Reading Notes

MATH 334

---

Table of contents

1. Chapter 1: Setting the Stage
   1. 1.2: Subsets of the Euclidean Space
   2. 1.3: Limits and Continuity
   3. 1.5: Completeness
   4. 1.6: Compactness
   5. 1.7: Connectedness
2. Chapter 2: Differential Calculus
   1. 2.1: Differentiability in One Variable
   2. 2.2: Differentiability in Several Variables
   3. 2.7: Taylor’s Theorem
   4. 2.8: Critical Points
   5. 2.10: Vector-Valued Functions and Their Derivatives
3. Chapter 4: Integral Calculus
   1. 4.1: Integration on the Line
   2. 4.2: Integration in Higher Dimensions
   3. 4.3: Multiple Integrals and Iterated Integrals
   4. 4.4: Change of Variables for Multiple Integrals
   5. 4.5: Functions defined by integrals
   6. 4.6: Improper Integrals

---

Textbook link

Chapter 1: Setting the Stage

Page 13

1.2: Subsets of the Euclidean Space

• Sphere: set of all points whose distance from a fixed point $\mathbf{a}$ is a fixed number $r$
• Ball: set of all points whose distance from a fixed point $\mathbf{a}$ is less than a fixed number $r$

$B(r, \mathbf{a}) = \{ \mathbf{x} \in \mathbb{R}^n \Vert \mathbf{x} - \mathbf{a} \Vert < r \}$

• A set $S \subset \mathbb{R}^n$ is bounded if it is contained in some ball $B(r, \mathbf{a})$
• Complement of $S$: $S^C = \mathbb{R}^n \setminus S$
• A point $\mathbf{x} \in \mathbb{R}^n$ is an interior point of $S$ if all points sufficiently close to $\mathbf{x}$ are also in $S$. That is, $S$ contains a ball centered at $\mathbf{x}$.

$S^{\text{int}} = \{ \mathbf{x} \in S : (B, r \mathbf{X}) \subset S \text{ for some } r > 0 \}$

• A point $\mathbf{x} \in \mathbb{R}^n$ is a boundary point of $S$ if every ball centered at $\mathbf{x}$ contains points in $S$ and points not in $S$.

$\partial S = \{ \mathbf{x} \in \mathbb{R}^n : \text{ every ball centered at } \mathbf{x} \text{ contains points in } S \text{ and points not in } S \}$

• $S$ is open if it contains none of its boundary points
• $S$ is closed if it contains all of its boundary points
• Closure of $S$ is the union of $S$ and its boundary points, denoted $\overline{S} = S \cup \partial S$.
• Neighborhood of a point $\mathbf{x} \in \mathbb{R}^n$ is a set of which $\mathbf{x}$ is an interior point
• The boundary points of $S$ are the same as the boundary points of $S^C$.
• If $\mathbf{x}$ is either an interior point of $S$ nor an interior point of $S^C$, it must be a boundary point of $S$.
• Proposition 1.4. Suppose $S \subset \mathbb{R}^n$.
  • $S$ is open $\iff$ every point of $S$ is an interior points
  • $S$ is closed $\iff$ $S^C$ is open
• Sets defined by strict inequalities are open; sets defined by equalities or weak inequalities are closed.

1.3: Limits and Continuity

• $\mathbb{C}$ can be regarded as $\mathbb{R}^2$.
• $f$ is a real-valued function defined across $\mathbb{R}^n$.
• $\lim_{x\to\mathbf{a}} f(\mathbf{x}) = L$ is the limit of $f(\mathbf{x})$ as $\mathbf{x}$ approaches $\mathbf{a}$.

$\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = L \text{ means }\forall \epsilon_{\epsilon \in \mathbb{R}^+} \exists \delta_{\delta \in \mathbb{R}^+} : \vert f(\mathbf{x} - L \vert < \epsilon \text{ whenever } 0 < \vert \mathbb{x} - \mathbb{a} \vert < \delta$

• In general, consider function $f$ which are only defined on a subset $S$ of $\mathbb{R}^n$ and points $\mathbf{a}$ that lie in the closure of $S$.

$\lim_{\mathbf{x} \to \mathbf{a}, \mathbf{x} \in S} f(\mathbf{x})$

• $f$ is continuous at $\mathbf{a}$ if $\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = f(\mathbf{a})$
• If $f$ is continuous at every point of $U \subset \mathbb{R}^n$, then $f$ is continuous on $U$.

$\forall \epsilon_{\epsilon \in \mathbb{R}^+} \forall \mathbf{a}_{\in U} \exists \delta_{\delta \in \mathbb{R}^+} : \vert f(\mathbf{x}) - f(\mathbf{a}) \vert < \epsilon \text{ whenever } \vert \mathbf{x} - \mathbf{a} \vert < \delta$

• Limits on vector-valued functions: $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^m$

$\lim_{\mathbf{x} \to \mathbf{a}} \mathbf{f}(\mathbf{x}) = L \iff \lim_{\mathbf{x} \to \mathbf{a}} f_j(\mathbf{x}) = L_j \text{ for } j = 1, ..., m$

• Limits are tricky in higher dimensions because there are many ways to approach a point.
• If $f$ is a continuous function, $\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = f(\mathbf{a})$
• Most functions are built up from continuous functions of one variable using artihemtic operations and composition, which all preserve unity except for division.

Theorem 1.9. Suppose $\mathbf{f} : \mathbb{R}^n \to \mathbf{R}^m$ is continuous on $U \subset \mathbf{R}^n$ and $\mathbf{g} : \mathbb{R}^m \to \mathbb{R}^k$ is continuous on $\mathbf{f} (U) \subset \mathbf{R}^m$. Then the composite function $\mathbf{g} \circ \mathbf{f} : U \to \mathbb{R}^k$ is continuous on $U$.

Theorem 1.10. Let $f_1(x,y) = x+y, f_2(x, y) = xy,$ and $g(x)=1/x$. Then $f_1$ and $f_2$ are continuous on $\mathbb{R}^2$ and $g$ is continuous on $\mathbb{R} \setminus \{0\}$.

Corollary 1.11. The function $f_3(x, y) = x-y$ is continuous on $\mathbb{R}^2$, and the function $f_4(x, y) = x / y$ is continuous on $\{(x, y) : y \neq 0 \}$.

Corollary 1.12. The sum, product, or difference of two continuosu functions is continuous; the quotient of two continuous functions is continuous on the set where the denominator is nonzero.

