Çok değişkenli Fonksiyon I: Kavramlar / Multivariable Calculus I: Concepts from Coursera

Ders çok değişkenli fonksiyonlardaki ikili dizinin birincisidir. Burada çok değişkenli fonksiyonlardaki temel türev ve entegral kavramlarını geliştirmek ve bu konulardaki problemleri çözmekteki temel yöntemleri sunmaktadır. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır.

Bölümler

Bölüm 1: Genel Konular ve Düzlemdeki Vektörler

Bölüm 2: Uzayda Vektörler, Doğrular ve Düzlemler; Vektör Fonksiyonları

Bölüm 3: Düzlem Eğrilerinden Hatırlatmalar ve Uzay Eğrileri, İki Değişkenli ve İkinci Derece Fonksiyonlar ve Karşıt Gelen Yüzeyler

Bölüm 4: Özel Yapıdaki İki Değişkenli Olarak Karmaşık Fonksiyonlar, İki Değişkenli Fonksiyonlarda Kısmi Türev ve İki Katlı Entegralin Temel Tanımları; Limit Kavramının Gerekliliği ve Anlatımı

Bölüm 5: Türev Hesaplama Yöntemleri

Bölüm 6: Türev Uygulamaları

Bölüm 7: İki Katlı Entegraller ve Uygulamaları

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The course is the first of the sequence of calculus of multivariable functions. It develops the fundamental concepts of derivatives and integrals of functions of several variables, and the basic tools for doing the relevant calculations. The course is designed with a “content-based” approach, i. e. by solving examples, as many as possible from real life situations.

Chapters

Chapters 1: General Topics and Vectors in the Plane

Chapters 2: Vectors in Space, Lines and Planes; Vector Functions

Chapters 3: Reminders of Plane Curves and Space Curves, Quadratic Functions and Variables, Surfaces

Chapters 4: Special Two Variables Complex Functions, the Basic Definition of Partial Derivatives and Two Storey Integrals in Two Unknown Functions ; Necessity and Details of Limits

Chapters 5: Methods of Derivative Calculations

Chapters 6: Application of Derivatives

Chapters 7: Two Storey Integrals and Applications

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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

What's inside

Syllabus

Genel Konular ve Düzlemdeki Vektörler

Fonksiyon kavramı: girdi – çıktı, bir değerin diğerine gönderimi, çizit, ve dönüşüm gösterimleri. Çok değişkenli fonksiyonların sınıflandırılması: uzayda eğriler, yüzeyler ve vektör alanları. Düzlemde karteziyen ve dairesel koordinatların, uzayda karteziyen, silindir ve küresel koordinatların tanıtılması. Fonksiyonların açık, kapalı ve parametrelerle gösterilmesi. Vektörler: düzlemde geometriden cebire. Düzlemde toplama, bir sayıyla çarpma, iç çarpım ve vektör çarpımı. Bu işlemlerin üç boyuta genellenmesi ve üçlü vektör çarpımları. Bu kavramların geometrideki anlamları ve uygulamaları. Uzayda doğrular ve düzlemler.

Uzayda Vektörler, Doğrular ve Düzlemler; Vektör Fonksiyonları

Uzayda eğriler: tek bağımsız ve üç bağımlı değişkenle vektör fonksiyonları. Düzlemdeki temel eğrilerin hatırlatılması ve uzaydaki bazı önemli eğrilerin tanıtılması. Düzlemde yay uzunluğu, eğrilik ile teğet ve dik vektörlerin hatırlatılması. Uzayda yay uzunluğu, teğet, dik ve ikinci dik (binormal) vektörleriyle eğrilik ve burulmanın tanımlanması. Uzaydaki yörüngelerde hız ve ivme.

