Part Four: CHANGE
Chapter 17. Change We Can Believe In
Differential calculus can show you the best path from A to B, and Michael Jordan’s dunks help explain why.
The fourth part of this book is about change, and specifically it’s an introduction to the main concepts of calculus. “Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball.” (page 131). In this chapter, the derivatives are introduced. The integrals are going to be explained in the next chapter. “Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much it’s accumulating.” (page 132)
Chapter 18. It Slices, It Dices
The lasting legacy of integral calculus is a Veg-O-Matic view of the universe.
This chapter is about integrals and how they came to life. Integrals are about the accumulation of change, so in order to do this, we can pretend we make infinite slices of a certain curvilinear figure and add all the result. “The fundamental theorem of calculus says something similar for functions – if you integrate the derivative of a function from one point to another, you get the net change in the function between the two points.” (page 144) Calculus has served us to understand many things in nature if not most of them, such as gravity, the orbits of planets, the motion of particles or the flow of heat, electricity, air, and water.
Chapter 19. All about e
How many people should you date before settling down? Your grandmother knows – and so does the number e.
“A few numbers are such celebrities…The most famous is π…Close behind is i…Next on the A list? Say hello to e. Nicknamed for its breakout role in exponential growth, e is now the Zelig of advanced mathematics. It pops up everywhere, peeking out from the corners of the stage, teasing us by its presence in incongruous places.” (page 147)
The numerical value of e is 2.71828, which is obtained by the limiting number of the sum of (1 + 1/1 + 1/1x2 + 1/1x2x3 + 1/1x2x3x4 + …)
Some of the uses of e can be observed in compounded interest rate, population growth, radioactive decay, theatre person distribution, the uncertainties of romance, etc.
Chapter 20. Loves Me, Loves Me Not
Differential equations made sense of planetary motion. But the course of true love? Now that’s confusing.
This chapter is a very interesting one, since it unites two apparently very unrelated areas; love and calculus. I guess that’s one strange sentences you’ll see only a couple of times in your life. Here’s an intro of the chapter: “While the laws of the heart may elude us forever, the laws of inanimate things are now well understood. They take the form of differential equations, which describe how interlinked variables change from moment to moment, depending on their current values. As for what such equations have to do with romance – well, at the very least they might shed a little light on why, in the words of another poet, “the course of true love never did run smooth.” (page 155)
The example Strogatz uses to explain this is the love between Romeo and Juliet. Like any two lovers, these two have times where one is more attracted to the other depending on the other’s feeling of love and hate towards them. This result in a cycle that can be described by a differential equation.
Chapter 21. Step Into the Light
A light beam is a pas de deux of electrical and magnetic fields, and vector calculus is its choreographer.
The opening story of this chapter is about the Maxwell’s equations for electricity and magnetism (Maxwell discovered what light is). This is an example of vector calculus.
“Calculus, you’ll recall, is the mathematics of change. And so whatever vector calculus is, it must involve vectors that change, either from moment to moment of from place to place. In the latter case, one speaks of a “vector field””. (page 164)
Four fundamental laws that the Maxwell’s equations express:
1. Divergence of the electric field
2. The curl of the electric field
3. Divergence of the magnetic field
4. The curl of the magnetic field
With these in mind, Maxwell calculated the speed of these hypothetical waves and realized they travel at the same speed of light (calculated by the French physicist Hippolyte Fizeau), thus the prediction that light is an electromagnetic wave.
Differential calculus can show you the best path from A to B, and Michael Jordan’s dunks help explain why.
The fourth part of this book is about change, and specifically it’s an introduction to the main concepts of calculus. “Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball.” (page 131). In this chapter, the derivatives are introduced. The integrals are going to be explained in the next chapter. “Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much it’s accumulating.” (page 132)
Chapter 18. It Slices, It Dices
The lasting legacy of integral calculus is a Veg-O-Matic view of the universe.
This chapter is about integrals and how they came to life. Integrals are about the accumulation of change, so in order to do this, we can pretend we make infinite slices of a certain curvilinear figure and add all the result. “The fundamental theorem of calculus says something similar for functions – if you integrate the derivative of a function from one point to another, you get the net change in the function between the two points.” (page 144) Calculus has served us to understand many things in nature if not most of them, such as gravity, the orbits of planets, the motion of particles or the flow of heat, electricity, air, and water.
Chapter 19. All about e
How many people should you date before settling down? Your grandmother knows – and so does the number e.
“A few numbers are such celebrities…The most famous is π…Close behind is i…Next on the A list? Say hello to e. Nicknamed for its breakout role in exponential growth, e is now the Zelig of advanced mathematics. It pops up everywhere, peeking out from the corners of the stage, teasing us by its presence in incongruous places.” (page 147)
The numerical value of e is 2.71828, which is obtained by the limiting number of the sum of (1 + 1/1 + 1/1x2 + 1/1x2x3 + 1/1x2x3x4 + …)
Some of the uses of e can be observed in compounded interest rate, population growth, radioactive decay, theatre person distribution, the uncertainties of romance, etc.
Chapter 20. Loves Me, Loves Me Not
Differential equations made sense of planetary motion. But the course of true love? Now that’s confusing.
This chapter is a very interesting one, since it unites two apparently very unrelated areas; love and calculus. I guess that’s one strange sentences you’ll see only a couple of times in your life. Here’s an intro of the chapter: “While the laws of the heart may elude us forever, the laws of inanimate things are now well understood. They take the form of differential equations, which describe how interlinked variables change from moment to moment, depending on their current values. As for what such equations have to do with romance – well, at the very least they might shed a little light on why, in the words of another poet, “the course of true love never did run smooth.” (page 155)
The example Strogatz uses to explain this is the love between Romeo and Juliet. Like any two lovers, these two have times where one is more attracted to the other depending on the other’s feeling of love and hate towards them. This result in a cycle that can be described by a differential equation.
Chapter 21. Step Into the Light
A light beam is a pas de deux of electrical and magnetic fields, and vector calculus is its choreographer.
The opening story of this chapter is about the Maxwell’s equations for electricity and magnetism (Maxwell discovered what light is). This is an example of vector calculus.
“Calculus, you’ll recall, is the mathematics of change. And so whatever vector calculus is, it must involve vectors that change, either from moment to moment of from place to place. In the latter case, one speaks of a “vector field””. (page 164)
Four fundamental laws that the Maxwell’s equations express:
1. Divergence of the electric field
2. The curl of the electric field
3. Divergence of the magnetic field
4. The curl of the magnetic field
With these in mind, Maxwell calculated the speed of these hypothetical waves and realized they travel at the same speed of light (calculated by the French physicist Hippolyte Fizeau), thus the prediction that light is an electromagnetic wave.