The Usefulness of Calculus
 |
Description
Book Introduction
A series that explains why math is necessary.
“The book that helped me overcome my fear of calculus.”
With richer examples and easier explanations!
The expanded and revised edition of the best-selling math series "The Usefulness of Calculus" has been published.
The second installment of the best-selling math series, "The Usefulness of Calculus," which helped teenagers overcome their fear of math and rekindled their interest in math in adults, has returned in a revised and expanded edition.
The world has changed dramatically in the past year.
The changes taking place in the world, such as drones ushering in a new era of delivery, space engineering attempting another leap forward with civilian space travel, and computer graphics constantly creating virtual worlds, are viewed and explained through a richer calculus.
Additionally, I have provided supplementary explanations for some of the pictures that were difficult to understand in the first edition.
I hope that even people who do not major in engineering can understand planimetry and the like.
The fact that calculus is difficult has not changed.
However, through this book, many students who had given up on mathematics became familiar with calculus and other mathematics.
Contrary to popular belief, the concept of calculus is quite understandable to most people.
Even if you can't solve calculus equations or create artificial intelligence programs, you can still use calculus.
It's the same principle as being able to use a computer even if you're not a computer expert, or being able to use a smartphone even if you don't know its structure.
I hope that this book, which richly explains how calculus is used, will help you overcome your fear of mathematics.
“The book that helped me overcome my fear of calculus.”
With richer examples and easier explanations!
The expanded and revised edition of the best-selling math series "The Usefulness of Calculus" has been published.
The second installment of the best-selling math series, "The Usefulness of Calculus," which helped teenagers overcome their fear of math and rekindled their interest in math in adults, has returned in a revised and expanded edition.
The world has changed dramatically in the past year.
The changes taking place in the world, such as drones ushering in a new era of delivery, space engineering attempting another leap forward with civilian space travel, and computer graphics constantly creating virtual worlds, are viewed and explained through a richer calculus.
Additionally, I have provided supplementary explanations for some of the pictures that were difficult to understand in the first edition.
I hope that even people who do not major in engineering can understand planimetry and the like.
The fact that calculus is difficult has not changed.
However, through this book, many students who had given up on mathematics became familiar with calculus and other mathematics.
Contrary to popular belief, the concept of calculus is quite understandable to most people.
Even if you can't solve calculus equations or create artificial intelligence programs, you can still use calculus.
It's the same principle as being able to use a computer even if you're not a computer expert, or being able to use a smartphone even if you don't know its structure.
I hope that this book, which richly explains how calculus is used, will help you overcome your fear of mathematics.
- You can preview some of the book's contents.
Preview
index
Entering
Calculus: The Language of Understanding Change and Predicting the Future · 005
I.
Calculate the Instantaneous Velocity of a Revolution: Acceleration·011
Nothing in the world is still. · 015│The relationship between the discovery of the century and acceleration. · 017│Why did physicist Newton invent differentiation? · 020│Differential calculus to catch road thugs. · 029│Leaping kangaroo drivers, differentiation that flies over them. · 034│The point of passage is not important: a new method of measuring acceleration. · 036│Space travel with the law of acceleration. · 040│The secret to SpaceX's success: rotational motion and differentiation. · 045
The Future Created by Calculus │ The Coming of a New Delivery Era: Control Drones · 050
II.
A Human Language for Representing Nature's Curves: Slope·059
History of Limits and Infinity · 063 │ Calculus: Explaining Numbers Without Limits · 068 │ Differentiation Through Slopes · 071 │ Curved Geometry Connecting Cities · 075 │ Types of Spirals That Shape the World · 080 │ Curved Designs that Capture the Beauty of Nature · 084
The Future Created by Calculus │Computer Painting that Draws Nature's Curves, CG·088
III.
How AI Learns Big Data: Optimization·093
Find a Realistic Compromise · 096 │ Conditions That Make Optimization Difficult: Multivariables · 102 │ The Link Between Amazon and AI: Optimization · 105 │ Models Reflecting Complex Reality: Artificial Neural Networks · 110 │ Optimization Methods for Training AI: Gradient Descent · 115 │ When Will We Meet an All-Powerful AI Assistant? · 123
The Future Shaped by Calculus │ Other Factors That Made AI Possible · 127
IV.
