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(ab)c = a(bc) for all a,b,c ∈ V A3 There exists an element of V, called the zero vector and denoted 0, such that a0 = 0a = a for all a ∈ V For any a ∈ V there exists an element of V, denoted −a, such that a(−a) = (−a)a = 0 A5 r(ab) = rarb for all r ∈ R and a,b ∈ V A6 (r s)a = rasa for all r,s ∈ R and6 If F and G are vector elds and r F = r G then F = G FALSE Fcan be Gplus any function whose curl is zero 7 The work done by a conservative force eld in moving a particle around a closed path is zero TRUE 8 There is a vector eld F such that r Fhx;y;ziPART 1 MODULE 2 SET INTERSECTION, SET UNION, SET COMPLEMENT SUMMARY The intersection of two sets denotes the elements that the sets have in common, or the "overlap" of the two sets S ∩ T = {xx∈ S and x∈ T} The union of two sets merges the two sets into one "larger" set S ∪ T = {xx ∈ S or x ∈ T}

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Line ƒAƒCƒRƒ" ‰Âˆ¤‚¢ ƒAƒCƒRƒ" ƒfƒBƒYƒj[-A(B C)=(AB)C Associative law for union A\(B \C)=(A\B)\C Associative law for intersection A(B \C)=(AB)\(AC) Distributive law for union A\(B C)=(A\B)(A\C) Distributive law for intersection De Morgan's Laws Let A and B be sets Then (AB)c = Ac \ (A\B)c = Ac 5 Write venn diagrams to represent each of the following sets (a) AHere r is the position vector from point O to any point on the line of action of F MOMENT OF A FORCE – VECTOR FORMULATION (continued) So, using the cross product, a r x F B) F x r C) r • F D) r * F CONCEPT QUIZ 1 If a force of magnitude F can be applied in four different 2D

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If every horizontal line in R2 intersects the graph of a function at most once, then the function is onetoone Examples Below is the graph of f R ⇥ R where f( x) = 2 There is a horizontal line that intersects this graph more than one point, so f is not onetoone 66 Inverse Functions Onetoone Suppose f A —* B is a functionCurves in R2 Three descriptions (1) Graph of a function f R !R (That is y= f(x)) Such curves must pass the vertical line test Example When we talk about the \curve" y= x2, we actually mean to say the graph of the function f(x) = x2That is, we mean the set⊇ Let y ∈ f(A) \ f(B) Then y ∈ f(A) and y /∈ f(B) Thus there exists x ∈ A with f(x) = y If x ∈ B, then y ∈ f(B), which contradicts the previous statement, so we must have x /∈ B This implies x ∈ A \ B, and hence y ∈ f(A\B) 1222 (c) Prove that f−1(f(A)) = A for all A ⊆ X iff f is injective Proof =⇒ Let x 1,x

Line z intersects plane S at point C Planes R and T intersect at line y Which two undefined geometric terms always describe figures with no beginning or end?B D G C F E q p r SOLUTION 1 Understand the Problem In the diagram, you are given three lines, p, q, and r, that intersect at point B You need to name two different planes that contain line r 2 Make a Plan The planes should contain two points on line r and one point not on line r 3 Solve the Problem Points D and F are on line r Point EAccess R d F a r Sid e O f B la n c, St o n e O a k P a r kw a y & Tu sca n y St o n e, St o n e O a k P kw y B t w n So n t er r a & O a k Cen t r e, St o n e O a k P a r kw a y a t Ma d iso n O a k, Th e 5 0 3 b u s lin e ( H u eb n er Cr o sst o w n ) h a s 2 r o u t es

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 It is usually best to see how we use these two facts to find a potential function in an example or two Example 2 Determine if the following vector fields are conservative and find a potential function for the vector field if it is conservative →F = (2x3y4 x)→i (2x4y3 y)→j F → = ( 2 x 3 y 4 x) i → ( 2 x 4 y 3 y) j →40 4 Differentiable Functions where A ⊂ R, then we can define the differentiability of f at any interior point c ∈ A since there is an open interval (a,b) ⊂ A with c ∈ (a,b) 411 Examples of derivatives Let us give a number of examples that illustrate differentiable andF i r e w o r ks ( b y C r e st o n S h o o t e r s / J& M D i sp l a ys) T H U R S D AY, J U LY 2 2 8 0 0 a m S w i n e S h o w – M i d A m e r i ca n A r e n a

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163 The Fundamental Theorem of Line Integrals One way to write the Fundamental Theorem of Calculus ( 721) is ∫ a b f ′ ( x) d x = f ( b) − f ( a) That is, to compute the integral of a derivative f ′ we need only compute the values of f at the endpoints Something similarSo A C and thus A ˘C 2Consider the set S = R where x ˘y if and only if x2 = y2 (a)Find all the numbers that are related to x = 1 Repeat this exercise for x = p 2 and x = 0 Solution (c)Draw a number line Draw a picture of the equivalence class of 1 Repeat this for x = 0, x = pFor all x in a,b Furthermore, there are numbers c and d in a,b such that f(c) = m andf(d) = M The intermediate value theorem Let f(x) be a continuous function in an interval a,b, andlet N be any number between f(a) andf(b) Thenthere is a number c in a,b such thatf(c) = N 3A preliminary result about the definite integral

