Warwick MA259 Multivariable Calculus: Syllabus, Topics and Practice

MA259 Multivariable Calculus is a Level 2 module at the University of Warwick that takes the one-variable analysis of Year 1 into several dimensions. The catalogue calls it "the final in a series of modules" in which analysis is developed rigorously in a many-dimensional setting. It is worth 10 credits and runs for 10 weeks. This guide maps each MA259 topic to free practice, so you can build the computational skills while your lectures supply the theory.

What Warwick MA259 covers

The official module catalogue gives this indicative outline syllabus:

• different notions of continuity for functions of several variables,
• quantitative linear algebra in terms of norms,
• different notions of differentiability for functions of several variables,
• the chain rule, the (generalised) mean value inequality and other properties of differentiable functions,
• the Inverse Function Theorem and Implicit Function Theorem, with applications to regular curves and hypersurfaces,
• vector fields and the theorems of Green, Gauss and Stokes, with some applications to PDEs,
• maxima, minima and saddles, and constrained critical points.

The learning outcomes add that you should be able to analyse and classify critical points using Taylor expansions. The catalogue lists 30 lectures and 9 seminars, but it does not state formal prerequisites or the term.

In US terms, MA259 covers the core of Calculus 3: partial derivatives, the multivariable chain rule, optimisation and the big theorems of vector calculus, with the proofs of an analysis course on top.

MA259 topic map

The Inverse and Implicit Function Theorems are proof-based results we do not teach directly. The implicit differentiation lesson gives you the calculation those theorems justify.

Example problems at the MA259 level

1. Classifying a critical point. For f(x, y) = x^2 - y^2, the only critical point is (0, 0). The Hessian determinant is f_{xx}f_{yy} - f_{xy}^2 = (2)(-2) - 0 = -4 < 0, so the origin is a saddle.

2. A constrained maximum. Maximise f(x, y) = xy on x + y = 10. Lagrange's condition \nabla f = \lambda \nabla g gives y = \lambda and x = \lambda, so x = y = 5 and the maximum is 25.

3. Green's theorem. For \mathbf{F} = (-y, x) around the unit circle,

How to do well in MA259

• Keep the calculations sharp. Theory questions often hide a partial derivative or a line integral. Practise those until they are routine.
• Draw pictures. Level curves, gradient vectors and the orientation of a boundary make the integral theorems far easier to apply correctly.
• Revise your linear algebra. Derivatives in several variables are linear maps, so matrices and norms are everywhere.
• Review Year 1 analysis. The module explicitly extends MA142 and MA143, so gaps there show up quickly.

Tip: before using Green's or Stokes' theorem, check the orientation of the curve. A wrong direction flips the sign of the answer.