When you accumulate small movements, you see the axis of change: Geometry 131
Ancient Mathematics for Calculating the Area of a Circle · 134 │ Special Mission: Determine the Rate of Confirmed COVID-19 Cases · 141 │ Today's Measurement Methods Using Integral Calculus · 147 │ Medical Advances Driven by Integral Calculus: CT Scanning · 155
The Future Shaped by Calculus │The Secret to Massive Data Compression: The Fourier Transform · 162
V.
How Disney Movies Captivate the World: The Navier-Stokes Flow Equations · 171
The Equation That Best Represents Fluid Change · 175 │ Use Unsolved Equations! Computational Fluid Dynamics · 179 │ A Mathematician Who Utilized Flow Equations Wins an Oscar · 183
Useful Calculus Concepts │ Differential Equations Explaining Natural Phenomena · 188
Ⅵ.
What Future Are We Moving Toward?: The Predictive Power of Calculus · 197
Marginal Utility: The Moment of Most Satisfactory Outcomes · 199 │ To Which Segment Should Disaster Relief Funds Be Provided to Maximize Utility? · 201 │ How the Future Works · 203 │ Understanding Economic Trends with Calculus · 210 │ When Will My Future Assets Double? Approximation Methods · 219 │ Short-Term vs. Long-Term: Safe Investment Strategies Revealed by Calculus · 224
Useful Calculus Concepts │ Reading the "Curve of Life" with Calculus · 236
Calculus: The Language of Understanding Change and Predicting the Future · 005
I.
Calculate the Instantaneous Velocity of a Revolution: Acceleration·011
Nothing in the world is still. · 015│The relationship between the discovery of the century and acceleration. · 017│Why did physicist Newton invent differentiation? · 020│Differential calculus to catch road thugs. · 029│Leaping kangaroo drivers, differentiation that flies over them. · 034│The point of passage is not important: a new method of measuring acceleration. · 036│Space travel with the law of acceleration. · 040│The secret to SpaceX's success: rotational motion and differentiation. · 045
The Future Created by Calculus │ The Coming of a New Delivery Era: Control Drones · 050
II.
A Human Language for Representing Nature's Curves: Slope·059
History of Limits and Infinity · 063 │ Calculus: Explaining Numbers Without Limits · 068 │ Differentiation Through Slopes · 071 │ Curved Geometry Connecting Cities · 075 │ Types of Spirals That Shape the World · 080 │ Curved Designs that Capture the Beauty of Nature · 084
The Future Created by Calculus │Computer Painting that Draws Nature's Curves, CG·088
III.
How AI Learns Big Data: Optimization·093
Find a Realistic Compromise · 096 │ Conditions That Make Optimization Difficult: Multivariables · 102 │ The Link Between Amazon and AI: Optimization · 105 │ Models Reflecting Complex Reality: Artificial Neural Networks · 110 │ Optimization Methods for Training AI: Gradient Descent · 115 │ When Will We Meet an All-Powerful AI Assistant? · 123
The Future Shaped by Calculus │ Other Factors That Made AI Possible · 127
IV.
When you accumulate small movements, you see the axis of change: Geometry 131
Ancient Mathematics for Calculating the Area of a Circle · 134 │ Special Mission: Determine the Rate of Confirmed COVID-19 Cases · 141 │ Today's Measurement Methods Using Integral Calculus · 147 │ Medical Advances Driven by Integral Calculus: CT Scanning · 155
The Future Shaped by Calculus │The Secret to Massive Data Compression: The Fourier Transform · 162
V.
How Disney Movies Captivate the World: The Navier-Stokes Flow Equations · 171
The Equation That Best Represents Fluid Change · 175 │ Use Unsolved Equations! Computational Fluid Dynamics · 179 │ A Mathematician Who Utilized Flow Equations Wins an Oscar · 183
Useful Calculus Concepts │ Differential Equations Explaining Natural Phenomena · 188
Ⅵ.
