WISB212, Analyse in meer variabelen Assignment 1

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1 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment 1 F. Ziltener The exercises marked with a are inleveropdrachten. The exercises marked with a + are particularly important. As in the lecture, we adopt the following: Convention: Vector space refers to a finite-dimensional real vector space. Below k, m, n, n 1,..., n k, and p are natural numbers. The following exercise shows that we may identify the space of linear maps between two coordinate spaces with the space of matrices. + Exercise 1 (space of linear maps) We denote by e 1,..., e n the standard basis of R n. Prove that the map Lin(R n,r p ) T ( T e 1 T e n ) R p n = Mat(p n,r) is an isomorphism of vector spaces. Recall here what a vector space, a linear map, and an isomorphism are. For this recall first that a group is a pair (G, ), where G is a set and is a map, such that : G G G (x y) z = x (y z), x, y, z G, (1) and there exists an element e G with the following properties: (i) e x = x, x G. (ii) For every x G there exists y G satisfying We call a group (G, ) abelian (or commutative) iff y x = e. x y = y x, x, y G. Remarks: The condition (1) means that is associative. The element e as above is unique. It is called the identity (or neutral) element of G. It also satisfies x e = x, x G. The condition y x = e implies that x y = e. Such a y is uniquely determined and called the inverse of x. Recall that a real vector space is a triple ( V, +, ), where V is a set, and are maps, such that the following conditions hold: (i) The pair (V, +) is an abelian group. + : V V V, : R V V 1

2 (ii) Multiplication in R is compatible with scalar multiplication, i.e., (iii) Scalar multiplication with 1 is trivial, i.e., (iv) Distributivity holds, i.e., for every a, b R, x, y V. (ab) x = a (b x), a, b R, x V. 1 x = x, x V. (a + b) x = a x + b x, a (x + y) = a x + a y, Remark: The identity element of the abelian group (V, +) is usually denoted by 0. Let now ( V, +, ) and ( W, +, ) be vector spaces and f : V W a map. We call f linear iff for every pair of numbers a, a R and vectors v, v V we have f ( av + a v ) = af(v) + a f(v ). The map f is called an isomorphism (of real vector spaces) iff it is linear and bijective. Exercise (differentiability of components) Let U R n be an open subset, f : U R p a map, and x 0 U. Show that f is differentiable at x 0 if and only if the i-th component f i is differentiable at x 0 for every i = 1,..., p. Show that in this case D(f 1 )(x 0 ) Df(x 0 ) =. D(f p )(x 0 ). * Exercise 3 (derivative of bilinear map) Show that every bilinear map is differentiable, with derivative given by f : R m R n R p Df(x, y)(v, w) = f(v, y) + f(x, w), x, v R m, y, w R n. Here we canonically identify R m R n with R m+n. Exercise 4 (derivative of multilinear map) Let f : R n1 R nk R p be a k-linear map. (This means that f is linear in each argument.) Show that f is differentiable and calculate its derivative. Here we canonically identify R n1 R nk with R n1+ +nk.

3 Exercise 5 (reading) Read the following in the lecture notes: rest of Section 1. De (totale) afgeleide van een functie tussen coördinatenruimten, which was not treated in the lecture. Section 1.3 Kettingregel, richtings- en partiële afgeleiden, Jacobi-matrix, gradiënt Let, be a (real) inner product on R n. Recall from WISB11 (Lineaire Algebra) that the induced norm is defined by x := x, x. + Exercise 6 (derivative of norm) Prove that is differentiable on R n \ {0} and calculate its (total) derivative. Hint: Use the Chain Rule. + Exercise 7 (Leibniz Rule) Let U R m be an open subset, x 0 U, and f : U R n1 n and g : U R n n3 be maps that are differentiable at x 0. (Here R n1 n denotes the space of real n 1 n -matrices. We identify this space with R n1n.) Prove that the map fg : U R n1 n3 is differentiable at x 0 and compute its derivative at this point. Here (fg)(x) := f(x)g(x) denotes the product of the matrices f(x) and g(x), for every x U. Hint: Write fg as a composition of two maps. Use Exercise 3 and the Chain Rule. We denote by 1 the identity matrix and by GL(n,R) the real general linear group. (It consists of the invertible real n n-matrices.) Exercise 8 (derivative of determinant map) (i) Show that the determinant map det : R n n R is differentiable. (Here we identify R n n with R n.) (ii) Show that the derivative of det is given by D det(a)x = tr(a # X), A, X R n n. Here tr denotes the trace of a matrix, and A # the adjugate (or complementary) matrix of A, which is defined as follows. For every i, j {1,..., n} we denote by Aîj R (n 1) (n 1) the matrix obtained from A by deleting the i-th row and the j-th column. The (i, j)-th entry of A # is given by A # ij := ( 1)i+j det Aĵi. Hint for proving differentiability: Exercise 4. Hint for the computation of the derivative: calculate the partial derivative (iii) Conclude that a ij det. D det(1) = tr, D det(a) = det A tr(a 1 ), A GL(n,R). 3