Theorem 1.13. Suppose $\mathbf{f} : \mathbb{R}^n \to \mathbf{R}^k$ is continuous and $u \subset \mathbb{R}^k$. Let $S = \{ \mathbf{x} \in \mathbb{R}^n : \mathbf{f}(\mathbf{x}) \in U\}$. Then $S$ is open if $U$ is open and $S$ is closed if $U$ is closed.

1.5: Completeness

• The essential properties of the real number system which underlies all of calculus: $\mathbb{R}$ is a complete ordered field.
  • Field: operations of addition, subtraction, multiplication, and division are defined and subjec to usual laws of arithmetic.
  • Ordered field: field with the binary relation and antisymmetric
  • Completeness: there are no ‘holes’ in the real number line
• If $S$ is a subset of $\mathbb{R}$, an upper bound for $S$ is a number $u$ such that $x \le u$ for all $x \in S$, and a lower bound for $S$ is a number $l$ such that $l \le x$ for all $x \in S$.

The Completeness Axiom. Let $S$ be a nonempty set of real numbers. If $S$ has an upper bound, then $S$ has a least upper bound, the supremum of $S$, denoted $\sup S$. If $S$ has a lower bound, then $S$ has a greatest lower bound, the infimum of $S$, denoted $\inf S$. Examples:

• $S = (0, 1]$. $\sup S = 1$, $\inf S = 0$.
• $S = \{ 1, 1/2, 1/3, 1/4, ... \}$. $\sup S = 1$, $\inf S = 0$.
• $S = \{ 1, 2, 3, 4, ... \}$. $\sup S = \infty$, $\inf S = 1$.

If $S$ has an upper bound, the number $a = \sup S$ is the unique number such that

• $x \le a$ for all $x \in S$ ($a$ is an upper bound)
• For every $\epsilon > 0$, there exists $x \in S$ with $x > a - \epsilon$ (there is no smaller upper bound)
• Completeness of the real number system is important in establishing the convergence of numerical sequences
  • A sequence is bounded if its range is bounded
  • A sequence is increasing if $x_{n+1} \ge x_n$ for all $n$
  • A sequence is decreasing if $x_{n+1} \le x_n$ for all $n$
  • A sequence is monotonic if it is either increasing or decreasing

Theorem 1.16 (Monotone Sequence Theorem). Every bounded monotone sequence in $\mathbb{R}$ is convergent. The limit of an increasing/decreasing sequence is the supremum/infimum of its set of values.

Theorem 1.17 (Nested Interval Theorem). Let $I_1 = [a_1, b_1], I_2 = [a_2, b_2], ...$ be a squence of closed, bounded intervals in $\mathbb{R}$. Suppose that $I_1 \supset I_2 \supset I_3 \supset ...$ and the length $b_k - a_k$ of $I_k$ approaches 0 as $k \to \infty$. Then there is eactly one point contained in all of the intervals $I_k$. The intersection $\bigcap_{k=1}^\infty I_k$ is nonempty (a single point).

• A subsequence of $\{x_k\}$ is a sequence specified by a one-to-one map $j \to k_j$ from the set of positive integers into itself, e.g. $k_j = 2j$$ selected even-numbered terms.

Theorem 1.18 (form of Bolzano-Weierstrass theorem). Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence.

• A sequence $\{\mathbf{x}_k\} \in \mathbb{R}^n$ is a Cauchy sequence if $\mathbf{x}_k - \mathbf{x}_j \to 0$ as $k, j \to \infty$; that is, if for every $\epsilon > 0$ there exists an integer $K$ such that $\vert \mathbf{x}_k - \mathbf{x}_j \vert < \epsilon$ whenever $k > K$ and $j > K$.

Theorem 1.20. A sequence $\{ \mathbf{x}_k \} \in \mathbb{R}^n$ is convergent iff it is Cauchy.

1.6: Compactness

• A subset of $\mathbb{R}^n$ is compact if it is closed and bounded.
• Compactness is important because it yields existence theorems for limits.

Theorem 1.21 (Bolzano-Weierstrass Theorem). If $S$ is a subset of $\mathbb{R}^n$, $S$ is compact iff every sequence of points in $S$ has a convergent subsequence whose limit lies in $S$.

• Note: every finite subset of $\mathbb{R}^n$ is obviously compact
• Connection between compactness and continuity

Theorem 1.22. Continuous functions map compact sets to compact sets. Suppose that $S$ is a compact subset of $\mathbb{R}^n$ and $\mathbf{f} : S \to \mathbb{R}^m$ is continuous at every point of $S$. Then the set $\mathbf{f}(S) = \{ \mathbf{f}(\mathbf{x}) : \mathbf{x} \in S \}$ is also compact.

Theorem 1.23 (Extreme Value Theorem). Suppose $S \subset \mathbb{R}^n$ is compact and $f : S \to \mathbb{R}$ is continuous. Then $f$ has an absolute minimum value and an absolute maximum value on $S$; that is, there exist points $\mathbf{a}, \mathbf{b} \in S$ such that $f(\mathbf{a}) \le f(\mathbf{x}) \le f(\mathbf{b})$ for all $\mathbf{x} \in S$.

• $\mathcal{U}$ is a collection of subsets $\mathbb{R}^n$. It is a covering of $S$ if $S$ is contained within the union of the sets in $\mathcal{U}$.

Theorem 1.24 (Heine-Borel Theorem). If $S$ is a subset of $\mathbb{R}^n$, then $S$ is compact iff every open covering of $S$ has a finite subcovering.

• Metric spaces: general spaces equipped with a distance function.
• But Bolzano-Weierstrass and Heine-Borel may not be completely valid for other metric spaces.

1.7: Connectedness

• A set in $\mathbb{R}^n$ is connected if it is ``all in one piece’’
• A set is disconnected if it is the union of two nonempty sets, neither of which intersects the closure of the other.
• A set is connected if it is not disconnected.

Theorem 1.25. The connected subsets of $\mathbb{R}$ are precisely the intervals (open, half-open, or closed; bounded or unbounded).

Theorem 1.26. Cotninuous functions map connected sets to connected sets.

Corollary 1.27. (The Intermediate Value Theorem.) Suppose $f : S \to \mathbb{R}$ is continuous at every point of $S$ and $V \subset S$ is connected. If $\mathbf{a}, \mathbf{b} \in V$ and $f(\mathbf{a}) < t < f(\mathbf{b})$ or $f(\mathbf{b}) < t < f \mathbf{a})$, there is a point $\mathbf{c} \in V$ such that $f(\mathbf{c}) = t$.

• A set is arcwise/pathwise connected if any two points in $S$ can be joined by a continuous curve in $S$.

Theorem 1.28. If $S \subset \mathbb{R}^n$ is arcwise connected, then $S$ is connected.

