Notation. - the jth partial derivative of the ith coordinate function of a vector valued function f

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1 5.1 M(m, n) - the m n matrices with real entries, also denoted M when m and n are clear from context. For A M(m, n) we write [A] ij for the (i, j)th entry of A. If the entries are given as a ij, we also write A = (a ij ) 1 i m. 1 j n L( n, m ) - the linear transformations from n to m, also denoted L when n and m are clear from context. T A for A M(m, n) - the linear transformation from n to n induced by the standard action of an m n matrix on an n-dimensional vector. T - the isomorphism from M(m, n) to L( n, m ) defined by A T A. 5.2 U - usually an open subset of n. p - usually a point in U. x j - the jth coordinate of a vector x (Df) p - the derivative of a function f : U m at a point p U (v) - the Taylor remainder (implicitly depending on a function f : U m and a point p U), defined by the formula f(p + v) = f(p) + (Df) p (v) + (v) f i - the ith coordinate function of a vector valued function f fi(p) x j - the jth partial derivative of the ith coordinate function of a vector valued function f Df - the (total) derivative or Fréchet derivative of f, usually thought of as a map Df : U L( n, m ). 5.3 D r f - the rth derivative of f, thought of as a map D r f : U L r ( n, m ). L r ( n, m ) - r-linear maps from n {{ n to m, isomorphic to L( nr, m ). r times f k f - for a function f and a sequence of functions (f k ) k N, with f k, f : U n, this notation means the sequence converges uniformly to f on U. f r - the C r norm for an function f : U m of class C r, defined by { f r = max sup f(p),..., sup (D r f) p. p U p U C r (U, m ) - the set of C r functions f : U m with f r < a rectangle in 2, usually given by = [a, b] [c, d]. Has area denoted = (b a)(d c). 1 c Brent Nelson 2018

2 G - a grid on a rectangle = [a, b] [c, d] given by G = P Q where P = {a = x 0 < x 1 < < x m = b is a partition of [a.b] and is a partition of [c, d] Q = {c = y 0 < y 1 < < y n = d ij - a subrectangle of determined by a grid G. If G is as above, then ij = [x i 1, x i ] [y j 1, y j ]. Note that ij = (x i x i 1 )(y j y j 1 ) = x i y j. S - sample points in determeind by a grid G. If G is as above, then S = {s ij : 1 i m, 1 j n where s ij is any point in ij. (f, G, S) - the iemann sum of f corresponding to the grid G and samples points S. It is the number (f, G, S) = f(s ij ) ij L(f, G) - the lower (Darboux) sum for a bounded function f with respect to a grid G. It is the number where m ij = inf{f(x, y): (x, y) ij. L(f, G) = m ij ij, U(f, G) - the upper (Darboux) sum for a bounded function f with respect to a grid G. It is the number where M ij = sup{f(x, y): (x, y) i,j. U(f, G) = M ij ij, f - the iemann integral of the function f over the rectangle f, f - the lower and upper integrals of f, respectively, defined as the f = sup L(f, G) G f = inf U(f, G). G For bounded functions f, these always exist and f f with equality if and only if f is iemann integrable. χ S for a subset S 2 - the characteristic function of S, defined by χ S (x, y) = 1 if (x, y) S and χ S (x, y) = 0 otherwise. S for a bounded subset S 2 - the area of the set S, which exists if S is iemann measurable. It is defined as S = χ S for any rectangle S. S f - for a iemann measurable set S 2 and f iemann integrable on some rectangle S, this integral is defined as fχ S Jac z (ϕ) - the Jacobian of C 1 function ϕ: U n (defined on an open subset U n ) at a point z, given by the number det((dϕ) z ). 2 c Brent Nelson 2018