4 Exercise 9 (inversion map differentiable) (i) Show that GL(n,R) is an open subset of R n n. (Here we identify R n n with R n.) (ii) Prove that the inversion map for matrices, is differentiable. GL(n,R) A A 1 GL(n,R), () Hint: Use Cramer s rule for the inverse of a matrix, Exercise 8(i), and the Chain Rule. * Exercise 10 (derivative of inversion map) Calculate the derivative of the map (). Justify your calculation. Hint: Use the identity A 1 A = 1. Let U R n be an open subset, f : U R p a map, and x 0 U. Assume that f is differentiable at x 0. We define the map ϕ x0 : R n R p, ϕ x0 (x) := f(x 0 ) + Df(x 0 )(x x 0 ). The following exercise shows that ϕ x0 is the best affine approximation of f. Exercise 11 (best affine approximation) Let ϕ ϕ x0 : R n R p be an affine map. (Recall from WISB11 that such a map is a linear map plus a constant.) Assume that l V is a line through x 0 on which ϕ and ϕ x0 are not identical. Show that there exists a neighbourhood U U of x 0 such that Choosing such a U, prove that the function f(x) ϕ(x), x (l U ) \ {x 0 }. (l U ) \ {x 0 } x (f ϕx0 )(x) (f ϕ)(x) R tends to zero as x tends to x 0. Here denotes the Euclidian norm. Hint: Draw a picture in the case n = p = 1. 4

5 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment The exercises marked with a are inleveropdrachten. F. Ziltener The exercises marked with a + are particularly important. As in the lecture, we adopt the following convention. Convention: Vector space refers to a finite-dimensional real vector space. * Exercise 1 (smoothness) (a) Show that the following maps are smooth (C ): (i) Every multilinear map (ii) f : V 1 V k W, where k N, ( ) ( ) V i, Vi, i = 1,..., k, and W, W are normed vector spaces. Here we endow the product V 1 V k with the norm ( ) v 1,..., v k V1 V k := max v i Vi. i=1,...,k V : V \ {0} R, where V is a vector space and V the norm induced by some inner product on V. (iii) The determinant map (iv) The inversion map for matrices, det : R n n R. GL(n, R) A A 1 GL(n, R). Hints: Prove and use that polynomial functions are smooth. Use Exercises 8 and 4 below, and the theorem from the lecture stating that the composition of C k -maps is C k. In parts (i,ii) first consider the case in which each vector space is some coordinate space R n together with the Euclidean norm. (b) For l N and f as in (i) with k = find a formula for the l-th derivative of f. + Exercise (Hessian matrix) Calculate the Hessian matrix (=matrix of second partial derivatives) of the function f : R R, f(x) := x 1 x, and compare to Exercise 1(b). The following exercise was stated as a proposition in the lecture. Exercise 3 (multilinear maps) Let k N, and V i, i = 1,..., k, and W be vector spaces. We define W 0 := W and recursively, for j = 1,..., k, W j := Lin ( V k j+1, W j 1 ). We denote Lin ( V 1,..., V k ; W ) := { f : V 1 V k W f k-linear }. 1

6 (i) For T W k we define Φ T k : V 1 V k W, Φ T ( ) k v1,..., v k := T (v1 ) (v k ). Show that this map is k-linear. (ii) We define Φ k : W k = Lin ( V 1,... Lin(V k ; W )... ) Lin ( V 1,..., V k ; W ), Φ k (T ) := Φ T k. (1) Show that this map is an isomorphism of vector spaces. Hint: Consider first the cases k = 1 and k =. The following exercise was stated as a lemma in the lecture. It was used in the proof of a proposition regarding the composition of k times (continuously) differentiable maps. Exercise 4 (composition with bilinear map preserves k-fold differentiability) Consider the statement: B(k): Let V, X, Y, Z be normed vector spaces, U V open, b : X Y Z bilinear, and F : U X, G : U Y k times differentiable. Then the following holds: The map is k times differentiable. b (F, G) : U Z If D k F and D k G are continuous then D k( b (F, G) ) is continuous. Show that B(k) holds for every k N 0 = N {0}. Hint: Use induction over k and the Chain Rule for b (F, G). * Exercise 5 (nonlinear system of equations) (i) Prove that there exist numbers r, r > 0 with the following properties: For every y R satisfying y (1, 1) < r there exists a unique solution x = x y R of the equations x 1 + e x = y 1 e x1 + x = y, satisfying x y < r. Furthermore, the map y x y is smooth. (ii) Calculate the derivative of the map y x y at the point (1, 1). Exercise 6 (local inverse of a holomorphic function) Let U 0 C be an open subset, f : U 0 C a holomorphic (=complex differentiable) function, and z 0 U 0 a point. Assume that the complex derivative f does not vanish at z 0. (i) Show that there exists an open neighbourhood U U 0 of z 0, such that f U is injective with an inverse f 1 U : f(u) U that is complex differentiable. (ii) Can we choose U = U 0?