Theorem 1.30. If $S \subset \mathbb{R}^n$ is open and connected, then $S$ is arcwise connected.

---

Chapter 2: Differential Calculus

2.1: Differentiability in One Variable

• A more useful notion of the derivative than in elemetnary calculus books
• $f : \mathbb{R} \to \mathbb{R}$ is differentiable at $x = a$ if it is approximately linear near $x = a$.
• That is, there exists a linear function $l(x) = mx + b$ satisfying $l(a) = f(a)$, i.e. $l(x) = f(a) + m(x-a)$.
  • The linear approximation $f(x) - l(x)$ must go to zero faster than $x - a$ as $x \to a$ (i.e. faster than $x$ approaches $a$), so we have

$\frac{f(x) - l(x)}{x - a} \to 0, x \to a$

• Let $h = x - a$. Then

$f(x) - l(x) = f(a + h) - f(a) - mh$

• We have the error function $E(h) = f(x) - l(x) = f(a + h) - f(a) - mh$, which is the difference between the function and its linear approximation.
• Formal definition. $f$ is a real-valued function on an open interval in $\mathbb{R}$ containing $a$. $f$ is differentiable at $a$ if there exists some number $m$ such that

$f(a + h) = f(a) + mh + E(h),$\lim_{h \to 0} \frac{E(h)}{h} = 0$$

• We can compute $m$ as follows into the standard form:

$m = \frac{f(a + h) - f(a) - E(h)}{h} = \frac{f(a + h) - f(a)}{h} - \frac{E(h)}{h} \to m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

• Differentiability at $a$ implies continuity at $a$.
• $E(h)$ is little-oh of $h$, i.e. it is of a smaller order of magnitude than $h$.
• $f(a + h)$ is the sum of a linear function of $h$ and an error term which is $o(h)$.
• We can work out standard differentiation rules from this definition.

The Product Rule. Suppose $f$ and $g$ are differentiable at $x = a$. Then $f(a + h) = f(a) + f'(a) h + E_1(h)$ and $g(a + h) = g(a) + g'(a) h + E_2(h)$, where $E_1(h)$ and $E_2(h)$ are both $o(h)$. Then we get

$f(a + h) g(a + h) = f(a) g(a) + [f'(a) g(a) + f(a) g'(a)]h + E_3(h)$

Left-hand derivative $f'_{-}(a)$ and right-hand derivative $f'_{+}(a)$:

$f'_{\pm}(a) = \lim_{h \to 0^{\pm}} \frac{f(a + h) - f(a)}{h}$

Mean Value Theorem. Definition of derivative: passing from local information given by values of $f(x)$ for $x$ near $a$ to the infinitesimal information $f'(a)$. How to go from infinitesimal information to local information? i.e. explain information about $f$ given $f'$?

Proposition 2.5. Suppose $f$ is defined on an open interval $I$ and $a \in I$. If $f$ has a local maximum or minimum at the point $a \in I$ and $f$ is differentiable at $a$, then $f'(a) = 0$.

Lemma 2.6 – Rolle’s Theorem. Suppose $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. If $f(a) = f(b)$, then there is at least one point $c \in (a, b)$ such that $f'(c) = 0$.

Theorem 2.7 – Mean Value Theorem I. Suppose $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. There is at least one point $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b-a}$

Theorem 2.8. Suppose $f$ is differentiable on the open interval $I$.

1. If $\vert f'(x) \vert \le C$ for all $x \in I$, then $\vert f(b) - f(a) \vert \le C \vert b - a \vert$ for all $a, b \in I$.
2. If $f'(x) = 0$ for all $x \in I$, then $f$ is constant on $I$.
3. If $f'(x) \ge 0$ for all $x \in I$, then $f$ is increasing on $I$.

Theorem 2.9 – Mean Value Theorem II. Suppose $f, g$ are continuous on $[a, b]$ and differentiable on $(a, b)$; and $g'(x) \neq 0$ for all $x \in (a, b)$. Then there is a point $c \in (a, b)$ such that $\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}$

Theorem 2.10 – L’Hopital’s Rule I. Suppose $f, g$ are differentiable functions on $(a, b)$ and $\lim_{x \to a^+} f(x) = \lim_{x \to a^+ g(x) = 0}$. If $g'$ never vanishes on $(a, b)$ and the limit $\lim_{x \to a^+} \frac{f'(x)}{g'(x)} = L$ exists, then $g$ never vanishes on $(a, b)$ and $\lim_{x \to a^+ \frac{f(x)}{g(x)}} = L$. The same result holds for the left-handed limit, the two-sided limit, and limits to infinity or negative infinity.

Theorem 2.11 – L’Hopital’s Rule II. Thoerem 2.10 remains value when the limits of $f(x)$ and $g(x)$ go to infinity.

Corollary 2.12 – Rates of Growth. For any $a > 0$:

$\lim_{x \to +\infty} \frac{x^a}{e^x} = \lim_{x \to +\infty} \frac{\log x}{x^a} = \lim_{x \to 0+} \frac{\log x}{x^-a} = 0$

Vector-valued functions.

• The derivative of $\mathbf{f} = (f_1, ..., f_n)$ is

$\mathbf{f}'(a) = \lim_{h \to 0} \frac{\mathbf{f}(a + h) - \mathbf{f}(a)}{h} = \left( \lim_{h \to 0} \frac{f_1(a + h) - f_1(a)}{h}, ..., \lim_{h \to 0} \frac{f_n(a + h) - f_n(a)}{h} \right)$

• The mean value theorem is not valid for vector-valued functions.

2.2: Differentiability in Several Variables

2.7: Taylor’s Theorem

• Taylor expansions in their finite form
• Taylor’s theorem – higher-order version of the tangent line approximation.
• A function $f$ of class $C^k$ on an interval $I$ containing the point $x = a$ is the sum of a certain polynomial of degree $k$ and a remainder term that vanishes more rapidly than $\vert x - a \vert^k$ as $x \to a$
• The polynomial $P = P_{a, k}$ of order $k$ such that $P^{(j)}(0) = f^{(j)} (a)$ for $0 \le j \le k$; the $k$th-order Taylor polynomial for $f$ based at $a$:

$P_{a, k}(h) = \sum_{j = 0}^k \frac{f^{(j)}(a)}{j!} h^j$

• The $k$-th order taylor remainder is given by

$R_{a, k}(h) = f(a + h) - P_{a, k}(h) = f(a + h) - \sum_{j = 0}^k \frac{f^{(j)(a)}}{j!} h^j$

• The Taylor polynomial is a good approximation of $f$ near $a$.

