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@ 2f @y@x = 1 @f @x@y = 2x These functions are continuous and unequal, but by Clairaut's Theorem, if a function has continuous second partial derivatives then its mixed second partials must be equal) 2 TRUE or FALSE There is a function f R2!R such that @f @x = x and @f @y = y2 Solution TRUE (An example is f(x;y) = x2 2 y3 3 80 0 1 10 Miles 10 Kilometers 1 North 40 40 40 143 143 416 28 28 28 32 32 73 73 73 19 74 74 19 74 23 23 19 129 129 441 441 441 441 276 321 321 411 441 441 321 441 321PART 1 MODULE 2 Now suppose we merge all of the elements of A with all of the elements of B to form a single, larger set {Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia, The Godfather Part 2, The Wizard of Oz, To Kill A Mockingbird}

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3 / 3 (c) Yes Suppose (u, v) is a particular but arbitrarily chose element in the codomainThen, u and v are both real numbers Let x = 1v and y = (u1)/3Then, x and y are also both real numbers So, (x, y) is in the domainBy the definition of F, we have F( , )=F(1− , 1 3)=(3∙ 1 3 −1,1−(1− ))=( , ) (d) Yes, because F is both injective and surjectiveSuppose X = {a,b,c} and Y = {u,v,w,x} and suppose f X → Y is a function Then we can define f by simply specifying the images of a, b, and c, and to make it injective, we need to make sure none of the images are the same But this will simply be equal to the number of 3 permutations from the set Y, and thus there will be a total ofN) is a Cauchy sequence in F, then (x n) is Cauchy in Xand x n!xfor some x2Xsince Xis complete Then x2Fsince Fis closed, so Fis complete (b) Suppose that FˆXwhere Fis closed and Xis compact If (x n) is a sequence in F, then there is a subsequence (x n k) that converges to x2Xsince Xis compact Then x2F since F is closed, so F is compact

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Question f (X,τX) → (Y,τY) is continuous ⇔ ∀x0 ∈ X and any neighborhood V of f(x0), there is a neighborhood U of x0 such that f(U) ⊂ V Proof "⇒" Let x0 ∈ X f(x0) ∈ Y For each open set V containing f(x0), since f is continuous, f−1(V ) which containing x0 is open Then, there is a neighborhood U of x0 such thatF is perpendicular to the plane that contains both v and B (Review the vector or cross product!) The magnitude of the Lorentz force F is F = qvB sinθ, where θ is the smallest angle between the directions of the vectors v and B If v and B are parallel or antiparallel toF(x,y) = 8xy, 0 < x < y < 1 = 0 elsewhere (a) Verify that the f(x,y) given above is indeed a pdf Answer Z 1 0 Z y 0 8xydxdy = Z 1 0 4yx2y 0 dy = Z 1 0 4y3dy = y41 0 = 1 So, f(x,y) is a pdf (b) Find the marginal probability density of X, f 1(x) Answer f 1(x) = Z 1 x 8xydy = 4xy21 x = 4x(1−x2),0 < x < 1 (c) Find the marginal

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101 Constructing Topologies and Checking Continuity 121 Remark 1010 Let f X→Ybe a function Given a topology τY⊂2Y,the topology τX= f−1(τY) is the smallest topology on Xsuch that fis (τX,τY) continuous Similarly, if τXis a topology on Xthen τY= f∗τXis the largest topology on Ysuch that fis (τX,τY) continuous Definition 1011Continuous random variables We de ned the conditional density of X given Y to be fXjY (xjy) = fX;Y (x;y) fY (y) Then P(a X bjY = y) = Z b a fX;Y (xjy)dx Conditioning on Y = y is conditioning on an event with probability zero This is not de ned, so we make sense of the left side above by a limiting procedure P(a X bjY = y) = lim !0 P(a X bjjY yj < )This list of all twoletter combinations includes 1352 (2 × 26 2) of the possible 2704 (52 2) combinations of upper and lower case from the modern core Latin alphabetA twoletter combination in bold means that the link links straight to a Wikipedia article (not a disambiguation page) As specified at WikipediaDisambiguation#Combining_terms_on_disambiguation_pages,