7 Remarks: Recall from WISB11 that f is called or complex differentiable at z 0 U 0 iff the difference quotient f(z) f(z 0 ) z z 0 converges, as z z 0 tends to z 0. The limit is denoted by f (z 0 ) C and called the complex derivative of f at z 0. The function f is called complex differentiable (or holomorphic) iff it is complex differentiable at every point in U 0. You may use results from Functies en Reeksen. If you find this exercise too hard, then consider the concrete example f : U 0 := C C, f(z) := z. The following result shows that spherical coordinates provide a smooth bijective map between suitable open subsets of R 3. + Exercise 7 (spherical coordinates) We define f : R 3 R 3, f(r, ϕ, θ) := r ( cos ϕ cos θ, sin ϕ cos θ, sin θ ). (i) Draw the images under f of the planes r =constant, ϕ =constant, and θ =constant, and those of the lines (ϕ, θ), (r, θ), and (r, ϕ) =constant, respectively. (ii) Prove that f is surjective, but not injective. (iii) We define ( U := (0, ) ( π, π) π, π ). Prove that the restriction of f to U is injective. (iv) Determine V := f(u). (v) Prove that f : U V is a C -diffeomorphism. (vi) Find a formula for the inverse map. The following exercise was stated as a theorem in the lecture. Exercise 8 (characterization of C k ) Let k, n, p N, U R n be open, and f : U R p be a map. Show that f lies in C k (U, R p ) if and only if its partial derivatives up to order k exist and are continuous, by reducing to the case k = 1. (In that case the exercise was a result in WISB11.) 3

8 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment 3 The exercises marked with a are inleveropdrachten. F. Ziltener The exercises marked with a + are particularly important. + Exercise 1 (nonlinear equation) (i) Prove that there exist numbers a > 0 and b > 0 with the following properties: For every x ( a, a) there exists a unique solution y = y x ( b, b) of the equation sin(x + y) + y = 0. Furthermore, the function x y x is smooth. (ii) Calculate the derivative of this function at 0. + Exercise (inversion map smooth) Prove that the inversion map GL(n, R) A A 1 GL(n, R) is smooth in a short and elegant way without using Cramer s Rule. Remark: Here we equip R n n with an arbitrary norm. Hint: Use a theorem from the lecture. Remark: A proof using Cramer s Rule was given in the solution to an exercise in Assignment (smoothness). Exercise 3 (second derivative of implicit function) (i) Find a formula for the second derivative of the implicit function g in the Implicit Function Theorem at a point x. (ii) Interpret your formula in the case n = p = 1. (iii) Calculate the second derivative of the function x y x of Exercise 1 at the point 0. Hint: If you feel more comfortable with partial derivatives, you may solve (i) by finding a formula for the second partial derivatives. The following exercise was stated as a lemma in the lecture. It was used in the proof of the Inverse Function Theorem. Exercise 4 (inverse map differentiable) Let U, V R n be open, x U, f : U V, and y := f(x). Assume that f is differentiable at x with invertible derivative, f is bijective, and that f 1 (y ) x sup y V \{y} Show that f 1 is differentiable at y, with y y D(f 1 )(y) = Df(x) 1. <. (1) Remark: Condition (1) means that f 1 is Lipschitz-continuous at y. This condition is satisfied if f 1 is Lipschitz-continuous. 1

9 * Exercise 5 (smooth dependence of simple eigenvalues) Let A 0 R n n and λ 0 R be an algebraically simple eigenvalue of A 0. This means that it is a simple zero of the characteristic polynomial p(λ) := det ( λ1 A 0 ). Prove that there exist open neighbourhoods U R n n of A 0 and V R of λ 0, such that every A U has a unique eigenvalue λ A in V, and the map is smooth. U A λ A V Exercise 6 (smooth dependence of simple eigenvalues and -vectors) Let A 0 R n n, λ 0 R be an algebraically simple eigenvalue of A 0, and v 0 R n an eigenvector of A 0 for the eigenvalue λ 0, such that v 0 = 1. Prove that there exist open neighbourhoods U R n n of A 0 and V R R n of (λ 0, v 0 ) with the following properties: For every A U there exists a unique pair (λ A, v A ) V consisting of an eigenvalue and a corresponding eigenvector, such that v A = 1. The map is smooth. Hints: Define U A (λ A, v A ) V f : R n n R R n R R n, f(a, λ, v) := ( v 1, λv Av ). Calculate D (λ,v) f ( A 0, λ 0, v 0 ), the derivative of the map (λ, v) f ( A0, λ, v ) at (λ 0, v 0 ). Consider first the case in which v 0 = e 1. Calculate the matrix of D (λ,v) f ( A 0, λ 0, v 0 ) with respect to the basis e 1,..., e n. Show that it is invertible. Reduce the general situation to the case v 0 = e 1. Exercise 7 (roots of a matrix) Let k, n N and A 0 R n n be a symmetric matrix. (This means that A T 0 = A 0.) We call A 0 positive definite iff v T A 0 v 0, v R n and v T A 0 v = 0 = v = 0. (i) Prove that for every positive definite symmetric matrix A 0 there exists a positive definite symmetric matrix B 0 R n n satisfying B k 0 = A 0. Remark: The last condition means that B 0 is a k-th root of A 0. (ii) Let B 0 R n n be a positive definite symmetric matrix and A 0 = B0 k. Prove that there exist open neighbourhoods U and V R n n of A 0 and B 0 with the following properties. For every A U there exists a unique solution B = B A V of the equation Furthermore, the map A B A is smooth. B k = A. Remark: This means that every matrix A close enough to A 0 has a unique k-th root close to B 0, and this root depends smoothly on A. Hint: First consider the case A 0 = 1, and in (ii) consider the case B 0 = 1 (identity matrix). Next consider the case in which A 0 is diagonal. Reduce the general situation to this case. (iii) Does every real n n-matrix have a real square root? (iv) Is (ii) true without assuming that B 0 is positive-definite?