Theorem 2.55. – Taylor’s Theorem with Integral Remainder, I. Suppose that $f$ is of class $C^{k+1}$, with $k \ge 0$ on an interval $I \subset \mathbb{R}$, and $a \in I$. Then the remainder $R_{a, k}$ defined by 2.53 - 2.54 is given by

$R_{a, k}(h) = \frac{h^{k+1}}{k!} \int_0^1 (1 -t )^k f^{(k+1)} (a + th) dt$.

Theorem 2.58. – Taylor’s Theorem with Integral Remainder, II. Suppose that $f$ is of class $C^k$, $k \ge 1$ on an interval $I \subset \mathbb{R}$, and $a \in I$. Then the remainder $R_{a, k}$ is given by

$R_{a, k}(h) = \frac{h^k}{(k-1)!} \in_0^1 (1 - t)^{k-1} [ f^{(k)}(a + th) - f^{(k)(a)} ] dt$

Corollary 2.60. If $f$ is of class $C^k$ on $I$, then $R_{a, k}(h) / h^k \to 0$ as $h \to 0$.

• If $f$ is $C^k$ near $x = a$, we can write $f(x)$ as the sum of a $k$-th order polynomial

Corollary 2.61. If $f$ is of class $C^{k+1}$ on $I$ and $\vert f^{(k+1)} (x) \vert \le M$ for $x \in I$, then

$\vert R_{a, k} (h) \vert \le \frac{M}{(k+1)!} \vert h \vert^{k+1}, a + h \in I$

Lemma 2.62. suppose $g$ is $k + 1$ times differentiable on $[a, b]$. If $g(a) = g(b)$ and $g^{(j)(a) = 0} for 441 \le j \le k$, then there is a point $c \in (a, b)$ such that $g^{(k+1)}(c) = 0$

Theorem 2.63. – Taylor’s Theorem with Lagrange’s Remainder. Suppose $f$ is $k + 1$ times differentiable on an interval $I \in \mathbb{R}$, and $a \in I$. For each $h \in \mathbb{R}$ such that $a + h \in I$, there is a point $c$ between $0$ and $h$ such that

$$R_{a, k}(h) = f^{(k + 1)}(a + c) \frac{h^{k+1}}{(k + 1)!}

Proposition 2.65. The Taylor Polynomials of degree $k$ about $a = 0$ are:

• For $e^x$: $\sum_{0 \le j \le k} \frac{x^j}{j!}$
• For $\cos x$: $\sum_{0 \le j \le k/2} \frac{(-1)^j x^{2j}}{(2j)!}$
• For $\sin x$: $\sum_{0 \le j \le (k-1)/2} \frac{(-1)^j x^{2j + 1}}{(2j + 1)!}$

• For $(1 - x)^{-1}$: $\sum_{0 \le j \le k} x^j$

• Taylro polynomials can approximate complicated functions with easier computations
• Theoretically, importantly, the behavior of any function near some point is determined by the first nonvanishing term
• Suppsoe $f : \mathbb{R}^n \to \mathbb{R}$ is of class $C^k$ on a convex open set $S$. We can derive a Taylor expansion for $f(\mathbf{x})$ about a point $\mathbf{a} \in S$ by looking at the restriction of $f$ and the line joining $\mathbf{a}$ and $\mathbf{x}$.
• With $\mathbf{h} = \mathbf{x} - \mathbf{a}$ and $g(t) = f(\mathbf{a} + t(\mathbf{x} _ \mathbf{a})) = f(\mathbf{a} + t \mathbf{h})$; so $g'(t) = \mathbf{h} \cdot \nabla f(\mathbf{a} + t \mathbf{h})$

Theorem 2.68. – Taylor’s Theorem in Several Variables. Suppose $f : \mathbb{R}^n \to \mathbb{R}$ is a class $C^k$ on an open convex set $S$. If $\mathbf{a} \in S$ and $\mathbf{a} + \mathbf{h} \in S$. Then

$f(\mathbf{a} + \mathbf{h}) = \sum_{\vert \alpha \vert \le k} \frac{\partial^\alpha f(\mathbf{a})}{\alpha!} \mathbf{h}^\alpha + R_{\mathbf{a}, k}(\mathbf{h})$

Corollary 2.75. If $f$ is of class $C^k$ on $S$, then $R_{\mathbf{a}, k}(\mathbf{h}) / \vert \mathbf{h} \vert^k \to 0$ as $\mathbf{h} \to 0$.

Lemma 2.76. If $P(\mathbf{h})$ is a polynomial of degree $\le k$ that vanishes to order $> k$ as $\mathbf{h} \to 0$.

Theorem 2.77. Suppose $f$ is of class $C^{(k)}$ near $\mathbf{a}$. If $f(\mathbf{a} + \mathbf{h}) = Q(\mathbf{h}) + E(\mathbf{h})$ where $Q$ is a polynomial of degree $\le k$ and $E(\mathbf{h}) / \vert \mathbf{h} \vert^k \to 0$ as $\mathbf{h} \to \mathbf{0}$, then $Q$ is the Taylor polynomial $P_{\mathbf{a}, k}$.

2.8: Critical Points

• $\mathbf{a} \in S$ is a critical point for $f$ if $\nabla f(\mathbf{a}) = \mathbf{0}$.
• $f$ has a local maximum at $\mathbf{a}$ if $f(\mathbf{x}) \le f(\mathbf{a})$ for all $\mathbf{x}$ in some neighborhood of $\mathbf{a}$

Proposition 2.78. If $f$ has a local max or min at $\mathbf{a}$ and $f$ is differentiable at $\mathbf{a}$, then $\nabla f(\mathbf{a}) = \mathbf{0}$.

• How can we tell if a function has a local maximum or minimum?
• If $f$ is of class $C^2$, then $f$ has a local min at $a$ if $f''(a) > 0$.

Definition. Suppose $f$ is a real-valued function of class $C^2$ on some open set $S \subset \mathbb{R}^n$ and $f$ has a critical point at $\mathbf{a}$. One needs an $n \times n$ matrix $H$

$H = H(\mathbf{a}) = \begin{pmatrix} \partial_1^2 f(\mathbf{a}) & \partial_1 \partial_2 f(\mathbf{a}) & ... & \partial_1 \partial_n f(\mathbf{a}) \\ \partial_2 \partial_1 f(\mathbf{a}) & \partial_2^2 f(\mathbf{a}) & ... & \partial_2 \partial_n f(\mathbf{a}) \\ \vdots & \vdots & \ddots & \vdots \\ \partial_n \partial_1 f(\mathbf{a}) & \partial_n \partial_2 f(\mathbf{a}) & ... & \partial_n^2 f(\mathbf{a}) \end{pmatrix}$

• The Hessian is always a symmetric matrix
• Spectral theorem: every symmetric matrix has an orthonormal eigenbasis

Theorem 2.81. Suppose $f$ is of class $C^2$ at $\mathbf{a}$ and that $\nabla f(\mathbf{a}) = 0$, and let $H$ be the Hessian matrix. For $f$ to have a local min at $\mathbf{a}$, it is necessary for the eigenvalues of $H$ all to be nonnegative and sufficient for them all to be strictly positive. For $f$ to hav ea local max at $\mathbf{a}$, it is necessary for the eigenvalues of $H$ all to be nonpositive and sufficient for them all to be strictly negative.

