Multivariable & Vector Calculus

Calculus in more than one dimension, ending in one theorem

About this course

Partial derivatives, multiple integrals and vector calculus, culminating in the unification of Green, Stokes and the divergence theorem.

What you will be able to do

  • Differentiate functions of several variables and interpret the gradient
  • Optimise multivariable functions, with and without constraints
  • Evaluate double and triple integrals, and change variables with the Jacobian
  • Compute line and surface integrals over vector fields
  • State and apply Green's, Stokes' and the divergence theorems as one result

What this course covers

  • Critical points in two variables
  • Failure of the naive second-derivative test
  • The discriminant D and the second-derivative test
  • The Hessian matrix
  • Principal directions and eigenvalues of the Hessian
  • Lagrange multipliers for constrained optimisation
  • Complete worked example and practical summary
  • Functions of two variables as surfaces
  • Contour maps and level curves
  • Limits in the plane: infinitely many paths
  • Polar coordinates as a proof tool for limits
  • Continuity and surviving theorems in two variables
  • Worked limit examples: failure and proof side by side
  • Double integral as a limit of columns
  • Fubini's theorem and order of integration
  • Describing and swapping the order of integration over non-rectangular regions
  • Applications: area, average value, mass, and centroid
  • Worked example: volume under z = x + 2y, both orders
  • Green's Theorem: Statement and Orientation
  • Proof by Cell Cancellation
  • Flux Form and Divergence
  • Area from a Line Integral (Planimeter and Shoelace Formula)
  • Regions with Holes and Loop Deformation
  • Strategy, Cautions, and Broader Context
  • Worked Verification: Both Sides of Green's Theorem
  • Vector fields: arrows at every point
  • Historical origin of vector-field notation
  • Divergence: flux per unit volume in the limit
  • The del (∇) operator: three jobs, one symbol
  • Curl: the paddle-wheel measurement
  • Two vector-calculus identities via mixed partials
  • Maxwell's equations read through divergence and curl
  • Partial derivative definition and notation
  • Geometric meaning: slopes of surface slices
  • Partial derivatives as physical/applied rates of change
  • Second-order partial derivatives and their geometric meaning
  • Clairaut's theorem and the order-of-mixed-differentiation counterexample
  • Partial derivatives as the language of physical laws
  • Weakness of partial-derivative existence: no continuity guarantee
  • Line integral of a scalar field
  • Line integral of a vector field (work)
  • Closed-loop integrals: circulation and flux
  • Parametrisation invariance and orientation
  • The vortex field and the failure of curl-zero implying zero circulation
  • Computational procedure and error-checking habits
  • Preview of conservative fields and Green's theorem
  • The Unifying Shape of All Five Theorems
  • Surface Integrals and the Area Element
  • Stokes' Theorem
  • The Divergence Theorem
  • Exterior Calculus and the Single Generalized Theorem
  • The Boundary of a Boundary is Empty
  • Worked Examples and Practical Strategy
  • Why changing coordinates requires an area correction
  • The polar area element and the origin of the factor r
  • The general change-of-variables rule and Jacobian determinant
  • Standard coordinate changes and their Jacobians
  • The Jacobian as a local area scale factor
  • Choosing coordinates to convert a region into a rectangle
  • The Gaussian integral evaluated exactly via polar coordinates
  • Failure of partial derivatives to guarantee continuity
  • Tangent plane formula and its derivation
  • Definition of differentiability via the error-to-distance ratio
  • Sufficient condition: continuous partials imply differentiability
  • Linearization and the total differential
  • Error propagation using partial derivatives
  • The gradient vector hidden in the tangent plane
  • The 'integral of a derivative over a region equals boundary values' pattern
  • Three equivalent conditions for conservative fields
  • Fundamental theorem for line integrals
  • The curl test and simply connected regions
  • The vortex as a multivalued potential
  • Finding a potential function: the three-step recipe
  • Full workflow: conservative field worked examples
  • Rolling-race outcome and coordinate-system selection guide
  • Triple integral: definition and iteration
  • Cylindrical coordinates and their Jacobian
  • Spherical coordinates and derivation of dV
  • Physical quantities as weighted triple integrals
  • Moment of inertia of a solid sphere: two-fifths derivation
  • Worked examples: ice-cream cone and flywheel
  • Directional derivative: definition and formula
  • The gradient vector: definition and notation
  • Two famous facts: steepest ascent direction and steepest slope magnitude
  • Gradient perpendicular to level curves
  • Extension to three (and more) variables
  • Gradient descent and its practical limitations
  • Capstone worked example and summary rules
  • Chain rule along a path: one parameter
  • Notational trap: partial vs. total derivative
  • Tree diagram and general chain rule
  • Implicit differentiation via the chain rule
  • Implicit function theorem
  • Jacobian matrix and chain rule as matrix multiplication
  • Backpropagation as associativity of matrix multiplication

Syllabus

  1. Derivatives in Many Variables

    1. L1 · Leaving the Line
    2. L2 · The Partial Derivative
    3. L3 · The Tangent Plane
    4. L4 · The Gradient
    5. L5 · The Chain Rule, Rebuilt

  2. Optimisation

    1. L6 · Optimisation in Many Variables

  3. Multiple Integrals

    1. L7 · The Double Integral
    2. L8 · Change of Variables and the Jacobian
    3. L9 · Triple Integrals

  4. Vector Fields and Line Integrals

    1. L10 · Vector Fields
    2. L11 · Line Integrals and Work
    3. L12 · Conservative Fields

  5. The Fundamental Theorems

    1. L13 · Green's Theorem
    2. L14 · Stokes, Divergence, and One Theorem