Course

Multivariable calculus

Course under construction · 37 of 38 concepts published; more are added as they are reviewed.

You will learn to differentiate and integrate functions of several variables, optimize under constraints and compute line and surface integrals up to the theorems of Green, Stokes and divergence. It is used to describe fields, flows and volumes in physics, engineering and data science.

37concepts

6topics

370exercises

≈ 19 hof study approx.

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Course path

1
Vectors and curves in space 5 concepts
Cross product · Lines and planes in space · Parametric curve · Arc length · Quadric surface
2
Differentiation in several variables 9 concepts
Function of several variables · Limit of a function of several variables · Level curve · Partial derivative · Tangent plane · Differentiability in several variables · Gradient · Directional derivative · Chain rule in several variables
3
Optimization 5 concepts
Critical point of a multivariable function · Hessian matrix · Second derivative test · Lagrange multipliers · Absolute extrema on a region
4
Multiple integrals 7 concepts
Double integral · Fubini's theorem · Double integral over a general region · Triple integral · Polar coordinates · Cylindrical and spherical coordinates · Change of variables in multiple integrals
5
Line and surface integrals 6 concepts
Line integral of a scalar function · Vector field in the plane and space · Line integral of a vector field · Conservative field · Parametric surface · Surface integral
6
Integral theorems 5 concepts
Divergence · Curl · Stokes' Theorem · Green's Theorem · Divergence Theorem

Sources

James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, 9th edition, Cengage, 2021.
Jerrold E. Marsden, Anthony Tromba, Vector Calculus, 6th edition, W. H. Freeman, 2012.
Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr., Thomas' Calculus, 15th edition, Pearson, 2023.