What Future Are We Moving Toward?: The Predictive Power of Calculus · 197
Marginal Utility: The Moment of Most Satisfactory Outcomes · 199 │ To Which Segment Should Disaster Relief Funds Be Provided to Maximize Utility? · 201 │ How the Future Works · 203 │ Understanding Economic Trends with Calculus · 210 │ When Will My Future Assets Double? Approximation Methods · 219 │ Short-Term vs. Long-Term: Safe Investment Strategies Revealed by Calculus · 224
Useful Calculus Concepts │ Reading the "Curve of Life" with Calculus · 236
Detailed image
Into the book
Newton's dream was to understand universal gravitation through the movement of celestial bodies, that is, the change in position of celestial bodies over time in space called the universe.
Newton's remaining task was to understand this acceleration and describe it mathematically.
And the concept created to mathematically accurately express this acceleration is differentiation.
Differentiation is a mathematics that deals with motion that was born in modern times.
(Omitted) What is clear is that this mathematical concept, which Newton devised to achieve his dream, contributed to the scientific revolution, and 300 years later, it is being used in various fields through cutting-edge technology.
---「Ⅰ.
From "The Beginning of the Revolution: Calculate the Instantaneous Speed"
Unlike straight lines, I think curves are smooth.
But not all curves are smooth.
For a curve to be perfectly smooth and natural without any awkward parts, the first and second derivatives as well as higher order derivatives on the curve must be continuous.
(Omitted) When laying curved tracks, it is important to ensure that they can be smoothly connected to straight tracks.
Not only must the connecting tracks not be misaligned, but the slopes of the tangent lines must also be aligned in the same direction.
In other words, the function values must be continuous and the slopes must be the same.
---「Ⅱ.
From "Human Language for Realizing the Curves of Nature"
Oak barrels are bulged in the middle, so the amount of wine is not proportional to the depth of the barrel.
In the past, when trading wine, these facts were known.
Wine merchants used to measure the volume by inserting a long stick into an oak barrel and measuring the height at which the wine rose.
At this time, in order to avoid the trouble of integrating every time to measure the volume, the markings on the bar were set differently depending on the height.
The markings in the middle are marked more closely than those at the upper and lower ends.
---「Ⅳ.
From “When you collect small movements, you can see the axis of change”
When each grid value f(x,y) inside the body is given, the process of integrating the transmitted light and producing the Rf result is called the Radon transform, and the process of applying the Radon transform in reverse to extract the grid value f(x,y) is called the inverse Radon transform. CT can be said to be an algorithm that reconstructs three-dimensional spatial information inside the body by performing the inverse Radon transform on multiple two-dimensional sinograms that have been taken.
There is a huge amount of mathematical calculations involved here, involving integration.
---「Ⅳ.
From “When you collect small movements, you can see the axis of change”
Pixar's success was largely due to the mathematicians and computer scientists hired by Steve Jobs, as well as Disney's investment in producing feature-length animations.
Toy Story is the world's first feature-length theatrical animation film made entirely with CG.
At the time of its release, people were enthusiastic about this new visual that did not feel out of place at all in terms of action or characters.
It was all thanks to Pixar's mathematicians and computer scientists' 3D animation techniques and resolution control techniques that were designed to naturally create "moving" natural phenomena like snowflakes and tsunamis.
And behind all this manufacturing process, there is one differential equation.
---「Ⅴ.
From "How Disney Movies Captivated the World"
So how can we utilize equations whose theoretical solutions haven't yet been discovered? The answer is to use a computer to find approximate solutions.
As computers developed, methods were developed to interpret the dynamic movement of fluids using numbers rather than relying on mathematical formulas.
Among them, the numerical analysis of the NS equation using a computer is called computational fluid dynamics, CFD.
Computational fluid dynamics is widely used in various practical fields such as weather forecasting and aircraft design.
---「Ⅴ.
From "How Disney Movies Captivated the World"
When you're hungry, a slice of pizza can't be that delicious.
At this point, marginal utility reaches its maximum.
But with each additional slice of pizza you eat, your marginal utility decreases.
But even though marginal utility decreases, total utility continues to increase.
In other words, total utility is the integration of marginal utility, and marginal utility is the differentiation of total utility.
At first, total utility increases rapidly, but as marginal utility decreases, the rate of increase in total utility slows down.
---「Ⅵ.
From "What kind of future are we moving towards?"
One thing that can be explained by utilizing the price elasticity of supply is the rise in apartment prices in Seoul.
In a situation where demand for Seoul apartments not only exists but actually increases, the supply of Seoul apartments is inelastic, so apartment prices continue to rise.
