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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 variable9 > 4 > 9 1 < 2 1 5 $ % & ' $ ( ( u u u ^ L Q R S 9 < 2 1 5 F6 i8 9 5 A D U > V ?Let X ˘R and fn(x) ˘n Then for any †¨0, choose any –¨0 and we have jfn(x)¡ fn(y)j˘jn¡nj˘0 ˙† whenever jx ¡ yj ˙ –, so {fn} is equicontinuous If it were uniformly bounded then there would be some M ¨ 0 such that jfn(x)j ˙ M for all n 2 N and x 2 R, but this is clearly not possible by taking n ¨M Problem 7 (Supp HW2 #5)

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Therefore the work done by the force field F(x,y) = (y2/x2) i−(2y/x) j in moving an object from P(1,1) to Q(4,−2) is Z C F·dr = Z C ∇f ·dr = f(4,−2)−f(1,1) = −1−(−1) = 0 9 Show that if the vector field F = P i Q j R k is conservative and P,Q,R have continuousBaby a b c d e f g h i j k l m n o p q r s t u v w x y z,#B_S_bhojpuri_new_songD(x,y) = (1 if x6= y 0 if x=y (X,d) is a metric space, usually called the trivial metric space Remark 22 This example tells us that any nonempty set can be made into a metric space Example 22 Let X = Rn, x = (x 1, , xn), y = (y 1, ,yn) Define the following metrics on Rn d 1(x,y) = jx 1 y 1jjx2 y2j jxn ynj d2(x,y) = q jx

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Curves 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 setS is the surface of the solid bounded by the paraboloid z = 4 − x2 − y2 and the xyplane Solution The divergence of F is9 l h f, l h _ ^ b g _ g g u l Z l u h m l _ l _ g ^ h \ Z l v l, l h u g Z a u \ Z l v k y ^ _ f h d j Z l b _, _ j b d Z g p _ \ l _ f h d j Z l

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F wt = mg N or kgm/s 2 Force (gravity) N or kgm/s 2 Force (Coulomb) F = k q 1 q 2 / (d 2) N or kgm/s 2 Force (magnetic) F m = BqvAssignment 8 (MATH 215, Q1) 1 Use the divergence theorem to find RR S F ndS (a) F(x,y,z) = x3 i 2xz2 j 3y2z k;S ij f x y z dS f P S of x y z x y z u u u v v v w w w w w w w w w w w w r i j k r i j k SURFACE INTEGRALS If the components are continuous and r u and r v are nonzero and nonparallel in the interior of D, it can be shown from Definition 1²even b Ca ³³f x y z ds f t t dt rr 1 ( ) uv SD

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