Multilinear algebra, differential forms and Stokes' theorem Yakov Eliashberg April 2018 2 Contents I Multilinear Algebra 7 1 Linear and multilinear functions 9 1.1 Dual space . 9 1.2 Canonical isomorphism between (V ∗)∗ and V ...................... 11 1.3 The map A∗ ......................................... 11 1.4 Multilinear functions . 14 1.5 More about bilinear functions . 15 1.6 Symmetric bilinear functions and quadratic forms . 16 2 Tensor and exterior products 19 2.1 Tensor product . 19 2.2 Spaces of multilinear functions . 20 2.3 Symmetric and skew-symmetric tensors . 20 2.4 Symmetrization and anti-symmetrization . 22 2.5 Exterior product . 22 2.6 Spaces of symmetric and skew-symmetric tensors . 25 2.7 Operator A∗ on spaces of tensors . 26 3 More about tensors 31 3.1 Quotient space . 31 3.2 Tensor product of spaces . 32 3.3 Tensor square and bilinear functions . 33 3 4 Orientation and Volume 35 4.1 Orientation . 35 4.2 Orthogonal transformations . 36 4.3 Determinant and Volume . 37 4.4 Volume and Gram matrix . 39 5 Dualities 41 5.1 Duality between k-forms and (n − k)-forms on a n-dimensional Euclidean space V . 41 5.2 Euclidean structure on the space of exterior forms . 47 5.3 Contraction . 50 6 Complex vector spaces 55 6.1 Complex numbers . 55 6.2 Complex vector space . 58 6.3 Complex linear maps . 60 II Calculus of differential forms 63 7 Topological preliminaries 65 7.1 Elements of topology in a vector space . 65 7.2 Everywhere and nowhere dense sets . 67 7.3 Compactness and connectedness . 67 7.4 Connected and path-connected components . 70 7.5 Continuous maps and functions . 71 8 Vector fields and differential forms 75 8.1 Differential and gradient . 75 8.2 Smooth functions . 77 8.3 Gradient vector field . 77 8.4 Vector fields . 78 8.4.1 Gradient vector field . 79 4 8.5 Differential forms . 81 8.6 Coordinate description of differential forms . 81 8.7 Smooth maps and their differentials . 82 8.8 Operator f ∗ ......................................... 83 8.9 Coordinate description of the operator f ∗ ........................ 85 8.10 Examples . 86 8.11 Pfaffian equations . 87 9 Exterior differential 89 9.1 Coordinate definition of the exterior differential . 89 9.2 Properties of the operator d ................................ 91 9.3 Curvilinear coordinate systems . 95 9.4 Geometric definition of the exterior differential . 95 9.5 More about vector fields . 97 9.6 Case n = 3. Summary of isomorphisms . 99 9.7 Gradient, curl and divergence of a vector field . 100 9.8 Example: expressing vector analysis operations in spherical coordinates . 101 9.9 Complex-valued differential k-forms . 104 10 Integration of differential forms and functions 107 10.1 Useful technical tools: partition of unity and cut-off functions . 107 10.2 One-dimensional Riemann integral for functions and differential 1-forms . 110 10.3 Integration of differential 1-forms along curves . 113 10.4 Integrals of closed and exact differential 1-forms . 118 10.5 Integration of functions over domains in high-dimensional spaces . 119 10.6 Fubini's Theorem . 134 10.7 Integration of n-forms over domains in n-dimensional space . 137 10.8 Manifolds and submanifolds . 144 10.8.1 Manifolds . 144 10.8.2 Gluing construction . 146 5 10.8.3 Examples of manifolds . 153 10.8.4 Submanifolds of an n-dimensional vector space . 155 10.8.5 Submanifolds with boundary . 157 10.9 Tangent spaces and differential . 159 10.10Vector bundles and their homomorphisms . 162 10.11Orientation . 163 10.12Integration of differential k-forms over k-dimensional submanifolds . 163 III Stokes' theorem and its applications 169 11 Stokes' theorem 171 11.1 Statement of Stokes' theorem . 171 11.2 Proof of Stokes' theorem . 175 11.3 Integration of functions over submanifolds . 177 11.4 Work and Flux . 183 11.5 Integral formulas of vector analysis . 186 11.6 Expressing div and curl in curvilinear coordinates . 187 12 Topological applications of Stokes' formula 191 12.1 Integration of closed and exact forms . 191 12.2 Approximation of continuous functions by smooth ones . 192 12.3 Homotopy . 