10 + Exercise 8 (sphere is a submanifold) For n N prove that the unit sphere S n 1 is a smooth submanifold of R n of dimension n 1. + Exercise 9 ( graph map ) Let n N 0, d = 0,..., n, V R d be open, k N := N { }, and f C k (V, R n ). Show that the map is a C k -embedding. V y (y, f(y)) R d R n The following exercise was stated as part of a corollary in the lecture. * Exercise 10 (characterization of a submanifold) Let n N 0, d {0,..., n}, k N, M R n, and x 0 M. Show that (i) implies (iii) and (iii) implies (ii), where: (i) (submanifold) M is a C k -submanifold at x 0 of dimension d. (ii) (local parametrization) There exist an open subset V R d, an open neighbourhood U R n of x 0, and a C k -embedding ψ : V R n such that M U = ψ(v ). (iii) (straightening) There exist an open neighbourhood U R n of x 0, an open subset V R d, and a C k -diffeomorphism Φ : U Φ(U), such that Φ(M U) = V {0R n d}. Hint for (i)= (iii) : When using the definition of a submanifold, consider first the case in which the permutation is the identity. 3

11 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment 4 The exercises marked with a are inleveropdrachten. F. Ziltener The exercises marked with a + are particularly important. Exercise 1 (stereographic projection) Let n and ψ : V := R n 1 S n 1 be the inverse map of stereographic projection. (i) Derive a formula for ψ. (ii) Prove that ψ is a smooth embedding. Exercise (dimension well-defined) Let n N 0, M R n, x 0 M, k N, and d, d {0,..., n}. Show that if M is a C k -submanifold at x 0 of dimensions d and d then d = d. Exercise 3 (product of submanifolds) Let n, n N {0}, M R n, M R n, x 0 M, x 0 M, k N, d {0,..., n}, and d {0,..., n }. Assume that M is a C k -submanifold at x 0 of dimension d, and M is a C k -submanifold at x 0 of dimension d. Show that the Cartesian product M M is a C k -submanifold of R n R n at (x 0, x 0 ) of dimension d + d. * Exercise 4 (curve of degree four) Draw the set M := { x R x x 4 = 1 } and prove that it is a smooth submanifold of R of dimension 1. Exercise 5 (hyperboloid) Draw the hyperboloid of one sheet, M := { x R 3 x 1 + x = x }, and prove that it is a smooth submanifold of R 3 of dimension. + Exercise 6 (special linear matrices) Prove that for every n N the set SL(n, R) := { A R n n det A = 1 } is a smooth submanifold of R n n (which we identify with R n ), and compute its dimension. Exercise 7 (existence of immersion or embedding) Decide whether there exists (i) a smooth map, (ii) a smooth immersion, 1

12 (iii) an injective smooth immersion, (iv) a smooth embedding with domain R and image 1. [0, ) R,. ( [0, ) {0} ) ( {0} [0, ) ) R, 3. S 1 R. Does there exist an immersion R R? + Exercise 8 (submersion is open) Let n, p N 0, U 0 R n be open, and g C 1 (U 0, R p ) be a submersion. Show that g is open, i.e., it maps open sets to open sets. Hint: This is an easy consequence of a result of the lecture. The following exercise is part of a corollary stated in the lecture. Let n N 0, M R n, x 0 M, k N, and d {0,..., n}. Exercise 9 (submanifolds and local submersions) Prove that if M is a C k -submanifold of R n at x 0 of dimension d then there exist an open neighbourhood U R n of x 0 and a C k -submersion g : U R n d, such that M U = g 1 (g(x 0 )). + Exercise 10 (tangent spaces to linear subspace) Let M be a d-dimensional linear subspace of R n and x 0 M. Show that T x0 M = M. Exercise 11 (tangent spaces to open set) Let M R n be open and x 0 M. Show that T x0 M = R n. * Exercise 1 (tangent spaces to some submanifolds) Calculate the tangent space at every point to the following submanifolds of coordinate space: (i) the hyperbola (ii) the curve M := { x R x1 x = 1 }. M := { x R x x 4 = 1 }, (iii) the hyperboloid of one sheet, (iv) The set of special linear matrices M := { x R 3 x 1 + x = x }, SL(n, R) := { A R n n det A = 1 }

