Unit · year 2
MU-203 · Multivariable & Vector Calculus
Threads space · change26 lectures5 theorems
Calculus in many dimensions, culminating in the great integral theorems.
Lectures
| L01 | Euclidean Space, Curves, and Surfaces — |
| L02 | Limits and Continuity in Several Variables — |
| L03 | Partial Derivatives and Differentiability — |
| L04 | The Total Derivative and the Jacobian Matrix |
| L05 | The Chain Rule in Several Variables — |
| L06 | Directional Derivatives and the Gradient — |
| L07 | The Inverse Function Theorem — |
| L08 | The Implicit Function Theorem — |
| L09 | Taylor's Theorem in Several Variables — |
| L10 | Critical Points and the Hessian — |
| L11 | Constrained Optimisation: Lagrange Multipliers |
| L12 | Applications of Lagrange Multipliers |
| L13 | Multiple Integrals and Fubini's Theorem — |
| L14 | Change of Variables and the Jacobian |
| L15 | Polar, Cylindrical, and Spherical Coordinates |
| L16 | Line Integrals and Work — |
| L17 | Conservative Fields and Path Independence — |
| L18 | Vector Fields: Divergence and Curl — |
| L19 | Green's Theorem |
| L20 | Surface Integrals and Flux — |
| L21 | Stokes' Theorem |
| L22 | The Divergence Theorem |
| L23 | Applications: Flux, Circulation, and Field Equations |
| L24 | Differential Forms: A First Look — |
| L25 | The Generalised Stokes Theorem |
| L26 | Synthesis: One Theorem in Many Dresses |
Theorems in this unit
T-052
Change of variables and the Jacobian
How volumes transform under a smooth change of coordinates.
T-053
The method of Lagrange multipliers
Constrained extrema occur where gradients align.
T-054
Green's theorem
A planar circulation integral equals a double integral of curl.
T-055
Stokes' theorem
Circulation around a boundary equals the flux of curl through the surface.
T-056
The divergence theorem
Flux through a closed surface equals the integral of divergence inside.