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REAL ANALYSIS I HOMEWORK 3 2 then we have x= X n2N b n3 n X n2N c n 2 3 n X n2N b n3 n 1 2 X n2N c n3 n2C C=2 since (b n) and (c n) are sequences of 0's and 2'sAs xwas arbitrary above, we obtain 0;1 A B Hence m(A B) 1, but nd Bare closed sets of measure zero(A) S= 1=2 (B) S= 1=4 S= 1 (D) S= 3=4 Q 17 Let f R !R be an infinitely differentiable function such that for all a;b2R with aC f(x) (a,b) 7!(a,b) flip over the xaxis Examples • The graph of f(x)=x2 is a graph that we know how to draw It's drawn on page 59 We can use this graph that we know and the chart above to draw f(x)2, f(x) 2, 2f(x), 1 2f(x), and f(x) Or to write the previous five functions

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SOLUTIONS 1 a F (A,B,C) = A' B' C' A' B' C A B' C' A B' C A B C' A B C Distributive = A'B' (C' C) AB' (C' C) AB (C8 (a) Sketch the plane curve r(t) = (2sint)i(3cost)j (b) Find r0(t) (c) Sketch the position vector r(t) and the tangent vector r0(t) for t = ˇ=3 Solution (a) x = 2sint, y = 3cost =) sint = x=2;C FT ds= Z F(x(t);y(t);z(t)) r0(t) dt 5There are various notations for the vector line integral of a vector eld F = hf;g;hiover a smooth curve C Z C FT ds= Z F(r(t)) r0(t) dt= Z (fx0(t) gy0(t) hz0(t)) dt= Z C fdx gdy hdz= Z C F dr 6Properties of vector line integrals Let Cbe a smooth oriented curve and F and G are vector elds, then (a

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 P Q ^ _ L ` a b ` J U c da _ L T e f g J a _ h a c V _ a _ i j L T e f g J a _ P Q ^ _ L J _ k ^ _ M L ` a b J e a l mJ j nJ c J o h a c l i J db p ` q J rLet a,b,c, and d be numbers (a) You can break up a fraction from a sum in the numerator, but not in the denominator ab c = a c b c but a bc 6= a b a c (b) Cancellation of the c here requires that it appears in each additive term of the numerator and denominator cacb cd(a) f g, f g, and fgare continuous at x= a (b) If g(a) 6= 0, then f=gis continuous at x= a Theorem 3 If fis continuous at a, and if gis continuous at f(a), then f g is continuous at a Intermediate Value Theorem (IVT) Suppose fis continuous on a;b If kis any number between f(a) and f(b), then there exists a number c2a;b such that f(c

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If F(x,y) is a vector field, then its divergence is written as divF(x,y) = ∇·F(r) which in two dimensions is ∇·F(x,y) = (∂ ∂x i ∂ ∂y j)·(F 1(x,y)iF 2(x,y)j), = ∂F 1 ∂x ∂F 2 ∂y It is obtained by taking the scalar product of the vector operator ∇ appliedCo R N Line r Rd Ri v er d o un ty Trk G F R d 4th tS tree R d R o g e rsv i l d E u r e k a e R d W i o u w a s h 6 T r l C l a i r v i l l e f R d K o r o R d L m i ra D r A m it y R d S e v e n J o b n R d C lo ve rnd D WY C o Rd Y Br eakn ck R d Fremo t Rd Zan gl R d H mp R d H a z e l X S t ic k R d R i v r m o o r S R d C o R d P p WMEE 10 Fall 10 8 Problem 450 Using a case statement, develop and simulate a behavioral model of the 8421 to BCD code converter described in Problem 48

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Given ∠ABC is a right angle Prove ∠ABD is a right angle(a x b) x c ≠ a x (b x c) Cross Product Properties of the cross product If a, b, and c are vectors and c is a scalar, then 1 a x b =使用line備忘錄 絕不放過每個靈光乍現！使用line備忘錄即可中輕鬆保留並整理重要資訊。 有圖有真相！將照片加入備忘錄中也絕對不是問題！ 截取畫面輕鬆寫意 無論是可愛的寵物照片或是複雜的圖表，皆可輕鬆截取您的畫面存入備忘錄，或是與好友分享。