• If two eigenvalues have opposite signs, $f$ has a saddle point.
• A critical point for which zero is an eigenvalue of the Hessian is degenerate (like Vivek)

Theorem 2.8. Suppose $f$ is of class $C^2$ on an open set in $\mathbb{R}^2$ containing the point $\mathbf{a}$, and suppose $\nabla f(\mathbf{a}) = \mathbf{0}$. Let $\alpha = \partial_1^2 f(\mathbf{a}), \beta = \partial_1 \partial_2 f(\mathbf{a}), \gamma = \partial_2^2 f(\mathbf{a})$.

1. If $\alpha \gamma - \beta^2 < 0$, $f$ has a saddle point at $\mathbf{a}$.
2. If $\alpha \gamma - \beta^2 > 0$ and $\alpha > 0$, $f$ has a local min at $\mathbf{a}$.
3. If $\alpha \gamma - \beta^2 > 0$ and $\alpha < 0$, $f$ has a local max at $\mathbf{a}$.
4. If $\alpha \gamma - \beta^2 = 0$, the test is inconclusive.

2.10: Vector-Valued Functions and Their Derivatives

• It can be useful to consider vector-valued functions – i.e. mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$, $n, m > 0 \in \mathbb{Z}$.

$\mathbf{f}(\mathbf{x}) = (f_1(\mathbf{x}), f_2(\mathbf{x}), ..., f_m(\mathbf{x}))$

• A mapping $\mathbf{f}$ from $S \in \mathbb{R}^n$ to $\mathbb{R}^m$ is differentiable at $\mathbf{a} \in S$ if $\exists$ an $m \times n$ matrix $L$ such that

$\lim_{\mathbf{h} \to \mathbf{0}} \frac{\vert \mathbf{f}(\mathbf{a} + \mathbf{h}) - \mathbf{f}(\mathbf{a}) - L \mathbf{h}}{\vert \mathbf{h} \vert} = 0$

• There can only be one such derivative, the Frechet derivative. – $D\mathbf{f}(\mathbf{a})$.

Proposition 2.85. An $\mathbb{R}^m$-valued function $\mathbf{f}$ is differentiable at $\mathbf{a}$ precisely when each of its components $f_1, ..., f_m$ is differentiable at $\mathbf{a}$. $D\mathbf{f}(\mathbf{a})$ is a matrix whose $j$th row is the row vecvtor $\nabla f_j(\mathbf{a})$.

Theorem 2.86. – Chain Rule III. Suppose $\mathbf{g} : \mathbb{R}^k \to \mathbb{R}^n$ is differentiable at $\mathbf{a} \in \mathbb{R}^k$ and $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^m$ is differentiable at $\mathbf{g}(\mathbf{a}) \in \mathbb{R}^n$. $\mathbb{H} = \mathbf{f} \circ \mathbf{g} : \mathbb{R}^k \to \mathbb{R}^m$ is differentiable at $\mathbf{a}$, and

$D \mathbf{H}(\mathbf{a}) = D\mathbf{f}(\mathbf{g}(\mathbf{a})) D\mathbf{g}(\mathbf{a})$

where the expression on the right is the product of the matrices $D \mathbf{f}(\mathbf{g}(\mathbf{a}))$ and $D \mathbf{g}(\mathbf{a})$.

Definition. Norm of a linear mapping is the smallest constant $C$ such that $$\vert A\mathbf{x} \vert \le C \vert x \vert4$

Theorem 2.88. Suppose $\mathbf{f}$ is a differentiable $\mathbb{R}^m$-valued function on an open convex set $S \subset \mathbb{R}^n$, and suppose that $\Vert D\mathbf{f}(\mathbf{x}) \Vert \le M$ for all $\mathbb{x} \in S$. Then

\[\vert \mathbf{f}(\mathbf{b}) - \mathbf{f}(\mathbf{a}) \vert \le M \vert \mathbf{b} - \mthbf{a} \vert , \forall \mathbf{a}, \mathbf{b} \in S\]

Definition. The Jacobian of a mapping $\mathbf{f}$ is a scalar-valued function on $S$ and is the determinant of $D\mathbf{f}$.

---

Chapter 4: Integral Calculus

4.1: Integration on the Line

• You can interpret $\int_a^b f(x) dx$ as the area of the region between the graph of $f$ and the $x$-axis over the interval $[a, b]$ – Riemann sums, etc.
• Partition $P$: subdivision of $[a, b]$ into non-overlapping subintervals
• $P'$ is a refinement of $P$ if $P \subset P'$
• Let $f$ be a bounded real-valued function on $[a, b]$. Given a partition $P = \{ x_0, ..., x_J \}$ of $[a, b]$, with $1 \le j \le J$, then we set

$m_j = \inf \{ f(x) : x_{j - 1} \le x \le x_j \}$ $M_j = \sup \{ f(x) : x_{j - 1} \le x \le x_j \}$

• Lower Riemann sum $s_P f = \sum_{1}^J m_j (x_j - x_{j - 1})$

• Upper Riemann sum $S_P f = \sum_{1}^J M_j (x_j - x_{j - 1})$

• Lemma 4.3. If $P'$ is a refinement of $P$, then $s_{P'} f \ge s_P f$ and $S_{p'} f \le S_P f$.