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1 80 100 8 7 4 9 2 1 A H F B G ?Theorem 23 Let X;Y be topological spaces with f X!Y The following are equivalent fis continuous For all closed B Y, f 1(B) is closed For all x2X and each neighborhood V of f(x), there exists a neighborhood U of x such that f(U) V For all A X;f(A ) f(A) Theorem 24 There are various ways of constructing toplogical spacesY is a function and the topology on Y is generated by B;

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∂v ∂x = − ∂u ∂y = 0 (17) Thus, in Ω, f′(z) = ∂f ∂x = ∂u ∂x i ∂v ∂x = 0 0 = 0 (18) Thus f(z) is constant (b) Since Im(f) = constant, ∂v ∂x = 0, ∂v ∂y = 0 (19) By the CauchyRiemann equations, ∂u ∂x = ∂v ∂y = 0 (110) Thus in Ω, f′(z) = ∂f ∂x = ∂u ∂x i ∂v ∂x = 0 0 = 0 (111) 4Click on a word in the word list when you've found it This will gray it out and help you remember that you've found itMath 113 Homework 1 Solutions Solutions by Guanyang Wang, with edits by Tom Church Exercise 1 Show that 1 p 3i 2 is a cube root of 1 (meaning that its cube equals 1) Proof We can use the de nition of complex multiplication, we have

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3 3 9 8 7 5 i h i _ j _ q g u c i j h ^ m o 8 55 7 4 2 9 10 ;6 8 Points Suppose that f(x,y), a(u, v), and B(1,v) are functions with the following values (1,0) =1, ,(1,0) = 2Proof Suppose that the condition holds If > 0, then V = B (f(a)) is a neighborhood of f(a), so U = f 1(V) is a neighborhood of a Then B (a) ˆUfor some >0, which implies that f(B (a)) ˆB (f(a)), so fis continuous at a Conversely, if f is continuous at aand V is a neighborhood of f(a), then B (f(a)) ˆV for some >0 By continuity, there

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Map) f from A to B, denoted fA → B, is a subset of A × B such that ∀ xx ∈ A → ∃ yy ∈ B ∧ < x,y >∈ f and < x,y1 >∈ f∧ < x,y2 >∈ f→ y1 = y2 _____ Note f associates with each x in A one and only one y in B A is called the domain and B is called the codomainF (a;b) a < bg † The discrete topology on X is generated by ffxg x 2 Xg † Bases are NOT unique If ¿ is a topology, then ¿ = ¿ ¿ Theorem 18 If f X !Lake Mendota Lake Wingra Lake Monona Lake Waubesa Cherokee Marsh University Bay Upper Mud MLake Monona Bay Yahara River STA RKWEATHER CRE EK PATH C A P I T A L C I T Y

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Solution Let y be a limit point of fx f(x) = 0g So there is a sequence fy ngsuch that y n 2fx f(x) = 0gfor all nand lim n!1y n = y Since f is continuous, by Theorem 402 we have f(y) = lim n!1f(y n) = lim n!10 = 0 Hence y2fx f(x) = 0g, so fx f(x) = 0gcontains all of its limit points and is a closed subset of R 3A map f X!Y is said to be an open map is for every open set Uof X, the set f(U) is open in Y Show that ˇ y) 2B 1 B 2 U V, so that a2U By de nition of basis elements, B 1 is an open subset of X Thus for every x2U, there exists an open set B 1 of Xsuch that x2B 1 UA B C Ch D E F G H I J K L LL M N Ã O P Q R RR S T U V W X Y Read more about definition, alfabeto, letras, kilogramo, llamo and requires

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 I have downloaded php file of a website through path traversal technique, but when I opened the file with notepad and notepad I only get encrypted text IsX(a;b)(x a) f y(a;b)(y b) = 1 2(x 1) 2(y 1) (e) We can always use the theorem rf(a;b) ~uto compute the directional derivative at (a;b) in the direction of ~u SOLUTION False This formula only works if fis di erentiable at (a;b) (See Exercise 3 below) (f) Di erent parameterizations of the same curve result in identical tangent vectors at a 100 C in a D 1000 Y in a M 11 P in a F (S) T 12 M in a Y 12 S of the Z 13 is U F S 13 L in a B D 13 S in the U S F 15 M on a D M C 15 P in a R T 18 H on a G C 23 P of C in the H B 24 H in a D Ans 24 hours in a day 26 L of the A 27 B in the N T 29 D in F in a L Y 29 T in P F T 3 B M (S H T R) 3 W on a T 32 is the T in D F at which W F 365 D in

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