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2 29 if a ib=0 wherei= p −1, then a= b=0 30 if a ib= x iy,wherei= p −1, then a= xand b= y 31 The roots of the quadratic equationax2bxc=0;a6= 0 are −b p b2 −4ac 2a The solution set of the equation is (−b p 2a −b− p 2a where = discriminant = b2 −4ac 32Lines and Planes in R3 A line in R3 is determined by a point (a;b;c) on the line and a direction ~v that is parallel(1) to the line The set of points on this line is given by fhx;y;zi= ha;b;ci t~v;t 2Rg This represents that we start at the point (a;b;c) and add all scalar multiples of the vector ~vThe gradient vector of a differentiable function f D ⊂ R3 → R at any point (x,y,z) ∈ D is the vector ∇f = hf x,f y,f zi Notation I For two variable functions ∇f = f x i f y j I For two variable functions ∇f = f x i f y j f z k Theorem If f D ⊂ Rn → R, with n =

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And cost = y=3 So sin2 tcos2 t = x2 4 y2 9 = 1 The curve is an ellipse with center at (0;0) and the major axis along the yaxis x ( /3) (p5Suppose R 1 and R 2 are equivalence relations on a set A De ne the relation R on A by xRy if xR 1 y and xR 2 y Give the rst two steps of the proof that R is an equivalence relation by showing that R is re exive and symmetric Proof ReNotation We occasionally write M(n;F) for the ring of all n n matrices over the eld F, where F is either R or C For a real matrix Awe sometimes use that the adjoint is the transpose and write AT 1 Basics 1 At noon the minute and hour hands of a clock coincide a) What in the rst time, T

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B 1 5 0 a a a a a b d e g a a cy a b c y f f f aa c c g d d dd 1 3 yy y d d b b e b b g p p 13 elev r r s r p u 2 2 ils hold hs 2 j h t a g b b g g a a f f e b d b 40°46'n emas 3 1 7 0 0 3 x 1 5 0 7 elev 12 elev 40°47'n hs 1 b b a m b a m j b a d ils hold facilities & base maintenance aviation general nws terminal aviation general stationStandard Form ax by = c SlopeIntercept Form y = mx b PointSlope Form y y 1 = m(x x 1) where a;b;c are real numbers, m is the slope, b (di erent from the standard form b) is the yintercept, and (x 1;y 1) is any xed point on the line 5 Circles A circle, sometimes denoted J, is by de nition the set of all points X = (x;y) a xed distance r,J u n o r C r Sampson Cr B i g C a in C C r a in r F o u r m i l e C r B la c k C r H o g C r F r e e m a C r F e n w o o d C r L i t c iso Big Flowage Wausau Mission Bass L Lit Bass L i Pike L P e r u it r e l v l e l e r R L te B r B B Section Line Multilane Divided Wayside Rest Area T 59 18 94 Town of Dewey Town of Alban

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Restriction of a convex function to a line f Rn → R is convex if and only if the function g R → R, g(t) = f(xtv), domg = {t xtv ∈ domf} is convex (in t) for any x ∈ domf, v ∈ Rn can check convexity of f by checking convexity of functions of one variableA line and a plane What is the format of this proof?Section 04 Functions A function is a rule that assigns each input exactly one output We call the output the image of the input The set of all inputs for a function is called the domainThe set of all allowable outputs is called the codomainWe would write \(fX \to Y\) to describe a function with name \(f\text{,}\) domain \(X\) and codomain \(Y\text{}\)

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Of f(x) and f(y) Geometrically, the line segment connecting (x;f(x)) to (y;f(y)) must sit above the graph of f If f is continuous, then to ensure convexity it is enough to check the de nition with = 1 2 (or any other xed 2(0;1)) This is similar to the notion of midpoint convex sets that we saw earlier We say that fis concave if fis convex(c)Let f (a;b) !R be continuous Show that there exists a continuous function F a;b !R such that F(x) = f(x) for all x2(a;b) if and only if fis uniformly continuous Hint Given f, how should you de ne F(a) and F(b)?Intersection The intersection of two sets A and B, denoted A∩B, is the set of all elements found in both A AND B We define two sets to be "disjoint" if their intersection is the empty set (this means the two sets have no elements in common) ex) U={a,b,c,d,e,f,g} R={a} S={a,b} and T={b,d,e,f,g} a) R∪S ={a}U{a,b} = {a,b}

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Solved A Central Force Is Defined To Be A Force That Points Radially And Whose Magnitude Depends Only On R That Is F R A F R R Show That A Central Force Is Conservative

144 8 Di erentiable Functions is approximated near cby the linear function h7!f0(c)h Thus, f0(c) may be interpreted as a scaling factor by which a di erentiable function fshrinks or stretches lengths near c If jf0(c)j1, then f stretches the length of an intervalIf the function f R → R is differentiable, then f is continuous Remark I This Theorem is not true for the partial derivatives of a function f R2 → R I There exist functions f R2 → R such that f x(x 0,y 0) and f y (x 0,y 0) exist but f is not continuous at (x 0,yA parametrized curve C r(t) = (x(t);y(t)), with trunning from ato b Note since r = (x;y), we have dr = (dx;dy) De nition Theline integral of F along Cis de ned as Z C Fdr = Z C (M;N) (dx;dy) = Z C Mdx Ndy Comment The notation Fdr is common in physics and MdxNdyin thermodynamics (Though everyone uses both notations) We'll see what

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