Reading Notes

MATH 334

Table of contents

Chapter 1: Setting the Stage
Chapter 2: Differential Calculus
Chapter 4: Integral Calculus

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 $I$.

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 $a, b \in I$.
If $f'(x) = 0$ for all $x \in I$, then $f$ is constant on $I$.
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 $\alpha = \partial_1^2 f(\mathbf{a}), \beta = \partial_1 \partial_2 f(\mathbf{a}), \gamma = \partial_2^2 f(\mathbf{a})$.

If $\alpha \gamma - \beta^2 < 0$, $f$ has a saddle point at $\mathbf{a}$.
If $\alpha \gamma - \beta^2 > 0$ and $\alpha > 0$, $f$ has a local min at $\mathbf{a}$.
If $\alpha \gamma - \beta^2 > 0$ and $\alpha < 0$, $f$ has a local max at $\mathbf{a}$.
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.