Sigma at x0∈Σ\mathbf{x}_0 \in \Sigma is

x0+Tx0Σ.\begin{align*} \mathbf{x}_0 + T_{\mathbf{x}_0}\Sigma. \end{align*}

The normal space of the surface Σ\Sigma at x0∈Σ\mathbf{x}_0 \in \Sigma is

(Tx0Σ)⊥,\begin{align*} (T_{\mathbf{x}_0}\Sigma)^\perp, \end{align*}

where the vectors within this set are called normal vectors of Σ\Sigma at x0∈Σ\mathbf{x}_0 \in \Sigma.

The normal line / normal plane of the surface Σ\Sigma at x0∈Σ\mathbf{x}_0 \in \Sigma is

x0+(Tx0Σ)⊥.\begin{align*} \mathbf{x}_0 + (T_{\mathbf{x}_0}\Sigma)^\perp. \end{align*}

Theorem 3.5. Let Σ⊆Rn\Sigma \subseteq \mathbb{R}^n be an mm-dimensional Ck\mathscr{C}^k surface. Then for any x0∈Σ\mathbf{x}_0 \in \Sigma, the tangent space Tx0ΣT_{\mathbf{x}_0}\Sigma is an mm-dimensional linear subspace of Rn\mathbb{R}^n, and

  1. If Σ\Sigma is a regular level set of a Ck\mathscr{C}^k mapping FF, Σ={x∈Rn∣F(x)=a}\Sigma = \{\mathbf{x} \in \mathbb{R}^n \mid F(\mathbf{x}) = a\}, then
Tx0Σ=Ker dF(x0)={v∈Rn∣dF(x0)(v)=0}\begin{align*} T_{\mathbf{x}_0}\Sigma &= \text{Ker} \, \mathrm{d}F(\mathbf{x}_0) \\ &= \{\mathbf{v} \in \mathbb{R}^n \mid \mathrm{d}F(\mathbf{x}_0)(\mathbf{v}) = \mathbf{0}\} \end{align*}

Equation of the tangent plane:

dF(x0)(x−x0)=0\begin{align*} \mathrm{d}F(\mathbf{x}_0)(\mathbf{x} - \mathbf{x}_0) = \mathbf{0} \end{align*}

The gradient vector ∇F(x0)\nabla F(\mathbf{x}_0) is a normal vector of the surface Σ\Sigma at x0∈Σ\mathbf{x}_0 \in \Sigma.

  1. If f:U→Rnf: U \to \mathbb{R}^n is a Ck\mathscr{C}^k parametric representation of Σ\Sigma, then for x0=f(u0)\mathbf{x}_0 = f(\mathbf{u}_0),
Tx0Σ=Range df(u0)={df(u0)(v)∈Rn∣v∈Rm}\begin{align*} T_{\mathbf{x}_0}\Sigma &= \text{Range} \, \mathrm{d}f(\mathbf{u}_0) \\ &= \{\mathrm{d}f(\mathbf{u}_0)(\mathbf{v}) \in \mathbb{R}^n \mid \mathbf{v} \in \mathbb{R}^m\} \end{align*}

Equation of the tangent plane:

x=x0+df(u0)(v),v∈Rm\begin{align*} \mathbf{x} = \mathbf{x}_0 + \mathrm{d}f(\mathbf{u}_0)(\mathbf{v}), \quad \mathbf{v} \in \mathbb{R}^m \end{align*}

The normal space of the surface Σ\Sigma at x0∈Σ\mathbf{x}_0 \in \Sigma is Ker((df(u0))T)\text{Ker}((\mathrm{d}f(\mathbf{u}_0))^T).

