# Math 4153/5053 Spring 2016 Homework

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## Homework 1 (due on Friday Jan. 22)

3.5: 3.5.2, 3.5.7
3.6: 3.6.7, 3.6.8
4.4: 4.4.1, 4.4.4

3.6: 3.6.9
4.4: 4.4.6

Sorry but I just noticed that there is no class on monday (duh!) So I moved the 5.4 down to homework 2, and added a couple more problems to homework 1.

## Homework 2 (due on Friday Jan. 29)

5.4: 5.4.1, 5.4.5, 5.4.6
7.1: 7.1.1, 7.1.5, 7.1.6

5.4: 5.4.4 The first formula has a typo it should be $\frac{1}{1+t}-\frac{(-1)^{n+1}}{1+t} t^{n+1}$, the $t^{n+1}$ is missing
7.1: 7.1.7

## Homework 3 (due on Friday Feb. 5)

7.2: 7.2.1, 7.2.2, 7.2.7
7.3: 7.3.1, 7.3.2, 7.3.6

7.2: 7.2.11, 7.2.12
7.3: 7.3.9

## Homework 4 (due on Friday Feb. 12)

7.4: 7.4.3, 7.4.4, 7.4.10
7.5: 7.5.1, 7.5.5, 7.5.8

7.4: 7.4.9 (Hint: the metric does not need to be at all related to the standard metric)
7.5: 7.5.9

## Homework 5 (due on Friday Feb. 26)

8.1: 8.1.3, 8.1.5, 8.1.7, 8.1.8, 8.1.9, 8.1.12
8.2: 8.2.1, 8.2.3

7.6: 7.6.6
8.1: 8.1.14
8.2: 8.2.5

## Homework 6 (due on Monday Mar. 7 Wednesday Mar. 9)

8.2: 8.2.2, 8.2.6, 8.2.7, 8.2.8
8.3: 8.3.2, 8.3.3, 8.3.7
E1) Also compute the derivative of $f(x,y) = (1,x^2+y^2)$ at the point $(1,2)$.

8.2: 8.2.12, 8.2.11

## Homework 7 (due on Friday Mar. 25)

8.4: 8.4.1, 8.4.2, 8.4.3 (assume $f$ is differentiable, also on part b, it is same as part a, show it for all $(x,y) \in C(0,1)$)
8.5: 8.5.1, 8.5.2, 8.5.3 (here the function should be $f \colon {\mathbb{R}}^2 \to {\mathbb{R}}^2 \setminus \{ 0 \}$, similarly in part c, $(a,b)$ should not be $(0,0)$), 8.5.5 (there is a missing hypothesis, and another typo, let me just rewrite the problem here:

8.5.5: Consider $z^2 + xz + y =0$ in ${\mathbb{R}}^3$. Find an equation $D(x,y)=0$, such that if $D(x_0,y_0) \not= 0$ and $z^2+x_0z+y_0 = 0$ for some $z \in {\mathbb{R}}$, then for points near $(x_0,y_0)$ there exist exactly two distinct continuously differentiable functions $r_1(x,y)$ and $r_2(x,y)$ such that $z=r_1(x,y)$ and $z=r_2(x,y)$ solve $z^2 + xz + y =0$. Do you recognize the expression $D$ from algebra?

8.4: 8.4.5
8.5: 8.5.4, 8.5.7

## Homework 8 (due on Wednesday Apr. 6 Friday Apr. 8)

Make sure to use the new version of the notes (version 2 of chapter 8)

8.6: 8.6.2, 8.6.3
9.1: 9.1.2, 9.1.3, 9.1.4
9.2: 9.2.1, 9.2.3

8.6: 8.6.4
9.2: 9.2.6

## Homework 9 (due on Friday Apr. 15)

Make sure to use the new version of the notes (namely version 3 of chapter 9).

9.2: 9.2.10, 9.2.11, 9.2.14 (you don't have to do part e)
9.3: 9.3.3, 9.3.4, 9.3.5