Çok değişkenli Fonksiyon II: Uygulamalar / Multivariable Calculus II: Applications from Coursera

Ders çok değişkenli fonksiyonlardaki iki derslik dizinin ikincisidir. Birinci ders türev ve entegral kavramlarını geliştirmekte ve bu konulardaki problemleri temel çözme yöntemlerini sunmaktadır. Bu ders, birinci derste geliştirilen temeller üzerine daha ileri konuları işlemekte ve daha kapsamlı uygulamalar ve çözümlü örnekler sunmaktadır. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır.

Bölümler

Bölüm 1: Multivar 1'in Özeti, Dairesel Koordinatlarda Entegraller

Bölüm 2: Türev Uygulamalarından Seçme Konular

Bölüm 3: Çok Değişkenle Zincirleme Türev ve Jakobiyan

Bölüm 4: Uzayda Yüzey ve Hacım Entegralleri

Bölüm 5: Düzlemde Akı Entegralleri

Bölüm 6: Düzlemde Green, Uzayda Stokes ve Green-Gauss Teoremleri

Bölüm 7: Stokes ve Green-Gauss Teoremleri ve Doğanın Korunum Yasaları

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The course is the second of the two course sequence of calculus of multivariable functions. The first course develops the concepts of derivatives and integrals of functions of several variables, and the basic tools for doing the relevant calculations. This course builds on the foundations of the first course and introduces more advanced topics along with more advanced applications and solved problems. The course is designed with a “content-based” approach, i. e. by solving examples, as many as possible from real life situations.

Bölümler

Bölüm 1: Summary of Multivar I, Integral in Circular Coordinates

Bölüm 2: Topics of Derivative Applications

Bölüm 3: Chain Derivatives with Multi Variables and Jacobian

Bölüm 4: Surface and Volume Integrals in Space

Bölüm 5: Flux Integrals in the Plane

Bölüm 6: Green in Plane, Stokes in Space and Green-Gauss Theorems

Bölüm 7: Stokes and Green-Gauss Theorem and Nature Conservation Laws

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Kaynak: Attila Aşkar, “Çok değişkenli fonksiyonlarda türev ve entegral”. Bu kitap dört ciltlik dizinin ikinci cildidir. Dizinin diğer kitapları Cilt 1 “Tek değişkenli fonksiyonlarda türev ve entegral”, Cilt 3: “Doğrusal cebir” ve Cilt 4: “Diferansiyel denklemler” dir.

Source: Attila Aşkar, Calculus of Multivariable Functions, Volume 2 of the set of Vol1: Calculus of Single Variable Functions, Volume 3: Linear Algebra and Volume 4: Differential Equations. All available online starting on January 6, 2014

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Syllabus

Multivar 1'in Özeti, Dairesel Koordinatlarda Entegraller

İki değişkenli fonksiyonlardan hatırlatmalar: ikinci derece fonksiyonlar, kısmi türev ve iki katlı entegrallerdeki temel tanımlar ve geometrideki anlamları; iki değişkenli fonksiyonlarda türev ve entegrallerdeki temel hesaplama yöntemleri, iki katlı entegral hesaplamasında sıranın öneminin örneklerle hatırlatılması; teğet düzlem ve diferansiyel; tam türev ve zincirleme türev. Yöne göre türev. Gradyan, Bu sonuçların üç ve “n” değişkenli fonksiyonlara genellenmesi.

Türev Uygulamalarından Seçme Konular

İki değişkenli fonksiyonlarda Koordinat dönüşümleri ve Jakobiyan. Diverjans, Rotasyonel ve Laplasyen. Dairesel koordinatlarda gradyan. Doğanın dört temel kısmi türevli denklemi: dalga, sızma, Laplace denklemleriyle Schrödinger denkleminin tanıtılması. İki değişkenli sonuçların üç değişkene ve mümkün olan durumlarda “n” değişkene genellenmesi. Karmaşık değerli fonksiyonların özel yapısıyla kısmi türevler ve tam türev.

Çok Değişkenle Zincirleme Türev ve Jakobiyan

En büyük ve en küçük değerler: yerel, mutlak ve kısıtlama altında. Kısıtlama altında en iyiyi arama (optimizasyon) ve Lagrange çarpanı yöntemi. Kısmi türevlerin uygulaması ile değişimler hesabına giriş.