13 Remark: By an exercise in the lecture and Exercises 4,5,6 these sets are indeed submanifolds. + Exercise 13 (tangent spaces to logarithmic spiral, tangent map) (i) Draw a picture of the logarithmic spiral M := { e y( cos y, sin y ) y R }. (ii) Prove that this is a submanifold of R of dimension 1 and calculate its tangent space at any point. (iii) Let a R. We denote by R a : R R rotation by the angle a in counterclockwise direction. Show that the image of the map is M. f : M R, f(x) := e a R a x, (iv) Compute the tangent map (derivative) of f at x 0 M. (v) How are the tangent spaces at two points x 0, x 0 M related? What about the case in which x 0 and x 0 lie on the ray (0, ) {0}? 3

14 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment 5 The exercises marked with a are inleveropdrachten. F. Ziltener The exercises marked with a + are particularly important. In some exercises reference is made to the following book: [DK] Duistermaat, J. J.; Kolk, J. A. C., Multidimensional real analysis II, Integration, Cambridge Studies in Advanced Mathematics, 86, Cambridge University Press, Cambridge, 004. Exercise 1 (tangent space to a product) Let M R n and M R n be C 1 -submanifolds, and x 0 M and x 0 M points. What is the tangent space to M M at (x 0, x 0 )? Prove that your answer is correct. Remark: By an exercise in Assignment 4 (product of submanifolds) the product M M is a C 1 - submanifold of R n+n, hence the above question makes sense. Exercise (tangent map for stereographic projection) Calculate the tangent map for the stereographic projection S n 1 \ {( 0,..., 0, 1 )} R n 1. (Compare to an exercise about stereographic projection in Assignment 4.) Exercise 3 (Chain Rule for tangent maps) For i = 1,, 3 let n i N 0 and M i R ni be a C 1 - submanifold. Let f C 1 (M 1, M ) and g C 1 (M, M 3 ). Show that g f : M 1 M 3 is C 1 and D(g f)(x 1 ) = Dg(f(x 1 ))Df(x 1 ), x 1 M 1. Exercise 4 (inverse of embedding) Let V R d be open and ψ : V R n be a C k -embedding. Show that ψ 1 : ψ(v ) V is C k and give a formula for its derivative at any point. Hint: For the second part use the previous exercise. + Exercise 5 (curve as image of embedding) We define ψ : R R, ψ(y) := ( y + y 5, y + y 6). (i) Draw a picture of the image M of ψ. (ii) Show that ψ is a smooth embedding and that M is a smooth submanifold of R of dimension 1. (iii) Show that ψ 1 : M R is smooth. (iv) Calculate D(ψ 1 )(0, 0). 1

15 + Exercise 6 (an extremum is a critical point) Let M R n be a C 1 -submanifold, f C 1 (M, R), and x 0 M be a point at which f attains its maximum or minimum. Show that x 0 Crit f. * Exercise 7 (extrema) Consider the curve M := { x R x x 4 = 1 } and the function f : M R, f(x) := x 1 + x. (i) Prove the f attains its extreme values (maximum and minimum) on M. (ii) Calculate these values. (iii) Draw M and some level sets of f and compare to your answer to (ii). The following exercise is part of a theorem in the lecture. + Exercise 8 (Lagrange multiplier method) Let n, p N 0, U R n open, F C 1 (U, R), g C 1 (U, R p ), M := g 1 (0), f := F M, x 0 U, L : U Lin ( R p, R ) R, L(x, λ) := F (x) λg(x). Assume that g is a submersion and that there exists λ Lin(R p, R) such that (x 0, λ) Crit L. Show that x 0 Crit f. The following exercise was used in the proof of the Lagrange multiplier method in the lecture. Exercise 9 (factoring a linear map) Let V 1, V, V 3 be vector spaces, T Lin(V 1, V 3 ), and T Lin(V 1, V ), such that if T v = 0 = T v = 0, v V 1. Prove that there exists λ Lin(V, V 3 ) such that λt = T. * Exercise 10 (Riemann integral) Consider the function (i) Draw a picture of f. (ii) Guess what the integral of f is. f : [0, 1] [0, ] R, f(x) := x 1. (iii) Prove that f is properly Riemann-integrable and calculate its Riemann integral. The following exercise was stated as a lemma in the lecture. It implies that every step function can be written as a linear combination of characteristic functions over rectangles that form a partition of a given rectangle. Exercise 11 (refinement of collection of rectangles) Let R 0 R n be a rectangle and R be a finite collection of rectangles that are contained in R 0. Show the following: (i) There exists a finite collection R of rectangles that is a partition of R 0, such that every R R is open or has volume zero, and if it intersects R R then it is contained in R.