• Lemma 4.4. If $P$ and $Q$ are any partitions of $[a, b]$, then $s_P f \le S_Q f$

• Lower integral of $f$ on $[a, b]$: $L_{a}^b (f) = \sup_P s_P f$
• Upper integral of $f$ on $[a, b]$: $I_a^b(f) = \inf_P S_p f$
• (Supremum and infimum are taken over all partitions of $[a, b]$)
• Riemann vs. Lebesgue integral.
• Lemma 4.5. (important!): If $f$ is a bounded function on $[a, b]$, the following conditions are equivalent:
  • $f$ is integrable on $[a, b]$
  • For every $\epsilon > 0$ there is a partition $P$ of $[a, b]$ such that $S_P f - s_P f < \epsilon$
• For any partition $P$, we have

$s_P f \le \int_a^b f(x) dx \le S_P f$

• If $S_P f - s_P f < \epsilon$, $S_P f$ and $s_P f$ are both within $\epsilon$ of $\int_a^b f(x) dx$.
• Theorem 4.6.
  • Suppose $a < b < c$. If $f$ is integrable on $[a, b]$ and on $[b, c]$, then $f$ is integrable on $[a, c]$, and $\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$
  • If $f, g$ are integrable on $[a, b]$, then so is $f + g$, and $\int_a^b [f(x) + g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx$.
• Observe negation and ordering of bounds:

$\int_b^a f(x) dx = -\int_a^b f(x) dx$

• Theorem 4.9. Properties of functions integrable on $[a, b]$
  • If $c \in \mathbb{R}$, then $cf$ is integrable on $[a, b]$, and $f_a^b c f(x) dx = c \int_a^b f(x) dx$
  • If $[c, d] \in [a, b]$, then $f$ is integrable on $[c, d]$
  • If $g$ is integrable on $[a, b]$ and $f(x) \le g(x)$ for $x \in [a, b]$, then $f_a^b f(x) dx \le f_a^b g(x) dx$
• Theorem 4.10. If $f$ is bounded and monotone on $[a, b]$, then $f$ is integrable on $[a, b]$. Proof sketch:
  1. Suppose $f$ is increasing on $[a, b]$.
  2. Consider the partition $P_k$ of $[a, b]$ into $k$ equal subintervals
     • The difference between the lower and upper Riemann sums is $\frac{(b - a) [f(b) - f(a)]}{k}$
  3. We can make $k$ sufficiently large, so $f$ is integrable
• Theorem 4.11. If $f$ is continuous on $[a, b]$, then $f$ is integrable on $[a, b]$.
  • We know that $f$ is uniformly continuous on $[a, b]$
• Theorem 4.12. If $f$ is bounded on $[a, b]$ and continuous at all except finitely many points in $[a, b]$, then $f$ is integrable on $[a, b]$
• A set $Z \in \mathbb{R}$ has zero content if for any $\epsilon > 0$, there is a finite collection of intervals $I_1, ..., I_L$ such that $Z \in \bigcup_1^L I_l$ and the sum of lengths of the $I_l$’s is less than $\epsilon$.
• Theorem 4.13. If $f$ is bounded on $[a, b]$ and the set of points in $[a, b]$ at which $f$ is discontinuous has zero content, then $f$ is integrable on $[a, b]$
• Proposition 4.14. Suppose $f$ and $g$ are integrable on $[a, b]$ and $f(x) = g(x)$ for all except finitely many points $x \in [a, b]$. Then $f_a^b f(x) dx = f_a^b g(x) dx$
• Theorem 4.15. (The Fundamental Theorem of Calculus)
  1. Let $f$ be an integrable function on $[a, b]$. For $x \in [a, b]$, let $F(x) = \int_a^x f(t) dt$. Then $F$ is continuous on $[a, b]$; moreover, $F'(x)$ exists and equals $f(x)$ at every $x$ at which $f$ is continuous.
  2. Let $F$ be a continuous function on $[a, b]$ that is differentiable except perhaps at finitely many points in $[a, b]$, and let $f$ be a function on $[a, b]$ that agrees with $F'$ at all points where the latter is defined. If $f$ is integrable on $[a, b]$, then $f_a^b f(t) dt = F(b) - F(a)$
• Given an integrable function $f$ on $[a, b]$, for which partitions $P$ do the sums $s_P f$ and $S_P f$ give a good approximation of $\int_a^b f(x) dx$?
• Proposition 4.16. Suppose $f$ is integrable on $[a, b]$. Given $\epsilon > 0$, there exists $\delta > 0$ such that if $P = \{ x_0, ..., x_J\}$ is any partition of $[a, b]$ satisfying $\max_{1 \le j \le J} (x_j - x_{j-1}) < \delta$, the sums $s_P f$ and $S_P f$ differ from $\int_a^b f(x) dx$ by at most $\epsilon$.
• The definite integral – good to understand as a sum of infinitely many infinitesimal terms.

4.2: Integration in Higher Dimensions

• Rectangle: a set of the form $R = [a, b] \times [c, d]$
• Partition of $R$: a subdivision of $R$ into rectangles by partitioning both sides of $R$
• We define the previous terms as follows:

$m_{jk} = \inf \{ f(x, y) : (x, y) \in R_{jk}\}$ $M_{jk} = \sup \{ f(x, y) : (x, y) \in R_{jk} \}$ $\delta A_{jk} = (x_j - x_{j-1}) (y_k - y_{k-1})$ $s_P f = \sum_{j = 1}^J \sum_{k=1}^K m_{jk} \delta A_{jk}$ $S_P f = \sum_{j=1}^J \sum_{k=1}^K M_{jk} \delta A_{jk}$ $_LI_R (f) = \sup_P s_P f$ $_RI_R (f) = \inf_P S_P f$

• $f$ is Riemann integrable on $R$ if the lower and upper integrals coincide: $\int \int_R f dA = \int \int_R f(x, y) dx dy$

• How can we integrate over regions other than rectangles?
• Draw a large rectangle containing $S$, redefine $f$ to be zero outside of $S$, and integrate over $R$.
• Characteristic / indicator function of $S$: $\chi_S(\mathbf{x}) = 1 \text{ if } \mathbf{x} \in S, 0 \text{ otherwise}$
• $f$ is integrable on $S$ if $f_{\chi S}$ is integrable on $R$

