Applications Of Calculus In Computer Science And Programming Pdf
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- Numerical Methods And Computer Programming Pdf
- How Much Math Do You Need for Computer Science?
- Application of Calculus in Computer Science
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Numerical Methods And Computer Programming Pdf
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. It only takes a minute to sign up. I'm wondering, how and when is calculus used in computer science? The CS content of a degree in computer science tends to focus on algorithms, operating systems, data structures, artificial intelligence, software engineering, etc. Are there times when Calculus is useful in these or other areas of Computer Science?
I can think of a few courses that would need Calculus, directly. I have used bold face for the usually obligatory disciplines for a Computer Science degree, and italics for the usually optional ones. And, besides that, one benefits indirectly from a Calculus course by learning how to reason and explain arguments with technical rigor. This is more valuable than students usually think. Finally -- you will need Calculus in order to, well, interact with people from other Exact Sciences and Engineering.
And it's not uncommon that a Computer Scientist needs to not only talk but also work together with a Physicist or an Engineer. This is somewhat obscure, but calculus turns up in algebraic data types. For any given type, the type of its one-hole contexts is the derivative of that type.
See this excellent talk for an overview of the whole subject. This is very technical terminology, so let's explain. You may have come across tuples being referred to as product types if not, it's because they are the cartesian product of two types.
We're going to take this literally and use the notation:. Next, you may have come across sum types these are types which can be either one type, or another known as unions , variants , or as the Either type kinda in Haskell. We're also going to take this one literally and use the notation:.
These types look like normal algebraic expressions and we can, in fact, manipulate them as such to a point. This says that a list is either empty or a tuple of a value and another list. Transforming that to algebraic notation, we get:. This definition says then that a list is either unit, or a tuple of one item, or a tuple of two items, or of three etc, which is the definition of a list!
Now on to one-hole contexts: a one-hole context is what you get when you 'take a value out' of a product type. Let's give an example:. But there are two different one-hole contexts of this type: namely the first and second values of the tuple. This is where the differentiation comes in to play.
Let's confirm this with another example:. Depending on where we put the hole. There is a proof of this in general here. On the surface this may seem like nonsense, but if you take the taylor series of this result you get the definition we derived earlier. Thus one list has become a pair of lists.
This in fact makes sense: the two lists produced correspond to the elements above and below the hole in the original list! Numerical Methods. There exist cumbersome calculus problems that are unique to specific applications, and they need solutions faster than a human can practically solve without a program. Someone has to design an algorithm that will compute the solution. Isn't that the only thing that separates programmers from scientists? Automation - Similar to robotics, automation can require quantifying a lot of human behavior.
Visualizations - Utilizing advanced algorithms requires calculus such as cos, sine, pi, and e. Especially when you're calculating vectors, collision fields, and meshing. Logistics and Risk analysis - Determining whether a task is possible, the risk involved, and possible rate of success. Security - Most security can be performed without calculus; however, many people who want explanations prefer it in mathematical expressions. Medical calculations - Visualizing most health data requires calculus such as an EKG reading.
Many people in programming can go their entire career without using calculus; however, it can prove invaluable if you're willing to do the work.
For me it has been most effective in automation, logistics, and visualization. By identifying specific patterns, you can simply ignore the pattern, imitate the pattern, or develop a superior method all together. The fact is that there's very little chance you'll ever use calculus. However, virtually every other scientific discipline DOES use calculus and you are working on a science degree.
There are certain expectations of what a university science degree is supposed to mean and one of those things is that you know calculus. Even if you'll never use it. It's okay if you do poorly in calculus, but make sure you put some effort into discrete mathematics. There are a lot of real-world programming problems where discrete math comes into play and ignorance of its principles can embarrass you in front of other coders.
Many people already provided applications in CS. But sometimes you'll find Calculus when you least expect:. Regular-expression derivatives reexamined. In creating test cases for some applications I have had to make use of calculus to predict expected running times, memory sizes, and choose optimal parameters when tuning data structures.
This includes understanding expected rounding error, etc. While statistics is mentioned in other answers, I would like to specifically mention Monte-carlo algorithms , such as optimization algorithms and some frugal streaming algorithms that are based on Mathematical principles that include calculus.
Insurance numerical integration of insurance policies in what-if scenarios to compute expected policy losses. Calculus -- the integral portion -- is used directly in CS as a foundation for thinking about summation.
If you work through any portion of Knuth's Concrete Mathematics section on summation, you will quickly recognize conventions common to calculus: understanding some of the continuous case gives you tools to consider the discrete. Many of the uses of your CS study involve programming systems which monitor change, or in some cases, attempt to predict the future. The mathematics around those systems is rooted in differential equations and linear algebra, and differential equations are There are teachers like Gibert Strang who advocate for moving more quickly into the differential equations part, but it is still a subset of calculus.
When change depends on change in any system, it starts to be unstable and stable in ways which are both non-intuitive and very well understood. To understand why your sensible linear system is behaving in nonlinear ways, you either need the tools of calculus or you need to re-invent them for your problem space.
And finally, CS often requires reading and understanding the work of others, and calculus is the first exposure to a lot of shared vocabulary, convention, and history. Sign up to join this community. The best answers are voted up and rise to the top. Ask Question. Asked 4 years, 11 months ago. Active 8 months ago. Viewed 83k times.