16 (ii) Let I 1 0,..., In 0 be bounded intervals, such that We may choose R as in (i) to be of the form R 0 = I 1 0 I n 0. R = { I 1 I n Ii R i, i = 1,..., n }, where R i is a finite collection of bounded intervals that is a partition of I i 0. Hint: First consider the case n = 1. The following exercise was used in the lecture to define the integral of a step function. Exercise 1 (integral of step function) Let R and R be finite collections of rectangles, c R R for R R, and c R R for R R. Assume that ϕ := c R χ R = c R χ R. R R R R Show that c R R = R R R R c R R. Remark: This equals by definition R 0 ϕ(x)dx, the integral of ϕ over R 0, where R 0 is a rectangle that contains R and R. Hint: Use Exercise 11. Exercise 13 (definition of Riemann integral) Let R R n be a closed rectangle and f : R R a bounded real-valued function. Show that the lower Riemann integrals of f over R defined in the lecture and in the book [DK] (. vol., p. 46, Definition 6..3) are equal. Show the analogous statement for the upper Riemann integrals. Deduce that the notions of (proper) Riemann integrability in the lecture and the book are equivalent, and that the corresponding integrals are the same. Exercise 14 (reading) Read the Section 5. Eigenschappen van de Riemann-integraal. 3

17 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment 6 The exercises marked with a are inleveropdrachten. F. Ziltener The exercises marked with a + are particularly important. The following exercise was used in the proof of the characterization of Riemann integrability in terms of the set of discontinuity. + Exercise 1 (countable union of sets of 0 Lebesgue measure) Prove that every countable union of sets of 0 Lebesgue measure has 0 Lebesgue measure. * Exercise (two-dimensional integrals) (i) Prove that the function f : [0, 1] [0, 1] R, f(x) := x 1 x, is properly Riemann integrable and calculate its integral. (ii) Prove that the function f : [0, 1] [1, 4] R, f(x) := e x1 x, is properly Riemann integrable and calculate its integral. (iii) Calculate 1 ( 1 ) arctan (e x1 x ) dx 1 dx. 1 0 Exercise 3 (tensor product) Let Q R m and R R n be rectangles, and f : Q R and g : R R be (properly) Riemann-integrable functions. Show that the function f g : Q R R, f g(y, z) := f(y)g(z) is Riemann-integrable and express its integral in terms of the integrals of f and g. Remark: This function is called the tensor product of f and g. Exercise 4 (graph negligible) Let n N 0, p N, K R n be compact, and f : K R p be continuous. Show that the graph of f is a negligible subset of R n+p. + Exercise 5 (rescaling and translation) (i) Let f RI(R n ), i.e., f is a properly Riemannintegrable function on R n. (Every such function is bounded and vanishes outside some bounded set.) Let c (0, ) and v R n. We define ( ) x v f : R n R, f( x) := f. c Show that f RI(R n ) and R n f( x)d x = c n 1 R n f(x)dx.

18 (ii) Let A R n be a Jordan-measurable set, c [0, ), and v R n. Prove that the set à := ca + v := { cx + v x A } is Jordan-measurable, with Jordan-measure ca + v = c n A. * Exercise 6 (volume of simplex (pyramid)) (i) Draw the set n := { x R n x1,..., x n 0, x x n 1 } for n = 1,, 3. (ii) Prove that n is Jordan-measurable. (iii) Calculate its Jordan-measure. Remark: This set is called the standard n-simplex. The following result will be used to calculate the volume of a ball in R n. Exercise 7 (integral of power of cosine) Prove that for n N 0 (n 1)(n 3) 1 π π, if n is even, cos n t dt = n(n ) (n 1)(n 3) π, if n is odd. n(n ) 3 Here we use the convention that the empty product equals 1. This product occurs in the cases n = 0, 1.

19 + Exercise 8 (volume of ball) Let n N 0. (i) Prove that the closed unit ball B n := B n 1 R n is Jordan-measurable. (ii) Calculate its Jordan-measure. Exercise 9 (Cantor set negligible) Prove that the Cantor set K is a negligible subset of R. Hint: Recall that where K k is recursively defined by K := K 0 := [0, 1], K k+1 := k N 0 K k, ( ) ( K k 3 K k + ). 3 Remark: The Cantor set is uncountable. (Why?) Hence it is an example of an uncountable negligible subset of R. Exercise 10 (integral of Gaußfunction) Show that x ± 0 e x dx converges, as x ± ±, and x+ 0 e x dx := lim e x dx + lim dx 3. R x + 0 x x e x Exercise 11 (goose problem, submanifold negligible) Prove or disprove: Let d < n N 0. Every bounded smooth submanifold of R n of dimension d is negligible. Remark: The first person (or group) who solves this problem will be awarded a metal goose. 3

