341 lines
12 KiB
Markdown
341 lines
12 KiB
Markdown
---
|
|
title: Area
|
|
TARGET DECK: Obsidian::STEM
|
|
FILE TAGS: geometry::area
|
|
tags:
|
|
- calculus
|
|
- geometry
|
|
---
|
|
|
|
## Overview
|
|
|
|
**Area** is a **set function** mapping from a class of so-called **measurable** sets $\mathscr{M}$ into the real numbers.
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is a set function?
|
|
Back: A function mapping a collection of sets to real numbers.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333289-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the first set function Apostol introduces?
|
|
Back: Area.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333310-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What kind of mathematical entity is area?
|
|
Back: A function.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333313-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the domain of the area function?
|
|
Back: The class of measurable sets.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333316-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the codomain of the area function?
|
|
Back: The real numbers.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333319-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the "function signature" of the area function $a$?
|
|
Back: $a \colon \mathscr{M} \rightarrow \mathbb{R}$ where $\mathscr{M}$ is the class of measurable sets.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333321-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does Apostol mean by a measurable set?
|
|
Back: A set that can be ascribed an area.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333324-->
|
|
END%%
|
|
|
|
## Axioms
|
|
|
|
We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and a set function $a$, whose domain is $\mathscr{M}$, with the following six properties:
|
|
|
|
### Nonnegative Property
|
|
|
|
For each $S \in \mathscr{M}$, $a(S) \geq 0$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does the nonnegative property of area state?
|
|
Back: For every set $S \in \mathscr{M}$, $a(S) \geq 0$.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333327-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
State the nonnegative property of area in FOL.
|
|
Back: $\forall S \in \mathscr{M}, a(S) \geq 0$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333329-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $a$ is an area function and $S \in \mathscr{M}$. Why can't $a(S) = -1$?
|
|
Back: This violates the nonnegative property of $a$.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333332-->
|
|
END%%
|
|
|
|
### Additive Property
|
|
|
|
If $S, T \in \mathscr{M}$, then $S \cup T$ and $S \cap T$ are in $\mathscr{M}$. Also $$a(S \cup T) = a(S) + a(T) - a(S \cap T).$$
|
|
|
|
Notice this last formulation is a special case of [[inclusion-exclusion|PIE]].
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $S, T \in \mathscr{M}$. What set(s) does the additive property of area state are also in $\mathscr{M}$?
|
|
Back: $S \cup T$ and $S \cap T$.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333334-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $a$ is an area function and $S, T \in \mathscr{M}$. Why is $S \cup T \in \mathscr{M}$?
|
|
Back: The additive property of $a$ states it is.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333337-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $a$ is an area function and $S, T \in \mathscr{M}$. Why is $S \cap T \in \mathscr{M}$?
|
|
Back: The additive property of $a$ states it is.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333340-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $a$ is an area function and $S, T \in \mathscr{M}$. What does $a(S \cup T)$ evaluate to?
|
|
Back: $a(S) + a(T) - a(S \cap T)$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333343-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The additive property of area uses what combinatorial concept?
|
|
Back: The principle of inclusion/exclusion.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333346-->
|
|
END%%
|
|
|
|
### Difference Property
|
|
|
|
If $S, T \in \mathscr{M}$ such that $S \subseteq T$, then $T - S \in \mathscr{M}$ and $$a(T - S) = a(T) - a(S).$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $S, T \in \mathscr{M}$. What set(s) does the difference property of area state are also in $\mathscr{M}$?
|
|
Back: N/A.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333349-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $S, T \in \mathscr{M}$ such that $S \subseteq T$. What set(s) does the difference property of area state are also in $\mathscr{M}$?
|
|
Back: $T - S$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333353-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $S, T \in \mathscr{M}$ such that $T \subseteq S$. What set(s) does the difference property of area state are also in $\mathscr{M}$?
|
|
Back: $S - T$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333357-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $a$ is an area function and $S, T \in \mathscr{M}$ s.t. $S \subseteq T$. Why is $T - S \in \mathscr{M}$?
|
|
Back: The difference property of $a$ states it is.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333361-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $a$ is an area function and $S, T \in \mathscr{M}$ s.t. $S \subseteq T$. What does $a(T - S)$ evaluate to?
|
|
Back: $a(T) - a(S)$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333365-->
|
|
END%%
|
|
|
|
### Invariance Under Congruence
|
|
|
|
If $S \in \mathscr{M}$ and $T$ is congruent to $S$, then $T \in \mathscr{M}$ and $a(S) = a(T)$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does the invariance of congruence property of area state?
|
|
Back: If $S \in \mathscr{M}$ and $T$ is congruent to $S$, then $T \in \mathscr{M}$ and $a(S) = a(T)$.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333368-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $S \in \mathscr{M}$ and $T$ is congruent to $S$. What set(s) does the invariance of congruence property of area state are also in $\mathscr{M}$?
|
|
Back: $T$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333372-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $S \in \mathscr{M}$ and $T$ is congruent to $S$. What does $a(T)$ evaluate to?
|
|
Back: $a(S)$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333376-->
|
|
END%%
|
|
|
|
### Choice of Scale
|
|
|
|
Every rectangle $R$ is in $\mathscr{M}$. If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
|
|
|
|
%%ANKI
|
|
Basic
|
|
What shape is the choice of scale property of area concerned with?
|
|
Back: Rectangles.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333380-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What sets does the choice of scale property of area state are also in $\mathscr{M}$?
|
|
Back: All rectangles.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333384-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $R$ is a rectangle. What property of area claims $R$ is measurable?
|
|
Back: Choice of scale.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333388-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Suppose $R$ is a rectangle. What does $a(R)$ evaluate to?
|
|
Back: If $R$ has edges of length $h$ and $k$, $a(R) = hk$.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333391-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the area of a line segment?
|
|
Back: $0$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333395-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The line segment is considered a special case of what other shape?
|
|
Back: A rectangle.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333399-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How does a rectangle relate to a line segment?
|
|
Back: A line segment is a rectangle with one dimension equal to zero.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333403-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the area of a point?
|
|
Back: $0$
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333409-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The point is considered a special case of what other shape?
|
|
Back: A rectangle.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333414-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How does a rectangle relate to a point?
|
|
Back: A point is a rectangle with both dimensions equal to zero.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333419-->
|
|
END%%
|
|
|
|
### Exhaustion Property
|
|
|
|
Let $Q$ be a set. If there exists exactly one $c$ such that $a(S) \leq c \leq a(T)$ for all step regions $S$ and $T$ satisfying $S \subseteq Q \subseteq T$, then $Q \in \mathscr{M}$ and $a(Q) = c$.
|
|
|
|
%%ANKI
|
|
Cloze
|
|
Let $Q$ be a set. The {exhaustion} property of area states that if there exists {exactly one} $c$ such that {$a(S) \leq c \leq a(T)$} for all {step regions} $S$ and $T$ satisfying {$S \subseteq Q \subseteq T$}, then {$Q \in \mathscr{M}$} and {$a(Q) = c$}.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333427-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The exhaustion property of area considers sets bounded by what?
|
|
Back: Step regions.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333433-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Why* does the exhaustion property of area require existence of exactly one satisfying real number?
|
|
Back: Area is a function, i.e. single-valued.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333438-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which axiom of area is typically used to prove ordinate sets are measurable?
|
|
Back: The exhaustion property.
|
|
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
|
|
<!--ID: 1732381333444-->
|
|
END%%
|
|
|
|
## Bibliography
|
|
|
|
* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). |