Improve this question. Victor Victor 1, 2 2 gold badges 9 9 silver badges 8 8 bronze badges. Please note also this and this discussion; you might want to improve your question as to avoid the problems explained there. If you are not sure how to improve your question maybe we can help you in Computer Science Chat? Sometimes is just about training you how to think in certain ways.
Add a comment. Active Oldest Votes. If you go down this path, you may also want to study some Differential Geometry which has multivariate Calculus as a minimum prerequisite. But you'll need Calculus here even for very basic things: try searching for "Fourier Transform" or "Wavelets", for example -- these are two very fundamental tools for people working with images.
Optimization , non-linear mostly, where multivariate Calculus is the fundamental language used to develop everything. These cannot be seriously studied without multivariate Calculus. Machine Learning , which makes heavy use of Statistics and consequently, multivariate Calculus Data Science and related subjects, which also use lots of Statistics; Robotics , where you will need to model physical movements of a robot, so you will need to know partial derivatives and gradients.
Discrete Math and Combinatorics yes! Similarly, Taylor Series and calculus can be useful in solving certain kinds of recurrence relations, which are used in algorithm analysis. Improve this answer. Jay Jay 1, 1 1 gold badge 9 9 silver badges 12 12 bronze badges. It was taught by rote and pattern matching pretty much like high school algebra and geometry. On the other hand, it was the prerequisite to several higher math classes that did teach these skills, so I suppose it wasn't entirely useless.
Working on programming languages theory, I rarely used calculus directly. Perhaps the most direct application was in probabilistic computational models e. Yet, my calculus course was mostly about proving things, and this was very, very valuable.
One or two calculus courses are IMHO needed in every serious CS program, along some more math discrete math, logic, linear algebra, numerical analysis, A little while ago I used this to derive some different interpolation polynomials for smoothing images.
Show 2 more comments. Algebraic Data Types You may have come across tuples being referred to as product types if not, it's because they are the cartesian product of two types. One-hole Contexts Now on to one-hole contexts: a one-hole context is what you get when you 'take a value out' of a product type.
How Much Math Do You Need for Computer Science?
On the other hand, Computer Science is quite interesting and students study it in hopes of becoming the next programming whizz-kid!!! We are very pleased to announce that the paper of Matilde Marcolli "Persistent Topology of Syntax" is cited in Caltech as best current research highlights. One needs to be fluent in it to work in many fields including data science, machine learning, and software engineering it is not a coincidence that math puzzles are often used for interviews. Learn more. Chapters 1 and 8 of Mathematics for Computer Science by E.
Application of Calculus in Computer Science
Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It is the theory, experimentation, and engineering that enables the use of algorithms to manipulate, store, and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems.
Computer science can be intimidating, but you can do it. Becoming a proficient computer scientist does, however, require an intermediate or advanced understanding of a couple of subjects, including math. In fact, some never use it at all.
I did some research and found out many different functional uses that Calculus has had since its discovery. Today, you can find Calculus within all manner of sciences such as chemistry, physics, computer science, and engineering, and even business and economics. Calculus is used for optimization, summation, and predicting trends through modeling change over time.
Все повернулись к экрану. Это был агент Колиандер из Севильи. Он перегнулся через плечо Беккера и заговорил в микрофон: - Не знаю, важно ли это, но я не уверен, что мистер Танкадо знал, что он пал жертвой покушения. - Прошу прощения? - проговорил директор. - Халохот был профессионалом высокого уровня, сэр. Мы были свидетелями убийства, поскольку находились всего в пятидесяти метрах от места. Все данные говорят, что Танкадо ни о чем таком даже не подозревал.
Черные всепроникающие линии окружили последний предохранительный щит и начали прорываться к сердцевине банка данных. Алчущие хакеры прорывались со всех уголков мира. Их количество удваивалось каждую минуту. Еще немного, и любой обладатель компьютера - иностранные шпионы, радикалы, террористы - получит доступ в хранилище секретной информации американского правительства. Пока техники тщетно старались отключить электропитание, собравшиеся на подиуме пытались понять расшифрованный текст. Дэвид Беккер и два оперативных агента тоже пробовали сделать это, сидя в мини-автобусе в Севилье. ГЛАВНАЯ РАЗНИЦА МЕЖДУ ЭЛЕМЕНТАМИ, ОТВЕТСТВЕННЫМИ ЗА ХИРОСИМУ И НАГАСАКИ Соши размышляла вслух: - Элементы, ответственные за Хиросиму и Нагасаки… Пёрл-Харбор.
Он держит нас в заложниках. Внезапно она встала. В голосе ее прозвучала удивительная решимость: - Мы должны установить с ним контакт. Должен быть способ убедить его не выпускать ключ из рук. Мы обязаны утроить самое высокое сделанное ему предложение. Мы можем восстановить его репутацию. Мы должны пойти на .
Я попробовал оказать ему помощь, но все было бесполезно. - Вы делали ему искусственное дыхание.
В служебных помещениях ТРАНСТЕКСТА было черно как глубокой ночью. Минуту он наслаждался полной темнотой. Сверху хлестала вода, прямо как во время полночного шторма. Стратмор откинул голову назад, словно давая каплям возможность смыть с него вину.
Стратмор смущенно посмотрел на труп, затем перевел взгляд на Сьюзан. Неужели она узнала. Этого не может .