20 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment 7 The exercises marked with a are inleveropdrachten. F. Ziltener The exercises marked with a + are particularly important. + Exercise 1 (piece of cake) (i) Let ϕ ϕ + ( π, π). Draw a picture of the sector S := { r ( cos ϕ, sin ϕ ) r [0, 1], ϕ [ϕ, ϕ + ] }. (ii) Show that this set is Jordan-measurable and calculate its area, i.e., two-dimensional Jordanmeasure. * Exercise (integral of rotationally symmetric function) Let r 0 > 0 and f : [0, r 0 ] R be a continuous function. We define f : B 3 r 0 R, f(x) := f( x ). (i) Find a formula for B 3 r 0 f(x) dx in terms of f. (ii) Use your formula to calculate the volume of B 3 r 0. Exercise 3 (area) (i) Let 0 < a < b, 0 < c < d. Draw a picture of the set S := { (x, y) R 0 < x, a y/x b, c xy d }. (ii) Prove that S is Jordan-measurable and calculate its area. Exercise 4 (volumes of cylinder, cone, and ball) (i) Draw pictures of the cylinder B 1 [0, 1] and the cone { x R } [0, 1] x 1 + x x 3. (ii) Show that these sets are Jordan-measurable and calculate their Jordan-measures. (iii) How are the volumes of the cylinder, cone, and the half-ball { 3 x B x3 0 } related? Why? 1 Exercises 5 through 7 below are used in the proof of the Change of Variables Theorem in the lecture. Exercise 5 (transforming sets of 0 Lebesgue measure) Let n N, S R n be a set of 0 Lebesgue measure, and f : S R n a Lipschitz-continuous map. Show that f(s) is of 0 Lebesgue measure. 1

21 * Exercise 6 (substitution for elementary transformations) We call T Aut(R n ) an elementary transformation iff one of the following conditions holds. (a) There exist j k {1,..., n} such that T (e i ) = e i, if i j, k, T (e j ) = e k, T (e k ) = e j. (b) There exist j {1,..., n} and a R \ {0} such that T (e i ) = e i, if i j, T (e j ) = ae j. (c) There exist j k {1,..., n} and c R, such that Show: T (e i ) = e i, if i j, T (e j ) = e j + ce k. (i) If T Aut(R n ) is an elementary transformation and S R n is Jordan measurable then T (S) = S det T. (ii) Every linear automorphism of R n can be written as a finite product of elementary transformations. Hint: For (i) in the case in which T has type (c), use Fubini. For (ii) use Gauß elimination. We say that T Aut(R n ) has the transformation property iff for every Jordan measurable S R n we have T (S) = S det T. + Exercise 7 (transformation property and composition) Assume that T, T Aut(R n ) have the transformation property. Show that the same holds for T T. Exercise 8 (goose problem, submanifold negligible) Prove or disprove: Let d < n N 0. Every bounded smooth submanifold of R n of dimension d is negligible. Remark: The first person (or group) who solves this problem will be awarded a metal goose.

22 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment 8 The exercises marked with a are inleveropdrachten. F. Ziltener The exercises marked with a + are particularly important. + Exercise 1 (integral of curl) Calculate the integral ( curl X(x) dx = D1 X D X 1) (x) dx B B for X(x) := x ( x x 1 ). * Exercise (line integral) Consider the ellipse C := { x R x 1 + 4x = 1 }. (i) Calculate the unit tangent vector field T on C that is positive with respect to the domain U := { x R x 1 + 4x < 1 }. (ii) Calculate the line integral for the vector field C,T X ds = C X T ds X(x) := 1 ( ) x. x 1 The following exercise was stated as part of a proposition in the lecture. Exercise 3 (well-definedness of line integral) Let C R n be a compact (i.e., closed and bounded) C 1 -curve. Let f : C R be continuous. We define the line integral of f along C to be C f ds := k j=1 I j f x j ẋ j (t) dt, (1) where for j = 1,..., k, I j is a compact interval of positive length and x j C 1 (I j, R n ) is an immersion, such that k x j (I j ) = C and the map is injective. j=1 {j} int I j (j, t) x j (t) C j Show that the RHS of (1) does not depend on the choice of this data. 1