In other words, the price elasticity of supply for Seoul apartments is very inelastic due to land limitations, so no matter how much the price rises, the supply cannot increase significantly.
Therefore, when demand increases, prices are directly affected.
Newton's remaining task was to understand this acceleration and describe it mathematically.
And the concept created to mathematically accurately express this acceleration is differentiation.
Differentiation is a mathematics that deals with motion that was born in modern times.
(Omitted) What is clear is that this mathematical concept, which Newton devised to achieve his dream, contributed to the scientific revolution, and 300 years later, it is being used in various fields through cutting-edge technology.
---「Ⅰ.
From "The Beginning of the Revolution: Calculate the Instantaneous Speed"
Unlike straight lines, I think curves are smooth.
But not all curves are smooth.
For a curve to be perfectly smooth and natural without any awkward parts, the first and second derivatives as well as higher order derivatives on the curve must be continuous.
(Omitted) When laying curved tracks, it is important to ensure that they can be smoothly connected to straight tracks.
Not only must the connecting tracks not be misaligned, but the slopes of the tangent lines must also be aligned in the same direction.
In other words, the function values must be continuous and the slopes must be the same.
---「Ⅱ.
From "Human Language for Realizing the Curves of Nature"
Oak barrels are bulged in the middle, so the amount of wine is not proportional to the depth of the barrel.
In the past, when trading wine, these facts were known.
Wine merchants used to measure the volume by inserting a long stick into an oak barrel and measuring the height at which the wine rose.
At this time, in order to avoid the trouble of integrating every time to measure the volume, the markings on the bar were set differently depending on the height.
The markings in the middle are marked more closely than those at the upper and lower ends.
---「Ⅳ.
From “When you collect small movements, you can see the axis of change”
When each grid value f(x,y) inside the body is given, the process of integrating the transmitted light and producing the Rf result is called the Radon transform, and the process of applying the Radon transform in reverse to extract the grid value f(x,y) is called the inverse Radon transform. CT can be said to be an algorithm that reconstructs three-dimensional spatial information inside the body by performing the inverse Radon transform on multiple two-dimensional sinograms that have been taken.
There is a huge amount of mathematical calculations involved here, involving integration.
---「Ⅳ.
From “When you collect small movements, you can see the axis of change”
Pixar's success was largely due to the mathematicians and computer scientists hired by Steve Jobs, as well as Disney's investment in producing feature-length animations.
Toy Story is the world's first feature-length theatrical animation film made entirely with CG.
At the time of its release, people were enthusiastic about this new visual that did not feel out of place at all in terms of action or characters.
It was all thanks to Pixar's mathematicians and computer scientists' 3D animation techniques and resolution control techniques that were designed to naturally create "moving" natural phenomena like snowflakes and tsunamis.
And behind all this manufacturing process, there is one differential equation.
---「Ⅴ.
From "How Disney Movies Captivated the World"
So how can we utilize equations whose theoretical solutions haven't yet been discovered? The answer is to use a computer to find approximate solutions.
As computers developed, methods were developed to interpret the dynamic movement of fluids using numbers rather than relying on mathematical formulas.
Among them, the numerical analysis of the NS equation using a computer is called computational fluid dynamics, CFD.
Computational fluid dynamics is widely used in various practical fields such as weather forecasting and aircraft design.
---「Ⅴ.
From "How Disney Movies Captivated the World"
When you're hungry, a slice of pizza can't be that delicious.
At this point, marginal utility reaches its maximum.
But with each additional slice of pizza you eat, your marginal utility decreases.
But even though marginal utility decreases, total utility continues to increase.
In other words, total utility is the integration of marginal utility, and marginal utility is the differentiation of total utility.
At first, total utility increases rapidly, but as marginal utility decreases, the rate of increase in total utility slows down.
---「Ⅵ.
From "What kind of future are we moving towards?"
One thing that can be explained by utilizing the price elasticity of supply is the rise in apartment prices in Seoul.
In a situation where demand for Seoul apartments not only exists but actually increases, the supply of Seoul apartments is inelastic, so apartment prices continue to rise.
In other words, the price elasticity of supply for Seoul apartments is very inelastic due to land limitations, so no matter how much the price rises, the supply cannot increase significantly.
Therefore, when demand increases, prices are directly affected.
---「Ⅵ.
