Calculus II: Multivariable Functions from edX

A strong foundation in mathematics is critical for success in all science and engineering disciplines. Whether you want to make a strong start to a master’s degree, prepare for more advanced courses, solidify your knowledge in a professional context or simply brush up on fundamentals, this course will get you up to speed.

Calculus is the branch of Mathematics that deals with differentiation and integration of functions; techniques that play a major role in solving many of the problems encountered in an engineer’s everyday life.

This course will lay the foundations of multivariable calculus as it introduces differentiation and integration techniques for functions of two or more variables. Among other things you will learn to differentiate and linearize multivariable functions, find critical points and extreme values, and integrate multivariable functions using a variety of coordinate systems.

Our courses in calculus offer enough depth to cover what you need to succeed in your engineering master’s or profession in areas such as modeling, physics, fluid dynamics, dynamical systems and more.

This is a review courseThis self-contained course is modular, so you do not need to follow the entire course if you wish to focus on a particular aspect. As a review course you are expected to have previously studied or be familiar with most of the material. Hence the pace will be higher than in an introductory course.

This format is ideal for refreshing your bachelor level mathematics and letting you practice as much as you want. Through the Grasple platform, you will have access to plenty of exercises and receive intelligent, personal and immediate feedback.

This course is part of our series Mastering Mathematics for Engineers , and together with the course Calculus I part of the program Mastering Calculus.

What's inside

Learning objectives

Visualize functions of two variables and read off qualitative information from graphs and contour plots.
Calculate (higher order) partial derivatives for a variety of multivariable functions.
Calculate and interpret directional derivatives and gradients of multivariable functions.

Find and classify critical points of functions of two variables and determine extreme values for functions of compact domains.
Evaluate double integrals over general integration domains using cartesian or polar coordinates.
Evaluate triple integrals over general integration domains using cartesian, cylindrical or spherical coordinates.

Visualize functions of two variables and read off qualitative information from graphs and contour plots.
Calculate (higher order) partial derivatives for a variety of multivariable functions.
Calculate and interpret directional derivatives and gradients of multivariable functions.
Find and classify critical points of functions of two variables and determine extreme values for functions of compact domains.
Evaluate double integrals over general integration domains using cartesian or polar coordinates.
Evaluate triple integrals over general integration domains using cartesian, cylindrical or spherical coordinates.

Syllabus

Week 1:

multivariable functions

graphs of functions of two variables

contour plots

Week 2:

partial derivatives

tangent planes and linearizations

higher order partial derivatives

chain rule

Week 3:

directional derivatives and gradients

maximizing directional derivatives

applications of gradients

Week 4:

extreme values

critical points

finding extreme values

Week 5:

double integrals

integrals over rectangles

double integrals over general regions

polar coordinates

Week 6:

triple integrals

triple integrals over general regions

cylindrical coordinates

spherical coordinates

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Identifies potential pitfalls and challenges in applied engineering

Emphasizes visualizing data and qualitative information for improved decision-making

Leverages the Grasple platform for personalized feedback through numerous exercises

Primarily targets working professionals or graduate students seeking a refresher in multivariable calculus

Assumes prior familiarity with calculus concepts, making it unsuitable for absolute beginners

Recommended for those seeking a comprehensive foundation in multivariable calculus for engineering applications

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 Calculus II: Multivariable Functions with these activities:

Review 'Multivariable Mathematics' by Hubbard and Hubbard

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Enhance your understanding of multivariable calculus by reviewing a comprehensive and well-written textbook.

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Read the relevant chapters covering the course topics.
Take notes and highlight important concepts.
Complete the practice problems at the end of each chapter.

Organize your notes and practice problems

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Prepare for the course by organizing your existing materials to facilitate effective review.

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Gather all notes, practice problems, and assignments from previous courses.
Create a filing system to organize the materials by topic.
Review the materials regularly to refresh your memory.

Review the chain rule

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Brush up on the chain rule to enhance your understanding of multivariable calculus.

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Read a short article or watch a video explaining the chain rule.
Practice applying the chain rule to simple examples.
Complete a set of practice problems using the chain rule.

Five other activities

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Practice finding partial derivatives

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Sharpen your partial derivative skills to strengthen your foundation for multivariable calculus.

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Review the concepts of partial derivatives.
Solve a set of practice problems to find partial derivatives of various functions.
Check your answers against provided solutions.

Explore applications of gradients

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Expand your understanding of gradients by exploring their practical applications.

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Find resources online or consult textbooks on applications of gradients.
Read articles or watch videos to learn about real-world uses of gradients.
Discuss the applications with a study group or mentor.

Form a study group with classmates

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Enhance your learning experience by collaborating with others through a study group.

Browse courses on Multivariate Calculus

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Find classmates with similar goals and schedules.
Establish meeting times and study plan.
Work together on problem sets, review concepts, and discuss ideas.

Create a presentation on the applications of double integrals

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Solidify your understanding of double integrals by explaining their applications to others in a presentation.

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Research various applications of double integrals.
Design slides and prepare a coherent presentation.
Practice presenting the material and answer questions.

