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Multivariable Calculus 2: Integrals
Sobre este cursoOmitir Sobre este curso
Variables are all around us: temperature, altitude, location, profit, color, and countless others. Multivariable Calculus is the tool of choice to shed light on complex relationships between 2, 3, or hundreds of variables simultaneously. Some of the multivariable questions considered in this course include:
- How can one quantify the efficiency of a power plant?
- How much snow can a roof safely hold?
- Over the next 100 years, how high will continually melting icebergs raise sea levels?
The key tool for answering each of these questions is multivariable integration.
In this course, you will learn how to set up, solve, and interpret many types of multivariable integrals:
- double integrals of scalar functions in any coordinate system,
- line integrals of scalar and vector-valued functions,
- triple integrals in cartesian, cylindrical, and spherical coordinates.
Physical applications will be highlighted, including the use of integrals to compute the work done by a force field, or the flux caused by a velocity field.
Finally, you will learn powerful tools for simplifying integral computations, including the Fundamental Theorem of Line Integrals and Green’s Theorem.
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Lo que aprenderásOmitir Lo que aprenderás
- How to compute double and triple integrals in different coordinates systems.
- How to compute flux through curves and surfaces, and work along a curve.
- How to parameterize surfaces and compute surface integrals.
- How to use integrals to solve word problems.
- How to relate different types of integrals using the Fundamental Theorem of Line Integrals, Green’s Theorem, and the Divergence Theorem.
Plan de estudiosOmitir Plan de estudios
Unit 1: Double Integrals
- Double integrals
- Polar coordinates
- Average value,
- Center of mass,
- Moment of inertia
- Change of variables
Unit 2: Introduction to Line Integrals
- Arc length and line integrals
- Force fields and work
- Conservative fields and path independence
- Fundamental Theorem for line integrals
Unit 3: Line Integrals and Green’s Theorem
- Gradient fields and potential functions
- Green’s Theorem
- Flux and Green’s Theorem in normal form
- Simply connected regions
Unit 4: Triple Integrals
- Volume integrals
- Cylindrical coordinates
- Spherical Coordinates
- Review all types of integrals
Acerca de los instructores
Preguntas frecuentesOmitir Preguntas frecuentes
Is 18.02.1x required for this course?
Yes. Although any first course on multivariable calculus will also work! The key here is that you must be comfortable with scalar and vector-valued functions of more than one variable, how to interpret them geometrically, and how to take derivatives of them.
Can I take this course concurrently with 18.01.2x?
We do not recommend this! We will expect that you are already familiar with many of the integration techniques introduced in the last unit of 18.01.2x at the beginning of this course.