Calculus Refresher & Multivariable Calculus

This module restores and extends calculus fluency to the level expected of postgraduate researchers across physical, life and social sciences. Learners revisit limits, differentiation and integration in one variable before advancing to partial derivatives, multiple integrals, vector calculus and the classical integral theorems. Mastery is demonstrated when a learner can set up and evaluate multivariable derivatives and integrals that arise in gradient-based learning, continuum models and probability densities, and can diagnose which theorem (Green, Stokes, divergence, Taylor, Lagrange) applies to a given problem.

Learning Outcomes

Recall and apply limit, continuity and differentiability criteria for scalar functions of one and several variables.

Compute partial derivatives, gradients, Jacobians and Hessians and interpret them geometrically.

Evaluate iterated and change-of-variable multiple integrals in Cartesian, polar, cylindrical and spherical coordinates.

Apply Green's, Stokes' and the divergence theorem to convert between line, surface and volume integrals.

Locate and classify constrained and unconstrained extrema using the Lagrange and second-derivative tests.

Expand smooth functions as multivariable Taylor series and bound the remainder.

Translate research problems in optimisation, probability densities or continuum models into multivariable calculus statements.

Judge when a numerical or symbolic approach is more appropriate for a given integral or derivative.

Topics Covered

Limits, continuity and the ε–δ view

Differentiation rules, mean-value theorem and L'Hopital

Taylor's theorem in one variable with Lagrange and integral remainder

Techniques of integration: substitution, parts, partial fractions, trigonometric

Improper integrals, convergence tests and the Γ and Β functions

Partial derivatives, differentiability and the chain rule for several variables

Gradient, directional derivative, Jacobian and Hessian matrices

Taylor series in several variables and quadratic approximation

Multiple integrals: Fubini, change of variables and the Jacobian determinant

Line and surface integrals of scalar and vector fields

Conservative fields, potentials and path independence

Green's theorem, Stokes' theorem and the divergence theorem

Unconstrained extrema and the second-derivative test

Constrained optimisation with Lagrange multipliers and the bordered Hessian

Parameter-dependent integrals and differentiation under the integral sign (Leibniz)

Skills Developed

Symbolic differentiation and integration Coordinate-system reasoning Vector-field analysis Series approximation Constrained optimisation Dimensional and sanity checking Translating applied problems into calculus Working across symbolic and numerical tools

Suggested Resources

Marsden, J.E. & Tromba, A.J. (2012). Vector Calculus, 6th ed., W.H. Freeman

Stewart, J. (2020). Calculus: Early Transcendentals, 9th ed., Cengage

Spivak, M. (1965). Calculus on Manifolds, Addison-Wesley

Apostol, T.M. (1969). Calculus, Volume II, 2nd ed., Wiley

Hubbard, J.H. & Hubbard, B.B. (2015). Vector Calculus, Linear Algebra and Differential Forms, 5th ed., Matrix Editions

MIT OpenCourseWare 18.02 Multivariable Calculus (Prof. Denis Auroux)

Khan Academy multivariable calculus lecture series

Assessment

Learners are judged on the ability to set up, execute and interpret multivariable calculus arguments relevant to research. The weighting rewards sustained problem-solving over single-sitting examination.

Written problem set portfolio across six sub-topics40%

Timed closed-book examination with multi-step computations35%

MCQ diagnostic bank (this module's 200+ item pool)25%

Practice bank: 100 MCQs · Timed mock & self-paced modes

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