Math 452 - Advanced Calculus II Maxima, Minima, Manifolds and Lagrange Multipliers Definition 1. Let D be a compact set of Rn . We say that the function f : D → R has a local maximum (respectively, local minimum) on D at the point p ∈ D if and only if there exists an open ball B ⊂ D centered at p such that f (x) ≤ f (p) [respectively, f (x) ≥ f (p) for all points x ∈ B. Recall the well-known result from single-variable calculus that if the differentiable function f : R → R has a local maximum or local minimum at p ∈ R, then f (p) = 0. Lemma 1. Let S ⊂ Rn , and ϕ : R → S be a differentiable curve with ϕ(0) = a. If f is a differentiable real-valued function defined on some open set containing S, and f has a local maximum (or local minimum) on S at a, then the gradient vector ∇f (a) is orthogonal to the velocity vector ϕ (0). Proof. On the blackboard. Corollary 1. If U is an open set of Rn and a ∈ U is a point at which the differentiable function f : U → R has a local maximum or local minimum, then ∇f (a) = 0. Definition 2. A set M ⊂ Rn is said to have a k-dimensional tangent plane at the point a ∈ M if the union of all tangent lines to differentiable curves on M passing through a is a k-dimensional plane. Definition 3. The projection mapping πi : Rn → Rn−1 is defined by removing the ith coordinate: πi (x1 , . . . , xn ) = (x1 , . . . , xi , . . . , xn ) = (x1 , . . . , xi−1 , xi+1 , . . . , xn ) ∈ Rn−1 . 1 1 Single-constraint Optimization Definition 4. The set P ⊂ Rn is called an (n − 1)-dimensional patch if and only if for some integer i, 1 ≤ i ≤ n, there exists a differentiable function h : U → R for U ⊂ Rn−1 open, such that P = {x ∈ Rn : πi (x) ∈ U and xi = h(πi (x)) }. NOTE: The concept of an (n − 1)-dimensional patch is equivalent to having a permutation xi1 , . . . , xin of the coordinates x1 , . . . , xn and a differentiable function h : U → R on an open set U ⊂ Rn−1 such that: P = {x ∈ Rn : (xi1 , . . . , xin−1 ) ∈ U and xin = h(xi1 , . . . , xin−1 ) } Definition 5. The set M ⊂ Rn is called an (n−1)-dimensional manifold if and only if each point a ∈ M lies in an open subset U ⊂ Rn such that U ∩ M is an (n − 1)-dimensional patch. Theorem 1. If M is an (n − 1)-dimensional manifold in Rn , then at each of its points M has an (n − 1)-dimensional tangent plane. Proof. On the blackboard. Theorem (Implicit Function Theorem). Let g : Rn → R be continuously differentiable and suppose that g(a) = 0 and Dn g(a) = 0. Then there exists a neighborhood U of a and a differentiable function f : V → R, with V ⊂ Rn−1 a neighborhood of (a1 , . . . , an−1 ), such that U ∩ g −1 (0) = {x ∈ Rn : (x1 , . . . , xn−1 ) ∈ V and xn = f (x1 , . . . , xn1 )}. Theorem 2. Suppose that g : Rn → R is continuously differentiable. If M is the set of all points x ∈ S = g −1 (0) at which ∇g(x) = 0, then M is an (n − 1)-manifold. Given a ∈ M , the gradient vector ∇g(a) is orthogonal to the tangent plane to M at a. Proof. On the blackboard. 2 Theorem 3. Suppose g : Rn → R is continuously differentiable and let M be the set of points x ∈ Rn at which g(x) = 0 and ∇g(x) = 0. If the differentiable function f : Rn → R attains a local maximum or minimum on M at the point a ∈ M , then ∇f (a) = λ∇g(a) for some number λ, denoted as the Lagrange multiplier. Proof. On the blackboard. 2 Multiple-constraint Optimization Definition 6. The set P ⊂ Rn is called a k-dimensional patch if and only if there exists a permutation xi1 , . . . , xin of x1 , . . . , xn , and differentiable function h : U → Rn−k for U ⊂ Rk open, such that P = {x ∈ Rn : (xi1 , . . . , xik ) ∈ U and (xik+1 , . . . , xin ) = h(xi1 , . . . , xik ) } Definition 7. The set M ⊂ Rn is called a k-dimensional manifold if and only if each point a ∈ M lies in an open subset U ⊂ Rn such that U ∩ M is a k-dimensional patch. Theorem 4. If M is an k-dimensional manifold in Rn then, at each of its points, M has a k-dimensional tangent plane. Proof. On the blackboard. Theorem (Implicit Mapping Theorem). Let g : Rn → Rm (m < n) be a continuously differentiable map. Suppose that g(a) = 0 and that the rank of the derivative matrix g (a) is m. Then there exists a permutation xi1 , . . . , xin of the coordinates in Rn , an open set U ⊂ Rn containing a, an open subset V ⊂ Rn−m containing b = πn−m (ai1 , . . . , ain ), and a differentiable mapping h : V → Rm such that each point x ∈ U lies on S = g−1 (0) if and only if (xi1 , . . . , xin−m ) ∈ V and (xin−m+1 , . . . , xin ) = h(xi1 , . . . , xin−m ). 3 Theorem 5. Suppose that g : Rn → Rm is continuously differentiable. If M is the set of all points x ∈ S = g−1 (0) for which the rank of g (x) is m, then M is an (n − m)-manifold. Given a ∈ M , the gradient vectors ∇g1 (a), . . . , ∇gm (a) are all orthogonal to the tangent plane to M at a. Theorem 6. Suppose g : Rn → Rm (m < n) is continuously differentiable and let M be the set of points x ∈ Rn such that g(x) = 0 and the gradient vectors ∇g1 (a), . . . , ∇gm (a) are linearly independent. If the differentiable function f : Rn → R attains a local maximum or minimum on M at the point a ∈ M , then there exist real numbers λ1 , . . . , λm (called Lagrange multipliers) such that: ∇f (a) = λ1 ∇g1 (a) + . . . + λm ∇gm (a) 4