This course provides an in depth exposition of the theory of differential equations and vector calculus. Applications will be related to problems mainly from the Physical Sciences.

Topics to be covered include:

Ordinary Differential Equations - Linear and non-linear first and second order differential equations; existence and uniqueness of solutions; solution methods; the use of Green’s functions; power series solutions; Bessel and Legendre equations; the gamma function; systems of first and second order equations; equilibrium points and stability; phase portraits; elementary bifurcation theory; the van der Pol equation; normal modes of oscillation; boundary value problems; regular and singular Sturm-Liouville systems and eigenvalue problems; generalized Fourier series.

Advanced Vector Calculus - Curves and surfaces in three dimensions; parametric representations; curvilinear coordinate systems; surface and volume integrals; use of Jacobians; scalar and vector fields; gradient, divergence and curl in orthogonal curvilinear coordinates. identities involving vector differential operators; the Laplacian; Green’s theorem in the plane; Divergence and Stoke’s theorems; scalar and vector potentials.

Note: This is an HPC, taught at a level requiring greater conceptual understanding than MATH2305.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of differential equations and vector calculus and their role in modern mathematics and applied contexts.
2. Demonstrate accurate and efficient use of techniques involved in solving differential equations and applying vector differential operators.
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from the theory of differential equations.
4. Apply problem-solving using techniques in differential equations and vector calculus in diverse situations in physics, engineering and other mathematical contexts.

Assessment will be based on:

• Eight assignments (25% in total; LO 1-4)
• Mid-semester test (25%; LO 1-4)
• Final examination (50%; LO 1-4)

48 lectures and 10 tutorials

A mark of 60 or more in MATH1116; or a mark of 80 or more in MATH1014.

MATH2014, MATH2114, MATH3109, MATH3209, MATH2305 and ENGN2212.