Visualize multivariable functions

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Enhance your spatial reasoning and visualize multivariable functions to deepen your understanding.

Browse courses on Multivariate Calculus

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Choose a multivariable function to visualize.
Use software or online tools to create graphs, contour plots, or other visualizations.
Analyze the visualizations and identify key features of the function.

Career center

Learners who complete Calculus II: Multivariable Functions will develop knowledge and skills that may be useful to these careers:

Data Scientist

Data scientists use mathematical and statistical techniques to extract knowledge and insights from data. Calculus II: Multivariable Functions can provide data scientists with a strong foundation in mathematics, which is essential for understanding data science algorithms and techniques. The course also covers topics such as machine learning and artificial intelligence, which are used in data science.

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Operations Research Analyst

Operations research analysts use mathematical and analytical techniques to solve problems in a variety of industries, including manufacturing, transportation, and healthcare. Calculus II: Multivariable Functions can provide operations research analysts with a strong foundation in mathematics, which is essential for understanding and solving complex problems. The course also covers topics such as optimization and linear algebra, which are used in operations research.

See salaries and explore the career path for Operations Research Analyst

Data Analyst

Data analysts may be involved in a range of tasks, including collecting, cleaning, and processing data; performing statistical analysis; and creating data visualizations. Calculus II: Multivariable Functions can provide a strong foundation for data analysts by providing them with a deep understanding of mathematical concepts such as derivatives, integrals, and vector calculus. These concepts are essential for understanding data and solving problems in data analysis.

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Statistician

Statisticians collect, analyze, and interpret data to help businesses and organizations make informed decisions. Calculus II: Multivariable Functions can provide statisticians with a strong foundation in mathematics, which is essential for understanding statistical methods and data analysis. The course also covers topics such as probability and inference, which are used in statistics.

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Biostatistician

Biostatisticians use statistical methods to design and analyze studies in the health sciences. Calculus II: Multivariable Functions can provide biostatisticians with a strong foundation in mathematics, which is essential for understanding statistical methods and data analysis. The course also covers topics such as probability and inference, which are used in biostatistics.

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Investment Analyst

Investment analysts use mathematical and statistical techniques to evaluate investments and make recommendations to clients. Calculus II: Multivariable Functions can provide investment analysts with a strong foundation in mathematics, which is essential for understanding financial data and making sound investment decisions. The course also covers topics such as derivatives and integrals, which are used in financial modeling and risk analysis.

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Financial Analyst

Financial analysts use mathematical and statistical techniques to evaluate investments and make recommendations to clients. Calculus II: Multivariable Functions can provide financial analysts with a strong foundation in mathematics, which is essential for understanding financial data and making sound investment decisions. The course also covers topics such as derivatives and integrals, which are used in financial modeling and risk analysis.

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Risk Analyst

Risk analysts use mathematical and statistical techniques to assess risk and uncertainty. Calculus II: Multivariable Functions can provide risk analysts with a strong foundation in mathematics, which is essential for understanding risk and uncertainty. The course also covers topics such as probability and statistics, which are used in risk analysis.

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Quantitative Analyst

Quantitative analysts use mathematical and statistical techniques to analyze financial data and make investment decisions. Calculus II: Multivariable Functions can provide quantitative analysts with a strong foundation in mathematics, which is essential for understanding financial data and making sound investment decisions. The course also covers topics such as derivatives and integrals, which are used in financial modeling and risk analysis.

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Actuary

Actuaries use mathematical and statistical techniques to assess risk and uncertainty. Calculus II: Multivariable Functions can provide actuaries with a strong foundation in mathematics, which is essential for understanding risk and uncertainty. The course also covers topics such as probability and statistics, which are used in actuarial science.

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Economist

Economists study the production, distribution, and consumption of goods and services. Calculus II: Multivariable Functions can provide economists with a strong foundation in mathematics, which is essential for understanding economic models and data. The course also covers topics such as optimization and game theory, which are used in economics.

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Software Engineer

Software engineers design, develop, and maintain software applications. Calculus II: Multivariable Functions can provide software engineers with a strong foundation in mathematics, which is essential for understanding software design and development. The course also covers topics such as data structures and algorithms, which are used in software engineering.

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Market Researcher

Market researchers use statistical methods to collect and analyze data about consumer behavior. Calculus II: Multivariable Functions can provide market researchers with a strong foundation in mathematics, which is essential for understanding statistical methods and data analysis. The course also covers topics such as probability and inference, which are used in market research.

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Business Analyst

Business analysts use mathematical and statistical techniques to analyze business data and make recommendations to management. Calculus II: Multivariable Functions can provide business analysts with a strong foundation in mathematics, which is essential for understanding business data and making sound decisions. The course also covers topics such as optimization and game theory, which are used in business analysis.

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Teacher

Teachers at high schools or community colleges may find Calculus II: Multivariable Functions helpful, as this course covers the mathematical concepts that are typically taught in high school and college calculus courses. The course also provides a deeper understanding of these concepts, which can be helpful for teachers who want to improve their teaching skills.

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