Düzlem Eğrilerinden Hatırlatmalar ve Uzay Eğrileri, İki Değişkenli ve İkinci Derece Fonksiyonlar ve Karşıt Gelen Yüzeyler

Uzayda yüzeyler: iki bağımsız ve tek bağımlı değişkenle tanımlanan sayısal fonksiyonlar. Yüzeylerin anlaşılması ve temel yüzeylerde çizimler: perspektif görünüm, eşit değer eğrileri ve kesitlerin çizimi. İki değişkenli ikinci derece kuvvet fonksiyonlarıyla verilen temel yüzeyler. Silindir yüzeyleri ve dönel yüzeyler. İki değişkenli özel bir yapı olarak karmaşık değerli fonksiyonlar. Mathematica, Mathlab, Ghostview… gibi yazılımlarla bilgisayarda çizimlerden örnekler.

Özel Yapıdaki İki Değişkenli Olarak Karmaşık Fonksiyonlar, İki Değişkenli Fonksiyonlarda Kısmi Türev ve İki Katlı Entegralin Temel Tanımları; Limit Kavramının Gerekliliği ve Anlatımı

Tek değişkenli fonksiyonlarda türev ve entegralin hatırlatılması. Buradaki ana kavramların İki değişkenli fonksiyonlarda “kısmi türev” ve “iki katlı entegral” olarak genellenmesi. Kısmi türev ve iki katlı entegralin geometrideki anlamları. Temel tanımları pekiştiren az sayıda kısmi türev ve iki katlı entegrallerin hesabı.

Türev Hesaplama Yöntemleri

İki değişkenli sayısal açık fonksiyonlarla tanımlanan yüzeyde teğet düzlem ve diferansiyel. Zincirleme türev yöntemi ve tam türev. Yöne göre türev. Gradyan. Koordinat dönüşümü ve Jakobiyan. Taylor serileri. Kritik noktalar, en büyük ve en küçük değerler. Türev hesaplamalarının üç ve “n” değişkenli fonksiyonlara genellenmesi.

Türev Uygulamaları

İki Katlı Entegraller ve Uygulamaları

İki katlı entegrallerde hesaplama örnekleri. Kartezyen ve dairesel koordinatlarda hesaplamalar, uygulamalardan örnekler.

Dönem Sonu Sınavı (Final Exam)

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Covers foundational concepts in multivariate calculus, including derivatives and integrals

Provides a strong foundation for advanced topics in mathematics and its applications in various fields

Taught by an experienced instructor with a strong reputation in the field

Utilizes a hands-on approach with real-world examples to enhance understanding

Emphasizes practical applications, making it relevant for individuals pursuing careers in fields that utilize mathematics

Requires a solid understanding of pre-calculus concepts, including algebra, trigonometry, and analytic geometry

Reviews summary

Multivariable calculus: partial derivatives and integrals

Multivariable Calculus I: Concepts is a college-level course that focuses on the fundamental principles of calculus for functions with multiple variables, such as derivatives and integrals. The course takes a hands-on approach with many real-life examples to help students understand the concepts. Students who took this course found it to be difficult, but ultimately rewarding, with one reviewer stating that it was “useful for me.” It is recommended for students who have a strong foundation in single-variable calculus. Overall, this course is well-received by students.

Students appreciate the professor.

"Aynı hocadan diff dersini de bekliyoruz."

"Teşekkürler Koç"

Course provides several examples from real-life situations.

"The course is designed with a “content-based” approach, i. e. by solving examples, as many as possible from real life situations."

Requires a strong foundation in single-variable calculus.

Students found the course to be challenging.

"it was hard for me."

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 I: Kavramlar / Multivariable Calculus I: Concepts with these activities:

Çalışma Grubu

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Çalışma Grubu'na katılmak, kurs konularının daha iyi anlaşılmasını sağlayacaktır.

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Career center

Learners who complete Çok değişkenli Fonksiyon I: Kavramlar / Multivariable Calculus I: Concepts will develop knowledge and skills that may be useful to these careers:

Geophysicist

Geophysicists study the physical properties of the Earth and its surrounding environment. They use a variety of mathematical and computational techniques to analyze data and develop models of the Earth's structure and dynamics. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Geophysics. This course can help you to develop the skills you need to pursue a career as a Geophysicist.