From "What kind of future are we moving towards?"
From "What kind of future are we moving towards?"
Publisher's Review
A book that penetrates the essence of calculus.
Choi Young-gi, Professor of Mathematics Education at Seoul National University and author of "This Kind of Math Is My First Time"
If you integrate the past, you can see the present.
Differentiating the present reveals the future.
There is nothing in this world that does not change.
Not only the position and speed of the planets change, but people and time also change.
Calculus is the language that describes these changes in the world.
From the perspective of calculus, everything from the principles of advanced science and technology to natural phenomena and social changes is clearly revealed.
Differentiation allows us to capture the momentary changes and movements of the world, and integration allows us to understand the state in which small changes accumulate.
For example, the incidence rate of confirmed COVID-19 cases can be determined using calculus.
Daily confirmed cases are the combined amount, and cumulative confirmed cases are the combined result amount.
If you add up all the daily confirmed cases, you get the cumulative confirmed cases, and the change rate of the cumulative confirmed cases becomes the daily confirmed cases.
The number of daily confirmed cases fluctuates significantly from day to day, but the cumulative number of confirmed cases steadily increases.
The number of daily confirmed cases corresponds to the differential value indicating the rate of increase, and the cumulative number of confirmed cases corresponds to the integrated value of the daily increase.
In Korea, the number of daily confirmed COVID-19 cases suddenly increased and reached a peak, but by the end of March 2021, the number of daily confirmed cases remained high, but the rate of change gradually slowed.
Of course, the steady increase in the number of confirmed cases remains unchanged.
_Ⅵ.
When you collect small movements, you can see the axis of change.
The current cumulative number of confirmed cases can be determined through the integrated value, the resulting quantity, and the rate of confirmed cases tomorrow can be predicted through the differentiated value, the change (slope).
Unless a new variable emerges, it is possible to predict the incidence of confirmed cases even one month later.
The same goes for trends in climate change, the stock market, and apartment prices.
In other words, by integrating the past, we can understand the present, and by differentiating the present, we can predict the future.
Understanding calculus is reading change.
A new era opened by calculus, a future predicted by calculus.
A math textbook that fosters the ability to read change.
If humans had not understood calculus and been unable to properly utilize its usefulness, it would be difficult to imagine an era like ours today.
This book contains stories of people who used calculus to read and move the future.
SpaceX's rocket propellant recycling business has finally begun to see success after many failures.
(Omitted) Above all, the technology to safely land the rocket propulsion vehicle at the last moment is important, and that is the descent speed control and attitude control technology.
A nitrogen jet device sprays a small amount of nitrogen horizontally to create a tiny rotational force, while grid fins, a type of small wing, adjust the angle to finely control the direction.
All of this is possible only if we understand rotational motion differentially.
_Ⅰ.
Calculate the instantaneous speed of the revolution
Unlike conventional differential equation models, artificial neural network models are based on real data rather than scientific laws or rules.
However, the algorithm of artificial neural networks uses the concept of differentiation in the process of minimizing the loss function.
Although we do not use differential equations formulated based on physical laws such as Newton's laws or the law of conservation of mass as mentioned above, the concept of differentiation is inseparable from training artificial neural networks with massive amounts of data.
_Ⅲ.
How AI Learns Big Data
For accurately simulating irregular motions that collide with each other or interact with the surface of objects, such as a tsunami or a splashing flow, the Lagrangian method, which follows scattered particles, is more suitable than the Euler method.
In the Disney film Frozen, a related mathematical model was used to express vivid eye movements.
The MPM algorithm used primarily interprets particles as a continuum rather than viewing them individually, making it the optimal model for considering changes in snow properties depending on the degree of snow melting.
_Ⅴ.
How Disney Movies Captivate the World
Additionally, the principle of integration has made it possible to identify the location of inflammation and cancer without cutting into the human body, and with a single line of differential equations, the animation production company Pixar captivated the world.
Beyond this, the utility of calculus is limitless, as seen in road design, data compression, and more.
At the forefront of change is calculus, the flower of mathematics.
Today, the usefulness of calculus is further expanded with the advent of computers capable of high-speed calculations.
From artificial intelligence becoming much smarter to driverless cars, games, and AI funds, the future that calculus will bring us is unpredictable.