194 12.4 Winding and linking numbers . 201 12.5 Properties of k-forms on k-dimensional manifolds . 203 6 Part I Multilinear Algebra 7 Chapter 1 Linear and multilinear functions 1.1 Dual space Let V be a finite-dimensional real vector space. The set of all linear functions on V will be denoted by V ∗. Proposition 1.1. V ∗ is a vector space of the same dimension as V . Proof. One can add linear functions and multiply them by real numbers: (l1 + l2)(x) = l1(x) + l2(x) ∗ (λl)(x) = λl(x) for l; l1; l2 2 V ; x 2 V; λ 2 R It is straightforward to check that all axioms of a vector space are satisfied for V ∗. Let us now check that dim V = dim V ∗. 0 1 x1 B C B . C Choose a basis v1 : : : vn of V . For any x 2 V let B . C be its coordinates in the basis v1 : : : vn. @ A xn It is important to observe that each of the coordinates x1; : : : ; xn can be viewed as a linear function on V . Indeed, 9 1) the coordinates of the sum of two vectors are equal to the sum of the corresponding coordinates; 2) when a vector is multiplied by a number, its coordinates are being multiplied by the same number. ∗ ∗ Thus x1; : : : ; xn are vectors from the space V . Let us show now that they form a basis of V . ∗ Indeed, any linear function l 2 V can be written in the form l(x) = a1x1 + ::: + anxn which means ∗ that l is a linear combination of x1 : : : xn with coefficients a1; : : : ; an. Thus x1; : : : ; xn generate V . On the other hand, if a1x + ::: + anxn is the 0-function, then all the coefficients must be equal to 0; i.e. functions x1; : : : ; xn are linearly independent. Hence x1; : : : ; xn form a basis of V and therefore ∗ dim V = n = dim V: ∗ 1 The space V is called dual to V and the basis x1; : : : ; xn dual to v1 : : : vn. ∗ Exercise 1.2. Prove the converse: given any basis l1; : : : ; ln of V we can construct a dual basis w1; : : : ; wn of V so that the functions l1; : : : ; ln serve as coordinate functions for this basis. Recall that vector spaces of the same dimension are isomorphic. For instance, if we fix bases in both spaces, we can map vectors of the first basis into the corresponding vectors of the second basis, and extend this map by linearity to an isomorphism between the spaces. In particular, sending a ∗ ∗ basis S = fv1; : : : ; vng of a space V into the dual basis S := fx1; : : : ; xng of the dual space V ∗ we can establish an isomorphism iS : V ! V . However, this isomorphism is not canonical, i.e. it depends on the choice of the basis v1; : : : ; vn. If V is a Euclidean space, i.e. a space with a scalar product hx; yi, then this allows us to define another isomorphism V ! V ∗, different from the one described above. This isomorphism associates ∗ with a vector v 2 V a linear function lv(x) = hv; xi. We will denote the corresponding map V ! V by D. Thus we have D(v) = lv for any vector v 2 V . ∗ Exercise 1.3. Prove that D : V ! V is an isomorphism. Show that D = iS for any orthonormal basis S. 1It is sometimes customary to denote dual bases in V and V ∗ by the same letters but using lower indices for V ∗ 1 n and upper indices for V , e.g. v1; : : : ; vn and v ; : : : ; v . However, in these notes we do not follow this convention. 10 The isomorphism D is independent of a choice of an orthonormal basis, but is still not completely canonical: it depends on a choice of a scalar product. However, when talking about Euclidean spaces, this isomorphism is canonical. Remark 1.4. The definition of the dual space V ∗ also works in the infinite-dimensional case. Exercise 1.5. Show that both maps iS and D are injective in the infinite dimensional case as well. However, neither one of them is surjective if V is infinite-dimensional. 1.2 Canonical isomorphism between (V ∗)∗ and V The space (V ∗)∗, dual to the dual space V , is canonically isomorphic in the finite-dimensional case to V . The word canonically means that the isomorphism is \god-given", i.e. it is independent of any additional choices. When we write f(x) we usually mean that the function f is fixed but the argument x can vary. However, we can also take.