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Aerospace Engineer

Aerospace Engineers design, develop, and test aircraft, spacecraft, and other aerospace vehicles. They use a variety of mathematical and computational techniques to analyze the performance of these vehicles and to ensure that they are safe and efficient. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Aerospace Engineering. This course can help you to develop the skills you need to pursue a career as an Aerospace Engineer.

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Materials Scientist

Materials Scientists study the properties of materials and develop new materials with improved properties. They use a variety of mathematical and computational techniques to analyze the structure and properties of materials. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Materials Science. This course can help you to develop the skills you need to pursue a career as a Materials Scientist.

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Mechanical Engineer

Mechanical Engineers design, develop, and test mechanical systems. They use a variety of mathematical and computational techniques to analyze the performance of these systems and to ensure that they are safe and efficient. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Mechanical Engineering. This course can help you to develop the skills you need to pursue a career as a Mechanical Engineer.

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Biomedical Engineer

Biomedical Engineers apply engineering principles to the design and development of medical devices and systems. They use a variety of mathematical and computational techniques to analyze the performance of these devices and systems and to ensure that they are safe and effective. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Biomedical Engineering. This course can help you develop the skills you need to pursue a career as a Biomedical Engineer.

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Chemical Engineer

Chemical Engineers design, develop, and operate chemical processes. They use a variety of mathematical and computational techniques to analyze the performance of these processes and to ensure that they are safe and efficient. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Chemical Engineering. This course can help you to develop the skills you need to pursue a career as a Chemical Engineer.

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Civil Engineer

Civil Engineers design, build, and maintain infrastructure projects such as roads, bridges, and buildings. They use a variety of mathematical and computational techniques to analyze the performance of these projects and to ensure that they are safe and efficient. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Civil Engineering. This course can help you develop the skills you need to pursue a career as a Civil Engineer.

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Electrical Engineer

Electrical Engineers design, develop, and test electrical systems. They use a variety of mathematical and computational techniques to analyze the performance of these systems and to ensure that they are safe and efficient. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Electrical Engineering. This course can help you develop the skills you need to pursue a career as an Electrical Engineer.

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Computer Scientist

Computer Scientists design, develop, and test computer systems. They use a variety of mathematical and computational techniques to analyze the performance of these systems and to ensure that they are safe and efficient. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Computer Science. This course can help you develop the skills you need to pursue a career as a Computer Scientist.

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Data Scientist

Data Scientists use mathematical and computational techniques to analyze data and develop models. They use these models to solve problems in a variety of fields, including business, finance, healthcare, and manufacturing. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Data Science. This course can help you develop the skills you need to pursue a career as a Data Scientist.

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Statistician

Statisticians collect, analyze, and interpret data. They use this data to draw conclusions about populations and to make predictions about future events. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Statistics. This course can help you to develop the skills you need to pursue a career as a Statistician.

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Economist

Economists study the production, distribution, and consumption of goods and services. They use a variety of mathematical and computational techniques to analyze economic data and to develop economic models. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Economics. This course can help you to develop the skills you need to pursue a career as an Economist.

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Financial Analyst

Financial Analysts evaluate the financial performance of companies and make recommendations about investments. They use a variety of mathematical and computational techniques to analyze financial data and to develop financial models. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Financial Analysis. This course can help you develop the skills you need to pursue a career as a Financial Analyst.

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Mathematician

Mathematicians study the properties of numbers, shapes, and other mathematical objects. They also develop new mathematical theories and techniques. The concepts of derivatives and integrals, which are covered in this course, are fundamental to Mathematics. This course can help you develop the skills you need to pursue a career as a Mathematician.

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Actuary

Actuaries use mathematical and statistical techniques to assess risk and uncertainty. They use this information to develop insurance policies and other financial products. The concepts of derivatives and integrals, which are covered in this course, are essential for understanding the mathematical models used in Actuarial Science. This course can help you develop the skills you need to pursue a career as an Actuary.

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