In this rapidly changing society, wouldn't understanding calculus be a new kind of liberal arts?
No need to solve calculus formulas anymore
It is important to think in calculus.
Previous calculus textbooks focused on 'how easy it is to understand and solve calculus problems.'
《The Usefulness of Calculus》 is different.
This book contains methods for utilizing calculus that can be easily read and understood by those who have just discovered the usefulness of mathematics, as well as those who still feel a headache just hearing the word "calculation."
Above all, I made sure not to solve formulas that were difficult to read at all.
It may sound nice, but the reality is that calculus calculations are so complex that they can only be left to computers.
To understand the integration principle of CT, let's consider a simplified cross-section of the body.
When the body is divided into a 4X4 grid, assuming that the bones (2) and organs (1) are distributed (as shown in the figure), and light is transmitted in four directions through the body, four films can be obtained.
The image displayed on film is called a sinogram.
A sinogram shows the integral of the light quantity added in the direction of the light beam.
Here, we can mathematically calculate the integral results shown in the four sinograms to find the 16 grid values f(x,y) inside the body.
To put it simply, you can think of it as the process of solving 16 equations and calculating 16 unknowns.
_Ⅵ.
When you collect small movements, you can see the axis of change.
In addition, when explaining gradient descent, which is the method by which artificial intelligence learns big data, he uses everyday examples such as descending a mountain, and when explaining the basic concept of differentiation, he uses speed cameras to explain in the easiest-to-understand way possible.
Additionally, if there are formulas that you need to know, we have made it as intuitive as possible to understand them through graphs and various pictures.
This book was written by an engineer who has dealt with calculus his entire life.
The author, a leading expert in the application of calculus, demonstrates how ordinary people can utilize calculus in their daily lives, along with cutting-edge trends in financial engineering, medical engineering, aerospace engineering, and astrophysics.
Like "The Usefulness of Mathematics," which was a bestseller in liberal arts science in 2020, this book shows how useful mathematical thinking is.
I recommend this book to those who have had a vague yearning and fear of calculus but have not been able to properly experience its usefulness.
You will have eyes to read the changes in the world.
Choi Young-gi, Professor of Mathematics Education at Seoul National University and author of "This Kind of Math Is My First Time"
If you integrate the past, you can see the present.
Differentiating the present reveals the future.
There is nothing in this world that does not change.
Not only the position and speed of the planets change, but people and time also change.
Calculus is the language that describes these changes in the world.
From the perspective of calculus, everything from the principles of advanced science and technology to natural phenomena and social changes is clearly revealed.
Differentiation allows us to capture the momentary changes and movements of the world, and integration allows us to understand the state in which small changes accumulate.
For example, the incidence rate of confirmed COVID-19 cases can be determined using calculus.
Daily confirmed cases are the combined amount, and cumulative confirmed cases are the combined result amount.
If you add up all the daily confirmed cases, you get the cumulative confirmed cases, and the change rate of the cumulative confirmed cases becomes the daily confirmed cases.
The number of daily confirmed cases fluctuates significantly from day to day, but the cumulative number of confirmed cases steadily increases.
The number of daily confirmed cases corresponds to the differential value indicating the rate of increase, and the cumulative number of confirmed cases corresponds to the integrated value of the daily increase.
In Korea, the number of daily confirmed COVID-19 cases suddenly increased and reached a peak, but by the end of March 2021, the number of daily confirmed cases remained high, but the rate of change gradually slowed.
Of course, the steady increase in the number of confirmed cases remains unchanged.
_Ⅵ.
When you collect small movements, you can see the axis of change.
The current cumulative number of confirmed cases can be determined through the integrated value, the resulting quantity, and the rate of confirmed cases tomorrow can be predicted through the differentiated value, the change (slope).
Unless a new variable emerges, it is possible to predict the incidence of confirmed cases even one month later.
The same goes for trends in climate change, the stock market, and apartment prices.
In other words, by integrating the past, we can understand the present, and by differentiating the present, we can predict the future.
Understanding calculus is reading change.
A new era opened by calculus, a future predicted by calculus.
A math textbook that fosters the ability to read change.
If humans had not understood calculus and been unable to properly utilize its usefulness, it would be difficult to imagine an era like ours today.
This book contains stories of people who used calculus to read and move the future.