$\int \int_S f dA = \int \int_R f_{\chi S} dA$

• Theorem 4.17.
  • If $f_1, f_2$ are integrable on the bounded set $S$ and $c_1, c_2 \in \mathbb{R}$, then $c_1 f_1 + c_2 f_2$ is integrable on $S$, and $\int \int_S [c_1 f_1 + c_2 f_2] dA = c_1 \int \int_S f_1 dA + c_2 \int \int_S f_2 dA$
  • Let $S_1, S_2$ be bounded sets with no points in common, and let $f$ be a bounded function. If $f$ is integrable on $S_1$ and on $S_2$, then $f$ is integrable on $S_1 \cup S_2$, in which case $\int \int_{S_1 \cup S_2} f dA = \int \int_{S_1} f dA + \int \int_{S_2} f dA$.
  • If $f$ and $g$ are integrable on $S$ and $f(\mathbf{x}) \le g(\mathbf{x})$ for $\mathbf{x} \in S$, then $\int \int S f dA \le \int \int_S g dA$
  • If $f$ is integrable on $S$, then so is $\vert f \vert$, and $\vert \int \int_S f dA \vert \le \int \int_S \vert f \vert dA$
• A set $Z \subset \mathbb{R}^2$ has zero content if for any $\epsilon > 0$, there is a finite collection of rectangles which cover $Z$ and the sum of their areas is less than $$\epsilon
• Theorem 4.18. Suppose $f$ is a bounded function on the rectangle $R$. If the set of points in $R$ at which $f$ is discontinuous has zero content, then $f$ is integrable on $R$.
• Smooth curves can in fact have zero content
• Proposition 4.19.
  • If $Z \subset \mathbb{R}^2$ has zero content and $U \subset Z$, then $U$ has zero content.
  • If $Z_1, ..., Z_k$ have zero content, then so does $\bigcup_1^k Z_j$.
  • If $\mathbf{f} : (a_0, b_0) \to \mathbb{R}^2$ is of class $C^1$, then $\mathbf{f}([a, b])$ has zero content whenever $a_0 < a < b < b_0$
• Lemma 4.20. The function $\chi_S$ is discontinuous at $\mathbf{x}$ iff $\mathbf{x}$ is in the boundary of $S$.
• We need the boundary of a set to have zero content. A set $S \subset \mathbb{R}^2$ is Jordan measurable if it is boudned and its boundary has zero content.
  • Any bounded set whose boundary is a finite union of pieces of smooth curves is measurable
• Theorem 4.21. Let $s$ be a measurable subset of $\mathbb{R}^2$. Suppose $f : \mathbb{R}^2 \to \mathbb{R}$ is bounded adn the set of points in $S$ at which $f$ is discontinuous has zero content. Then $f$ is integrable on $S$.
• Proposition 4.22. Suppose $Z \subset \mathbb{R}^2$ has zero content. If $f : \mathbb{R}^2 \to \mathbb{R}$ is boounded, then $f$ is integrable on $Z$ and $\int \int_Z f dA = 0$.
• Corollary 4.23.
  • Suppose $f$ is integrable on $S \subset \mathbb{R}^2$. If $g$ is bounded and $g(\mathbf{x}) = f(\mathbf{x})$ except for $\mathbf{x}$ in a set of zero content, then $g$ is integrable on $S$ and $\int \int_S g dA = \int \int_S f dA$
  • Suppose $f$ is integrable on $S, T$, and \(S \intersect T\) has zero content. Then $f$ is integrable on $S \cup T$. We have $\int \int_{S \cup T} f dA = \int \int_S f dA + \int \int_T f dA$
• If $S$ is any Jordan measurable set in a plane, its area is the integral over $SS$ of the constant function $f(\mathbf{x}) \equiv 1$.

$\text{area}(S) = \int \int_S 1 dA = \int \int \chi_S dA$

• Theory of $n$-dimensional integrals – need to use $n$-dimensional rectangular boxes in $\mathbb{R}^n$

• A bounded set $Z \subset \mathbb{R}^n$ has zero content iff for any $\epsilon > 0$ there are rectangular boxes $R_1, ..., R_k$ whose total volume is less than $\epsilon$, where the union of $R_j$ is a cover for $Z$.

• Theorem 4.24. (The Mean Value Theorem for Integrals.) Let $S$ be a compact, connected, measurable subset of $\mathbb{R}^n$, and let $f, g$ be continuous functions on $S$ with $g \ge 0$. Then there is a point $\mathbf{a} \in S$ such that $\int ... \int_S f(\mathbf{x}) g(\mathbf{x}) d^n \mathbf{x} = f(\mathbf{a}) \int ... \int_S g(\mathbf{x}) d^n \mathbf{x}$.
• Corollary 4.25. Let $S$ be a compact, connected, measurable subset of $\mathbb{R}^n$. Let $f$ be a continuous function on $S$. Then there is a point $\mathbf{a} \in S$ such that $\int ... \int_S f(\mathbf{x}) d^n \mathbf{x} = f(\mathbf{a}) \vert S \vert$ – this is the average of mean value of $f$ on $S$

4.3: Multiple Integrals and Iterated Integrals

• In the case of $n = 2$, we should have that

$\int \int_R f dA = \int_c^d \left[ \int_a^b f(x, y) dx \right] dy$

• Integrability of $f$ on $R$ does not need to imply the integrability of $f(x, y_0)$ as a function of $x$ for fixed $y_0$ on $[a, b]$
• A line segment is a set of zero content, so in fact it could be discontinuous at every point in it

Theorem 4.26. Let $R$ be a rectangle bounded by $[a, b]$ in $x$ and $[c, d]$ in $y$. Let $f$ be an integrable function in $R$. Suppose that the “slices” in each dimension are integrable. Then

$\int \int_R f dA = \int_c^d \left[ \int_a^b f(x, y) dx \right] dy = \int_a^b \left[ \int_c^d f(x, y) dy \right] dx$

• Iterated integrals
• An integral over an $n$-dimensional rectangular solid can be evaluated as an $n$-fold iterated integral
• Under suitable conditions for the integrand $f$, the order of integration in an iterated integral can be reversed.

4.4: Change of Variables for Multiple Integrals

• If $g$ is a one-to-one function of class $C^1$ on the interval $[a, b]$, then for a continous function $f$,

$\int_a^b f(g(u)) g'(u) du = \int_{g(a)}^{g(b)} f(x) dx$

• Sometimes have to compensate for the ‘right order’ of the bounds because $g$ might reverse them

$\int_I f(x) dx = \int_{g^{-1}(I)} f(g(u)) \vert g'(u) \vert du$

• Suppose $\mathbf{G}$ is a one-to-one transformation from a region $R$ to another region $S$. $R = \mathbf{G}^{-1}(S)$
• Area of any matrix $\mathbb{A}$ as a transformation on the unit matrix is the absolute value of the determinant of $\mathbb{A}$

$\int \int_S f(x, y) dx dy = \vert ad - bc \vert \int \int_{G^{-1}(S)} f(au + bv, cu + dv) du dv$

Theorem 4.37. Let $A$ be an invertible $n \times n$ matrix, and let $\mathbf{G}(\mathbf{u}) = A \mathbf{u}$ be the corresponding linear transformation of $\mathbb{R}^n$. Suppose $S$ is a measurable region in $\mathbb{R}^n$ and $f$ is an integrable function on $S$. Then $\mathbf{G}^{-1}(S) = \{ A^{-1} \mathbf{x} : \mathbf{x} \in S \}$ is measurable and $f \circ \mathbf{G}$ is integrable on $\mathbf{G}^{-1}(S)$, and

