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Calculus III: Multivariable & Vector Calculus

Vectors and space curves, partial derivatives and the gradient, multiple integrals and the Jacobian, then vector fields up to Green's, Stokes' and the divergence theorem -- unified at the end as one theorem. Runnable Python throughout.

intermediate Free calculusmathematicsvector-calculusmultivariable
📖 32 readings ⚡ 32 exercises · 327 min total
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4 modules · 32 lessons
9. Vectors and Space Curves 7 lessons · 70 min
  1. 1 Vectors in the plane and in space 9 min
  2. 2 The dot product: angles and projections 10 min
  3. 3 The cross product: areas, volumes, and orientation 10 min
  4. 4 Lines, planes, and surfaces in space 10 min
  5. 5 Vector-valued functions and space curves 10 min
  6. 6 Velocity, acceleration, and arc-length parameterization 10 min
  7. 7 Curvature and the TNB frame 11 min
10. Partial Derivatives 8 lessons · 84 min
  1. 8 Functions of several variables: level curves and surfaces 10 min
  2. 9 Limits and continuity in two variables, and why paths matter 11 min
  3. 10 Partial derivatives 10 min
  4. 11 Tangent planes, linear approximation, and total differentials 10 min
  5. 12 The multivariable chain rule 10 min
  6. 13 Directional derivatives and the gradient 11 min
  7. 14 Local extrema, saddle points, and the second-derivative test 11 min
  8. 15 Lagrange multipliers 11 min
11. Multiple Integrals 6 lessons · 58 min
  1. 16 Double integrals over rectangles and general regions 9 min
  2. 17 Double integrals in polar coordinates 10 min
  3. 18 Mass, moments, centroids, and joint densities 9 min
  4. 19 Triple integrals 10 min
  5. 20 Cylindrical and spherical coordinates 10 min
  6. 21 Change of variables and the Jacobian 10 min
12. Vector Calculus 11 lessons · 115 min
  1. 22 Vector fields 10 min
  2. 23 Line integrals of scalar functions 10 min
  3. 24 Line integrals of vector fields: work and circulation 10 min
  4. 25 Conservative fields, potential functions, and path independence 10 min
  5. 26 Green's theorem 9 min
  6. 27 Divergence and curl 11 min
  7. 28 Parametrized surfaces and surface area 11 min
  8. 29 Surface integrals and flux 11 min
  9. 30 Stokes' theorem 11 min
  10. 31 The divergence theorem 11 min
  11. 32 Adv: one theorem in four disguises — a look at differential forms 11 min

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