The Euclidean vector space Rd March 23, 2017 1 / 14 Points and operations of Rd Definition 3.1 ∗ d Let d 2 N . We denote the Cartesian product R × ::: × R by R , and we | {z } d times call its elements points or vectors. If x = (x1;:::; xd ), then the number xi is called the ith coordinate or component of the point or vector x. 2 3 Examples: R , R . Definition 3.2 d We define addition, multiplication by scalars and scalar product in R as d follows: if v = (v1;:::; vd ); w = (w1;:::; wd ) 2 R and λ 2 R, then v + w := (v1 + w1;:::; vn + wd ); λv := (λv1; : : : ; λvd ); hv; wi := v1w1 + ··· + vnwd : March 23, 2017 2 / 14 Rd as a Euclidean vector space Theorem 3.3 d The set R , together with the addition, scalar multiplication and scalar product defined above, is a Euclidean vector space. The vectors e1 := (1; 0;:::; 0); e2 := (0; 1; 0 :::; 0);:::; en := (0;:::; 0; 1): d form an orthonormal basis of R , which is called its canonical basis. March 23, 2017 3 / 14 The Euclidean norm Definition 3.4 d By the Euclidean norm or length of a vector v 2 R we mean the number kvk := phv; vi. Theorem 3.5 (properties of the norm) d If v; w 2 R and λ 2 R, then (i) kvk ≥ 0, and kvk = 0 () v = 0; (ii) kλvk = jλjkvk; (iii) kv + wk ≤ kvk + kwk. March 23, 2017 4 / 14 The Euclidean distance Definition 3.6 d By the distance of the points p; q 2 R we mean the number d(p; q) := kp − qk. Theorem 3.7 (properties of the distance) d If p; q; r 2 R , then (i) d(p; q) ≥ 0, ´es d(p; q) = 0 () p = q; (ii) d(p; q) = d(q; p) (symmetry); (iii) d(p; r) ≤ d(p; q) + d(q; r) (triangle inequality). March 23, 2017 5 / 14 Sequences in Rd Definition 3.8 d A sequence in R is a mapping d a: N ! R notation: the nth point in the sequence is an := a(n). We refer to the whole sequence as (an)n2N. Example an = (n; 2n). March 23, 2017 6 / 14 Limits of sequences in Rd Definition 3.9 d A sequence (an)n2N = (an1; an2;:::; and ) of points in R is convergent if the sequences of its coordinates (an1)n2N; (an2)n2N;:::; (and )n2N (as sequences of real numbers) are convergent. Then the limit of (an)n2N is the vector ( lim an1; lim an2;:::; lim and ): n!1 n!1 n!1 Example 1 1 1 n an = ( n ; 1 − n ; ( 2 ) ). March 23, 2017 7 / 14 Limits of sequences in Rd Lemma 3.10 d A sequence (an)n2N = (an1; an2;:::; and ) of points in R converges to a d point A 2 R if and only if the sequence of the distances d(an; A) converges to zero. March 23, 2017 8 / 14 Multivariable functions and vector-valued functions Definition 3.11 By a (real-valued) function of d variables we mean a function d f : E ⊂ R ! R. Example: f (x; y; z) = x2 sin(y + z3) Definition 3.12 By a vector-valued function of one variable we mean a function s f : E ! R . where E is a subset of R. Remark: such functions are also called curves. Example: f (t) = (3t; t2; sin(t)). Definition 3.13 By a vector-valued function of d variables we mean a function d s f : E ⊂ R ! R . Example: f (x; y) = (x=y; xy; y). March 23, 2017 9 / 14 Limits of multivariable functions Accumulation points and limits of functions are defined the same way as in R. Definition 3.14 d d Let E ⊂ R . A point x 2 R is an accumulation point of E, if there is a sequence in E n fxg that converges to x. Definition 3.15 d Let E ⊂ R and let f : E ! R. We say that the limit of f at a point 0 x0 2 E is A, if for any sequence (xn)n2N in E n fx0g converging to x0, we have lim f (xn) = A: n!1 March 23, 2017 10 / 14 Continuity of multivariable functions Definition 3.16 d Let E ⊂ R and let f : E ! R. We say that f is continuous at a point 0 x0 2 E , if for any sequence (xn)n2N in E converging to x0, we have lim f (xn) = f (x0): n!1 Example The usual operations of R, as two-variable functions f (x; y) = x + y, f (x; y) = xy, x f (x; y) = y , f (x; y) = xy . are continuous on their domain. March 23, 2017 11 / 14 Continuity of vector-valued functions d s A vector-valued function f : E ⊂ R ! R can be represented with its coordinate functions: 0 1 f1(x) B . C f (x) = @ . A : fs (x) Definition 3.17 A vector-valued function is continuous if its coordinate functions are continuous. Lemma 3.18 Composition of continuous functions are continuous. So, if d s s t f : E ⊂ R ! R and g : F ⊂ R ! R are continuous, and f (E) ⊂ (so the composition g ◦ f makes sense) then g ◦ f is continuous. March 23, 2017 12 / 14 Open sets in Rd Definition 3.19 d The open ball of radius r > 0 with center p 2 R is the set d Br (p) = fx 2 R j d(p; x) < rg: Definition 3.20 d Let A ⊂ R . Then x 2 A is an inner point of A if there is an open ball with center x contained in A. Definition 3.21 d A ⊂ R is open if all of its points are inner points. March 23, 2017 13 / 14 Example { Br (p) is open if r > 0. d { ;, R are open. 1 { In R = R, the open intervals are open. { A set of a single point is not open. 2 3 { A line in R ; R ;::: is not open. Theorem 3.22 The union of open sets is open. The intersection of finitely many open sets is open. March 23, 2017 14 / 14.