$\int ... \int_S f(\mathbf{x}) d^n \mathbf{x} = \vert \det A \vert \int ... \int_{G^{-1}(S)} f(A \mathbf{u}) d^n \mathbf{u}$

Theorem 4.41. Given open sets $U, V$ in $\mathbb{R}^n$, let $\mathbf{G} : U \to V$ be a one-to-one transformation of class $C^1$ whose derivative $D \mathbf{G}(\mathbf{u})$ is invertible for all $\mathbf{u} \in U$. Suppose that $T \subset U$ and $S \subset V$ are measurable sets such that $\bar{T} \subset U$ and $\mathbf{G}(T) = S$. If $f$ is an integrable function on $S$, then $f \circ \mathbf{G}$ is integrable on $T$, and

$\int ... \int_S f(\mathbf{x}) d^n \mathbf{x} = \int ... \int_T f(\mathbf{G}(\mathbf{u})) \vert \det D \mathbf{G}(\mathbf{u}) \vert d^n \mathbf{u}$

4.5: Functions defined by integrals

• We can form functions out of integrating variables. How do properties of $f$ relate to properties of $F$?
• To limits commute across integral operations? In general, the answer is no.

Theorem 4.46. Suppose $S, T$ are compact subsets of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, and $S$ is measurable. If \(f(x, y)$ is continuous on the set\)T \times S$, then the function$F$defined by$F(x) = \int … \int_S f(x, y) d^n y$is continuous on$T$$.

Theorem 4.47. Suppose $S \subset \mathbb{R}^n$ is compact and measurable, and $T \subset \mathbb{R}^m$ is open. If $f$ and $\nabla_x f$ are continuous on $T \times S$, then the function $F$ is of class $C^1$ on $T$, and

$\frac{\partial F}{\partial x_j} (\mathbf{x}) = \int ... \int_S \frac{\partial f}{partial x_j} (\mathbf{x}, y) d^n y$

Theorem 4.52. (Bounded Convergence Theorem.) Let $S$ be a measurable subset of $\mathbb{R}^n$ and $\{ f_j \}$ be a sequence of integrable functions on $S$. SUppose $f_j(y) \to f(y)$ for each $y \in S$, where $f$ is an integrable function on $S$, and there is a constant $C$ such that $\vert f_j(y) \vert \le C$ for all $j$ and all $y \in S$. Then,

$\lim_{j \to \infty} \int ... \int_S f_j (y) d^n y = \int ... \int_S f(y) d^n y$

4.6: Improper Integrals

• Type I proper integrals: $\int_a^\infty f(x) dx$, $f$ integrable over every finite subinterval $[a, b]$
• Type II proper integrals: $\int_a^b f(x) dx$, $f$ integrable over $[c, b]$ for every $c > a$ but unbounded near $$x = a$4

$\int_a^\infty f(x) dx = \lim_{b \to \infty} \int_a^b f(x) dx$

• The integral converges if the RHS limit exists; otherwise, the limit diverges
• Does $\int_a^\infty f(x) dx$ converge?

Lemma 4.54. If $\phi$ is a bounded increasing function on $[a, \infty)$, then $\lim_{x \to \infty} \phi(x)$ exists and equals $\sup \{ \phi(x) : x \ge a \}$.

• The integral $\int_a^\infty f(x) dx$ converges iff $\int_a^b f(x) dx$ remains bounded as $b \to \infty$

Theorem 4.55. Suppose that $0 \le f(x) \le g(x)$ for all sufficiently large $x$. If $\int_a^\infty g(x) dx$ converges, so does $\int_a^\infty f(x) dx$. If $\int_a^\infty f(x) dx$ diverges, so does $\int_a^\infty g(x) dx$.

Corollary 4.56. Suppose $f > 0$, $g > 0$, $f(x) / g(x) \to l$ as $x \to \infty$. If $0 < l < \infty$, then $\int_a^\infty f(x) dx$ and $\int_a^\infty g(x) dx$ are both convergent or both divergent. If $l = 0$, the convergence of $\int_a^\infty g(x) dx$ implies the convergence of $\int_a^\infty f(x) dx$. If $l = \infty$, the divergence of $\int_a^\infty g(x) dx$ implies the divergence of $\int_a^\infty f(x) dx$.

$\int_1^b \frac{dx}{x^p} = \frac{b^{1-p} - 1}{1 - p} \to \begin{cases} \infty & p < 1 \\ (p - 1)^{-1} & p > 1 \end{cases}$ $\int_1^b x^{-1} dx = \log b \to \infty$

• $\int_1^\infty x^{-p}$ converges iff $p > 1$

Corollary 4.57. If $0 \le f(x) \le Cx^{-p}$ for all sufficiently large $x$, where $p > 1$, then $\int_a^\infty f(x) dx$ converges. If $f(x) \ge cx^{-1} (c > 0)$ for all sufficiently large $x$, then $\int_a^\infty f(x) dx$ diverges.

• There are functions whose rate of decay at infinity is faster than $x^{-1}$ but slower than $x^{-p}$ for any $p > 1$, and their integrals can converge or diverge.

Theorem 4.58. If $\int_a^\infty \vert f(x) \vert dx$ converges, then $\int_a^\infty f(x) dx$ converges.

• integral is absolutely convergent if the same integral with absolute value of the function converges.
• The integral may converge even if the absolute integral does not converge because of cancellation effects in positive and negative values.

Type II integrals

$\int_a^b f(x) dx = \lim_{c > a, c \to a} \int_c^b f(x) dx$

• $\int_a^b f(x) dx$ converges if the RHS limit and diverges otherwise

Theorem 4.59. Suppose that $0 \le f(x) \le g(x)$ fro all $x$ sufficiently close to $a$. If $\int_a^b g(x) dx$ converges, so does $\int_a^b f(x) dx$. If $\int_a^b f(x) dx$ diverges, so does $\int_a^b g(x) dx$.

$\int_c^b (x - a)^{-p} dx = \frac{(x-a)^{1-p}}{1-p} \bigg \vert_c^b \to \begin{cases} (1 - p)^{-1} (b - a)^{1 - p} & p < 1 \\ \infty & p > 1 \end{cases}$ $\int_c^b (x - a)^{-1} dx = \log(x - a) \vert_c^b \to \infty$

Corollary 4.60. If $0 \le f(x) \le C(x - a)^{-p}$ for $x$ near $a$ where $p < 1$, then $\int_a^bf(x) dx$ converges. If $f(x) > c(x - a)^{-1} (c > 0)$ for $x$ near $a$, then $\int_a^b f(x) dx$ diverges.

---