SpaceX's rocket propellant recycling business has finally begun to see success after many failures.
(Omitted) Above all, the technology to safely land the rocket propulsion vehicle at the last moment is important, and that is the descent speed control and attitude control technology.
A nitrogen jet device sprays a small amount of nitrogen horizontally to create a tiny rotational force, while grid fins, a type of small wing, adjust the angle to finely control the direction.
All of this is possible only if we understand rotational motion differentially.
_Ⅰ.
Calculate the instantaneous speed of the revolution
Unlike conventional differential equation models, artificial neural network models are based on real data rather than scientific laws or rules.
However, the algorithm of artificial neural networks uses the concept of differentiation in the process of minimizing the loss function.
Although we do not use differential equations formulated based on physical laws such as Newton's laws or the law of conservation of mass as mentioned above, the concept of differentiation is inseparable from training artificial neural networks with massive amounts of data.
_Ⅲ.
How AI Learns Big Data
For accurately simulating irregular motions that collide with each other or interact with the surface of objects, such as a tsunami or a splashing flow, the Lagrangian method, which follows scattered particles, is more suitable than the Euler method.
In the Disney film Frozen, a related mathematical model was used to express vivid eye movements.
The MPM algorithm used primarily interprets particles as a continuum rather than viewing them individually, making it the optimal model for considering changes in snow properties depending on the degree of snow melting.
_Ⅴ.
How Disney Movies Captivate the World
Additionally, the principle of integration has made it possible to identify the location of inflammation and cancer without cutting into the human body, and with a single line of differential equations, the animation production company Pixar captivated the world.
Beyond this, the utility of calculus is limitless, as seen in road design, data compression, and more.
At the forefront of change is calculus, the flower of mathematics.
Today, the usefulness of calculus is further expanded with the advent of computers capable of high-speed calculations.
From artificial intelligence becoming much smarter to driverless cars, games, and AI funds, the future that calculus will bring us is unpredictable.
In this rapidly changing society, wouldn't understanding calculus be a new kind of liberal arts?
No need to solve calculus formulas anymore
It is important to think in calculus.
Previous calculus textbooks focused on 'how easy it is to understand and solve calculus problems.'
《The Usefulness of Calculus》 is different.
This book contains methods for utilizing calculus that can be easily read and understood by those who have just discovered the usefulness of mathematics, as well as those who still feel a headache just hearing the word "calculation."
Above all, I made sure not to solve formulas that were difficult to read at all.
It may sound nice, but the reality is that calculus calculations are so complex that they can only be left to computers.
To understand the integration principle of CT, let's consider a simplified cross-section of the body.
When the body is divided into a 4X4 grid, assuming that the bones (2) and organs (1) are distributed (as shown in the figure), and light is transmitted in four directions through the body, four films can be obtained.
The image displayed on film is called a sinogram.
A sinogram shows the integral of the light quantity added in the direction of the light beam.
Here, we can mathematically calculate the integral results shown in the four sinograms to find the 16 grid values f(x,y) inside the body.
To put it simply, you can think of it as the process of solving 16 equations and calculating 16 unknowns.
_Ⅵ.
When you collect small movements, you can see the axis of change.
In addition, when explaining gradient descent, which is the method by which artificial intelligence learns big data, he uses everyday examples such as descending a mountain, and when explaining the basic concept of differentiation, he uses speed cameras to explain in the easiest-to-understand way possible.
Additionally, if there are formulas that you need to know, we have made it as intuitive as possible to understand them through graphs and various pictures.
This book was written by an engineer who has dealt with calculus his entire life.
The author, a leading expert in the application of calculus, demonstrates how ordinary people can utilize calculus in their daily lives, along with cutting-edge trends in financial engineering, medical engineering, aerospace engineering, and astrophysics.
Like "The Usefulness of Mathematics," which was a bestseller in liberal arts science in 2020, this book shows how useful mathematical thinking is.
I recommend this book to those who have had a vague yearning and fear of calculus but have not been able to properly experience its usefulness.
You will have eyes to read the changes in the world.
GOODS SPECIFICS
- Date of issue: May 21, 2022
- Format: Hardcover book binding method guide
- Page count, weight, size: 248 pages | 546g | 161*230*20mm
- ISBN13: 9791165219550
- ISBN10: 1165219557
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