3.5 Application: Constrained Extremum and Lagrange Multiplier Method​

Since the constraint g=0g = 0 is usually not easy to solve, Lagrange proposed considering the following augmented function:

F(x,λ1,…,λr)=f(x)−λ1g1(x)−⋯−λrgr(x).\begin{align*} F(\mathbf{x}, \lambda_1, \dots, \lambda_r) = f(\mathbf{x}) - \lambda_1 g_1(\mathbf{x}) - \dots - \lambda_r g_r(\mathbf{x}). \end{align*}

The critical points of FF satisfy:

{∇xF(x,λ1,…,λr)=∇f(x)−λ1∇g1(x)−⋯−λr∇gr(x)=0,∂F∂λi(x,λ1,…,λr)=−gi(x)=0,i=1,…,r.\begin{align*} \begin{cases} \nabla_{\mathbf{x}} F(\mathbf{x}, \lambda_1, \dots, \lambda_r) = \nabla f(\mathbf{x}) - \lambda_1 \nabla g_1(\mathbf{x}) - \dots - \lambda_r \nabla g_r(\mathbf{x}) = 0, \\ \frac{\partial F}{\partial \lambda_i}(\mathbf{x}, \lambda_1, \dots, \lambda_r) = -g_i(\mathbf{x}) = 0, \quad i = 1, \dots, r. \end{cases} \end{align*}

The first equation corresponds to the necessary condition for a constrained extremum point, while the subsequent equations are precisely the constraints themselves. In this way, the constrained extremum problem is transformed into an unconstrained extremum problem for the augmented function FF. This method is called the "Lagrange Multiplier Method," where λ\lambda is called the Lagrange Multiplier.

Theorem 3.6. Let ff be a Cr\mathscr{C}^r function, and let g:Rm→Rng: \mathbb{R}^m \to \mathbb{R}^n (m>nm > n) be a Cr\mathscr{C}^r mapping. Suppose x∗\mathbf{x}^* satisfies g(x∗)=0g(\mathbf{x}^*) = 0 and the Jacobian matrix Jg(x∗)Jg(\mathbf{x}^*) has full rank (i.e., n=rank dg(x∗)n = \text{rank } \mathrm{d}g(\mathbf{x}^*)). Then:

  1. (Fermat's Lemma, Necessary Condition for Constrained Extremum): If ff attains a (constrained) extremum at x∗\mathbf{x}^* under the constraint g(x)=0g(\mathbf{x}) = 0, then the gradient ∇f(x∗)\nabla f(\mathbf{x}^*) is orthogonal to the tangent space of the surface Σ:g(x)=0\Sigma : g(\mathbf{x}) = 0 at x∗\mathbf{x}^*. In this case, there exists a unique set of real numbers λ1∗,…,λn∗\lambda_1^*, \dots, \lambda_n^* such that:
∇f(x∗)=λ1∗∇g1(x∗)+⋯+λn∗∇gn(x∗)\begin{align*} \nabla f(\mathbf{x}^*) = \lambda_1^* \nabla g_1(\mathbf{x}^*) + \dots + \lambda_n^* \nabla g_n(\mathbf{x}^*) \end{align*}

  1. (Sufficient Condition for Constrained Extremum): If the Hessian matrix of f(x)−λ1∗g1(x)−⋯−λn∗gn(x)f(\mathbf{x}) - \lambda_1^* g_1(\mathbf{x}) - \dots - \lambda_n^* g_n(\mathbf{x}) with respect to x\mathbf{x} is positive definite (or negative definite) when restricted to the tangent space Tx∗ΣT_{\mathbf{x}^*} \Sigma, then x∗\mathbf{x}^* is a constrained local minimum point (or respectively, a constrained local maximum point) of ff restricted to Σ\Sigma.

  2. (Saddle Point): If the Hessian matrix of f(x)−λ1∗g1(x)−⋯−λn∗gn(x)f(\mathbf{x}) - \lambda_1^* g_1(\mathbf{x}) - \dots - \lambda_n^* g_n(\mathbf{x}) with respect to x\mathbf{x}, when restricted to the tangent space Tx∗ΣT_{\mathbf{x}^*} \Sigma, has both positive and negative eigenvalues, then x∗\mathbf{x}^* is not a constrained extremum point of ff restricted to Σ\Sigma.