Multivariable Calculus 3: Theorems and Applications

We live in a multivariable world. Explore elegant theorems connecting differentiation, integration, and geometry in higher dimensions, and learn how to apply them to solve real world problems. Part 3 of 3.

Multivariable Calculus 3: Theorems and Applications

We live in a multivariable world. Explore elegant theorems connecting differentiation, integration, and geometry in higher dimensions, and learn how to apply them to solve real world problems. Part 3 of 3.

How do fields like fluid flow, electricity, and gravity behave in three-dimensional space? What are the mathematical structures that describe circulation, outflow, and equilibrium in space?

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Multivariable Calculus provides a framework for answering these questions in multiple dimensions. Whether modeling the movement of air over a wing, the flow of water through a pipe, or the spread of heat in a solid, you’ll learn how to use the mathematical language that describes such systems.

This online course – Part 3 of a three-part series from the MIT Department of Mathematics -- builds on earlier topics to explore more advanced concepts and techniques in multivariable calculus. It also uncovers deep connections with earlier concepts that manifest prominently in three-dimensional contexts. Some of the questions addressed include:

  • How much fluid passes through a curved surface at a given time?
  • How can one determine whether a vector field is conservative?
  • Under what conditions does a field exhibit rotational behavior?
  • How are local divergence and curl related to total flux and circulation?

This course develops methods for computing surface integrals and line integrals in dimension three and introduces the theorems—Divergence and Stokes’—that relate these integrals under operations taking differentials of integrand fields and taking boundaries of integration domains.

What you'll learn

  • Compute surface integrals that measure flux through spheres, cylinders, and parametrized surfaces
  • Compute line integrals that represent work done by vector fields along curves in space
  • Determine whether a vector field is conservative and construct corresponding potential functions
  • Apply the Divergence Theorem to relate surface flux to volume integrals
  • Compute the curl of a vector field and interpret its physical meaning as local rotation
  • Apply Stokes’ Theorem to relate circulation around a curve to the flux of curl through a surface
  • Analyze physical problems involving fluid flow, electromagnetism, and diffusion

Prerequisites

Meet your instructors

  • Featured image for Lawrence Guth
    Claude Shannon Professor of Mathematics MacVicar Faculty Fellow
  • Featured image for Denis Auroux
    Herchel Smith Professor of Mathematics, Harvard University

Who can take this course?

Because of U.S. Office of Foreign Assets Control (OFAC) restrictions and other U.S. federal regulations, learners residing in one or more of the following countries or regions will not be able to register for this course: Iran, Cuba, Syria, North Korea and the Crimea, Donetsk People's Republic and Luhansk People's Republic regions of Ukraine.