Uzayda Yüzey ve Hacım Entegralleri

Uzayda yüzeylerin açık, kapalı ve parametrik fonksiyonlarla gösterilmeleri ve eğrisel koordinatlar. sonsuz küçük yüzey alanları seçeneklerinin birleştirilmiş bir yaklaşımla elde edilmesi. Uzayda küre, koni, paraboloitler gibi temel yüzeylerin tanıtılması ve bunları içeren alanlarla hesaplar.

Düzlemde Akı Entegralleri

Uzayda kapalı yüzeylerle tanımlanan hacımlar; sonsuz küçük hacımların birleştirilmiş yaklaşımla elde edilmesi. Jakobiyan ve sonsuz küçük hacım. Kartezyen, silindir ve küresel koordinatlarla uzaydaki yüzey ve cisimlerde üç katlı entegral hesaplamaları. Uzayda küre, koni, paraboloitler gibi temel yüzeylerle tanımlanan hacımlardan örnekler ve bunları içeren hacım hesapları.

Düzlemde Green, Uzayda Stokes ve Green-Gauss Teoremleri

Vektör alanlarının tanıtılması; vektör alanlarıyla türev ve entegral. Düzlem eğrilerinde entegraller. Entegralin yörüngeye bağlı ve yörüngeden bağımsız olması. Düzlem eğrilerinde birinci ve ikinci Green teoremleri. Düzlemdeki Green teoremlerinin vektörler, rotasyonel ve diverjansla gösterimi.

Stokes ve Green-Gauss teoremleri ve doğanın korunum yasaları

Düzlemdeki Green teoremlerinden uzayda Stokes ve Green – Gauss teoremlerine geçiş. Uzayda Green – Gauss ve Stokes teoremleriyle yüzey ve hacım entegralleri. Diverjans, rotasyonel ve Laplasyen’in anlamı. Uzayda Green – Gauss ve Stokes teoremleriyle doğadan temel korunum denklemlerinin elde edilmesi. Kütle, elektrik yükü ve ısı enerjisinin korunmasında uygulamalar.

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Temel ve ileri matematik alanında, özellikle çok değişkenli fonksiyonlarda uzmanlaşmak isteyen lisans öğrencilerine hitap eder

Mühendislik, fizik, ekonomi ve veri bilimi gibi alanlarda çok değişkenli fonksiyonları kullanan öğrenciler için faydalıdır

Kurs, gerçek yaşam uygulamaları üzerinden örnekler sunarak konuları daha anlaşılır hale getirir

Kursun eğitmeni Prof. Dr. Attila Aşkar, çok değişkenli fonksiyonlar alanında uzman bir akademisyendir

Kurs, çok değişkenli fonksiyonlar alanında derinlemesine bilgi edinmek isteyenlere uygun olabilir

Kursun bazı bölümleri, daha önce çok değişkenli fonksiyonlar alanında temel bilgilere sahip olmayı gerektirebilir

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Mixed calculus ii app

This course is a calculus course that focuses on the applications of multivariable functions. Reviews are mixed, with some students praising the course and others complaining of misinformation.

Students can watch the videos with English subtitles.

Students recommend this course.

"Excelente curso lo recomiendo mucho ya que se aprende muchas cosas del cálculo de varias variables lo cual ayuda bastante a los estudiantes de Ciencias e Ingeniería"

Some students complained of misinformation in the course.

"Misinformation\n"

Quizzes are only available in Turkish.

"English subtitles are available, but the quizzes are only in Turkish"

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Çok değişkenli Fonksiyon II: Uygulamalar / Multivariable Calculus II: Applications with these activities:

Review Precalculus Concepts

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Strengthens your mathematical foundation for Calculus.

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Review the basics of algebra and trigonometry
Familiarize yourself with graphs and functions

Organize Course Materials

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Sets you up for better note-taking and quicker recall of important information.

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Download, organize, and read the course syllabus
Review the chapters and corresponding videos
Compile and review practice problems

Explore Online Calculus Resources

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Enhances your learning and provides additional perspectives.

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Watch video lectures on YouTube or Khan Academy
Read articles and tutorials on Calculus websites

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Join a Study Group

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Fosters collaboration, diverse perspectives, and deeper understanding.

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Find classmates or online forums for discussions

Practice Derivative Calculations

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Sharpens your understanding of basic calculus techniques.

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Work through the examples in the textbook
Complete the practice problems at the end of each section

Develop a Study Guide

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Forces you to synthesize and condense information, improving retention.