23 Remarks: In the lecture we will prove this by showing that the RHS of (1) agrees with the general formula defining the integral of a function over a manifold. It will be shown in the lecture that that formula does not depend on the choices. The point of the present exercise is to show this for line integrals in a direct way. It was shown in the lecture that I j, x j as above exist. It follows that C f ds is well-defined. * Exercise 4 (area of spherical cap) Let a (0, 1). (i) Draw the spherical cap { x S x3 a }. (ii) Calculate its area, i.e., -dimensional volume. (iii) Calculate the area of S. The following exercise was used in the lecture to give a formula for the integral along a surface in R 3. Exercise 5 (Gram s determinant) Let A R 3. Show that det(a T A) = A 1 A, where A j := j-th column of A. Exercise 6 (intrinsic boundary) Let d n N 0, k N { } and M R n be a C k -submanifold of dimension d with boundary. Recall from the lecture that this means that for every x 0 M there exists a (relatively) open subset V R d 0 := R d 1 [0, ) and an injective C k -immersion ψ : V R n with continuous inverse, such that ψ(v ) is a relatively open neighbourhood of x 0 in M. Such a pair (V, ψ) is called a local C k -parametrisation for M. We define the intrinsic boundary of M to be Show: M := { ψ ( V (R d 1 {0}) ) (V, ψ) local C k -parametrisation of M }. (i) If (V, ψ) and (Ṽ, ψ) are local C k -parametrisations of M then ψ 1( ψ ( V (R d 1 {0}) )) R d 1 {0}. (ii) The intrinsic boundary M is well-defined, i.e., it does not depend on the choice of k.

24 Utrecht University, blok 4, 016/ 017 WISB1, Analyse in meer variabelen Assignment 9 The exercises marked with a are inleveropdrachten. F. Ziltener The exercises marked with a + are particularly important. + Exercise 1 (flux through hemisphere) Calculate the flux (surface-integral) of the vector field X through M := { x S x3 0 } with respect to the unit normal vector field for the vector fields (i) X(x) := x, (ii) X = Y = curl Y, where (iii) X(x) := e 3. ν : M R 3, ν(x) := x, Y (x) := sin(x 3 ) x 3 x 1 x, The following exercise was used in the lecture to make sense of the integral along a submanifold. + Exercise (Gram s matrix) Let A R n d be such that the map R d v Av R n is injective. Show that det(a T A) > 0. The following exercise was used in the lecture to give a formula for the integral along a surface in R 3. Exercise 3 (Gram s determinant) Let A R 3. Show that det(a T A) = A 1 A, where A j := j-th column of A. The next exercise was stated as a lemma in the lecture and used in the proof of Stokes Theorem. + Exercise 4 (curl) Let U R 3 be open and X C 1 (U, R 3 ). Show that ( DX(x)v1 ) v ( DX(x)v ) v1 = ( X)(x) (v 1 v ), x U, v 1, v R 3. Exercise 5 (Faraday s law of induction and Maxwell s equation) Let E and B be the electric and magnetic fields and t time. Faraday s law of induction states that the time rate of change of the flux of B through a co-oriented surface Σ equals minus the circulation of E around Σ. Derive the equation E = B (1) t from this. Remarks: Faraday s law means that a changing magnetic flux through a surface spanned by a wire induces a current in the wire. This is the underlying principle of electrical motors and generators. Equation (1) is one of the four Maxwell laws of electrodynamics. 1

25 + Exercise 6 (Newton s law of gravitation) Newton s law of gravitation states that any two bodies attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. Here we idealize and assume that the bodies are points and that there are no other forces, such as an electrical force. Derive this law from the following principles: 1. The gravitational force F on a mass m generated by a mass M is proportional to Mm. It points from x to x 0, where x and x 0 denote the positions of m and M.. Mass is the source for gravitation, and hence the divergence of the gravitational field generated by M vanishes away from x The laws of physics are invariant under translations. 4. They are also invariant under rotations. + Exercise 7 (Laplace operator, Green s identity) Let U R n be a C 1 -domain and f, g C (U, R). We denote by n f := D i D i f i=1 the Laplacian of f and by ν : U R n the outward pointing unit normal vector field. Show that ( ) ( ) f g g f dx = f g g f ν da. () U Hint: Consider (f g) and use a theorem from the lecture. U Remarks: This identity () is an important tool in the theory of partial differential equations. For example, it implies that f g dx = g f dx, U if f and g vanish on the boundary U. This means that the Laplace operator is symmetric w.r.t. the L -inner product on the space of C -functions vanishing on U. This inner product is defined by f, g := fg dx. Symmetric operators will be discussed in WISB315, Functionaalanalyse. They play the role of observables in quantum mechanics. (More precisely, those symmetric operators that are in fact self-adjoint play this role.) In particular, the Laplace operator plays the role of kinetic energy of a particle in R 3. U U

26 The following exercise is used in the proof of Gauß Theorem. (See the lecture notes.) Exercise 8 (Sylvester s determinant identity) Show that for every m, n N, A R m n and B R n m the identity det(1 m + AB) = det(1 n + BA) holds. Hint: Consider the matrix in block form M := ( ) 1m A. B 1 n Show that the determinant of M stays the same when subtracting B times the first block row from the second block row. Show that the determinant of the new matrix is det(1 n + BA). Do a similar computation by subtracting A times the second block row from the first block row. The following exercise provides examples of d-negligible sets. These sets occur in the singular version of Gauß Theorem, see the lecture notes. Exercise 9 (d-negligible set) Let m < d n N 0 and M R n be a compact m-dimensional C 1 -submanifold. Show that M is d-negligible (as defined in the lecture). 3