3 5.8 C k ( n ) - the set of k-cells in n, which are smooth functions ϕ: [0, 1] k n. The unit k-cube [0, 1] k may also be denoted I k. ϕ I u - for ϕ C k( n ) and I = (i 1,..., i k ) {1,..., n k this is the partial Jacobian of ϕ defined at u [0, 1] k by ϕ i1 ϕ u ϕ I 1 (u) i1 u k (u) (u) = det. u..... Also denoted ϕ I u = (ϕi 1,...,ϕi k ) (u 1,...,u k ). ϕ ik u 1 (u) ϕ ik u k (u) dy I - a basic differential k-form on n. For I = {1,..., n k this is the functional on C k ( n ) defined by ϕ I dy I (ϕ) = [0,1] u, k and this number is called the I-shadow area of ϕ. fdy I - a simple differential k-form on n. For I = {1,..., n k and f : n smooth this is the functional on C k ( n ) defined by fdy I (ϕ) = f ϕ ϕ I [0,1] u. k Ω k ( n ) - the set of (general) differential k-forms on n. k-forms: ω = f I dy I. These are linear combinations of simple C k ( n ) - the set of all functionals on C k ( n ); thus C k ( n ) Ω k ( n ). ϕ ω - for ϕ C k( n ) and ω Ω k ( n ) this is equivalent notation for ω(ϕ). α β - the wedge product of two forms α and β. For α = I a Idy I Ω k ( n ) and β = J b Jdy J Ω l ( n ), this is the k + l-form on n given by I,J a Ib J dy IJ. dω - the exterior derivative of form ω. For ω = f I dy I, we have dω = d(f I ) dy I, where d(f I ) is the 1-form given by f I d(f I ) = dy j. y j j=1 T ϕ - the pushforward of ϕ C k ( n ) by a smooth map T : n m. It is the k-cell in m given by T ϕ := T ϕ C k ( m ). T ω - the pullback of ω Ω k ( n ) by a smooth map T : m n. It is the k-form on m given by [T ω](ϕ) = ω(t ϕ). More explicitly, if ω = f I dy I then T ω = I f I T dt I, where if I = (i 1,..., i k ) then dt I = (dt i1 ) (dt ik ). 3 c Brent Nelson 2018

4 5.9 ϕ - the boundary of ϕ C k ( n ). It is a (k 1)-chain in n : a formal linear combination of (k 1)-cells. It is given by k ϕ = ( 1) j+1 ( ϕ ι j,1 ϕ ι j,0), j=1 where ι j,1, ι j,0 : [0, 1] k 1 [0, 1] k are the maps defined by ι j,1 (u 1,..., u k 1 ) := (u 1,..., u j 1, 1, u j,..., u k 1 ) ι j,0 (u 1,..., u k 1 ) := (u 1,..., u j 1, 0, u j,..., u k 1 ). δ j ϕ - the jth dipole of ϕ C k ( n ) for j = 1,..., k. It is the (k 1)-chain given by δ j ϕ = ϕ ι j,1 ϕ ι j,0. Ω k (U) - the set of differential k-forms on U, for U n an open subset. These are linear combinations of simple k-forms on U: fdy I for f : U a smooth function and I a k-tuple in {1,..., n. C k (U) - the set of k-cells in U, which are smooth maps ϕ: [0, 1] k U. B k (U) - the set of exact k-forms on U; that is, the ω Ω k (U) on U such that ω = dα for some α Ω k 1 (U). Z k (U) - the set of closed k-forms on U; that is, the k-forms ω Ω k (U) such that dω = 0. H k (U) - the kth de ham cohomology group of U, which is the vector space quotient of Z k (U)/B k (U). 6.1 B - the volume of a box B n. m (A) - The outer measure of a set A d, which is defined as the quantity: { m (A) = inf B k : {B k k N is a countable covering of A by open boxes. k=1 6.2 M( d ) - the collection of Lebesgue measurable subsets of d, also denoted M, which are those subsets E d satisfying the Carathéodory condition: m (X) = m (X E) + m (X E c ) X d. m(e) - the Lebesgue measure of a Lebesgue measurable set E M, which is just its outer measure. G δ - a class of set: we say G is a G δ set if it is the countable (or finite) intersection of open sets. F σ - a class of set: we say F is an F σ set if it is the countable (or finite) union of closed sets. Measurable Functions - the set { {+, called the extended real line. By convention, we take 0 = 0. 4 c Brent Nelson 2018

5 a.e. - almost everywhere. We write this when a condition/property/equality holds except possibly on a measure set; e.g. f = g a.e. means that the functions f and g agree except possibly on a measure zero set. The Lebesgue Integral L + ( d ) - the space of Lebesgue measurable functions f : d [0, ]. Also denoted L +. f dm - the Lebesgue integral of a function f. Integrating -valued Functions f ± - the positive and negative parts of a Lebesgue measurable function f : d. Defined as f + := fχ f 1 ([0, ]) f := fχ f 1 ([,0)]) L 1 ( d, m) - the space of Lebesgue integrable functions f : d. Also denoted L 1 (m). Later redefined to be the set of equivalence classes of such functions under the equivalence relation f g f = g a.e. f 1 - the L 1 -norm for a function f L 1 (m). Defined as f 1 = f dm. 5 c Brent Nelson 2018