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Identify the key concepts in each chapter
Write down definitions, theorems, and formulas
Include worked-out examples

Career center

Learners who complete Çok değişkenli Fonksiyon II: Uygulamalar / Multivariable Calculus II: Applications will develop knowledge and skills that may be useful to these careers:

Data Scientist

As a Data Scientist, you will be responsible for extracting knowledge from data for making informed decisions. Data Scientists use their expertise in statistics, programming, and machine learning to analyze data and identify patterns. The knowledge and skills gained in this course, such as multivariable calculus and its applications, can serve as a strong foundation for a career in Data Science. Notably, the course's coverage of topics like optimization and differential equations is directly relevant to the work of a Data Scientist.

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Operations Research Analyst

Operations Research Analysts use mathematical and analytical methods to solve complex problems in business, industry, and government. They use a variety of techniques, including multivariable calculus, to develop and implement solutions that improve efficiency and effectiveness. This course can help you build a strong foundation in the mathematical concepts and techniques used by Operations Research Analysts and prepare you for a successful career in this field.

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Actuary

Actuaries use mathematical and statistical techniques to assess risk and uncertainty. They use their knowledge of multivariable calculus to develop models that can predict the likelihood and severity of future events. This course can help you develop the skills and knowledge needed to succeed as an Actuary.

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Financial Analyst

Financial Analysts use mathematical and statistical techniques to analyze financial data and make investment recommendations. They use their knowledge of multivariable calculus to develop models that can predict the performance of stocks, bonds, and other financial instruments. This course can help you develop the skills and knowledge needed to succeed as a Financial Analyst.

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Software Engineer

Software Engineers design, develop, and maintain software applications. They use their knowledge of multivariable calculus to develop algorithms and data structures that can efficiently solve complex problems. This course can help you develop the skills and knowledge needed to succeed as a Software Engineer.

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Transportation Engineer

Transportation Engineers design, develop, and maintain transportation systems. They use their knowledge of multivariable calculus to develop models that can be used to optimize the flow of traffic. This course can help you develop the skills and knowledge needed to succeed as a Transportation Engineer.

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Statistician

Statisticians use statistical methods to analyze data in a variety of fields, including business, government, and healthcare. They use their knowledge of multivariable calculus to develop models that can be used to make informed decisions. This course can help you develop the skills and knowledge needed to succeed as a Statistician.

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Economist

Economists use mathematical and statistical techniques to analyze economic data and make policy recommendations. They use their knowledge of multivariable calculus to develop models that can predict the behavior of the economy. This course can help you develop the skills and knowledge needed to succeed as an Economist.

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Biostatistician

Biostatisticians use statistical methods to analyze data in the field of biology. They use their knowledge of multivariable calculus to develop models that can be used to understand the causes and treatments of diseases. This course can help you develop the skills and knowledge needed to succeed as a Biostatistician.

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Market Research Analyst

Market Research Analysts use statistical methods to analyze data on consumer behavior. They use their knowledge of multivariable calculus to develop models that can predict the demand for products and services. This course can help you develop the skills and knowledge needed to succeed as a Market Research Analyst.

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Physicist

Physicists use mathematical and experimental methods to study the laws of nature. They use their knowledge of multivariable calculus to develop models that can be used to understand the behavior of the universe. This course can help you develop the skills and knowledge needed to succeed as a Physicist.

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Mechanical Engineer

Mechanical Engineers design, develop, and maintain mechanical systems. They use their knowledge of multivariable calculus to develop models that can be used to analyze the forces and stresses on mechanical components. This course can help you develop the skills and knowledge needed to succeed as a Mechanical Engineer.

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Chemical Engineer

Chemical Engineers design, develop, and maintain chemical processes. They use their knowledge of multivariable calculus to develop models that can be used to optimize the production of chemicals and other materials. This course can help you develop the skills and knowledge needed to succeed as a Chemical Engineer.

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Electrical Engineer

Electrical Engineers design, develop, and maintain electrical systems. They use their knowledge of multivariable calculus to develop models that can be used to analyze the flow of electricity. This course can help you develop the skills and knowledge needed to succeed as an Electrical Engineer.

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Civil Engineer

Civil Engineers design, develop, and maintain civil infrastructure. They use their knowledge of multivariable calculus to develop models that can be used to analyze the forces and stresses on civil structures. This course can help you develop the skills and knowledge needed to succeed as a Civil Engineer.

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