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This subject covers the material presented in MAST20009 Vector Calculus plus additional material designed to provide deeper insight into interesting areas of calculus and has a greater emphasis on mathematical rigour and proof.
This subject studies the fundamental concepts of functions of several variables and vector calculus. It develops the manipulation of partial derivatives and vector differential operators. The gradient vector is used to obtain constrained extrema of functions of several variables. Line, surface and volume integrals are evaluated and related by various integral theorems. Vector differential operators are also studied using curvilinear coordinates.
Functions of several variables topics include: limits, continuity, differentiability, the chain rule, Jacobian, implicit and inverse function theorems, Taylor polynomials and Lagrange multipliers. Vector calculus topics include: vector fields, flow lines, curvature, torsion, gradient, divergence, curl and Laplacian. Integrals over paths and surfaces topics include line, surface and volume integrals; change of variables; applications including moments of inertia, centre of mass; Green's theorem, Divergence theorem in the plane, Gauss' divergence theorem, Stokes' theorem; and curvilinear coordinates. Possible additional topics include differential geometry of surfaces.
Intended learning outcomes
On completion of this subject, students should be able to:
- Apply calculus to the functions of several variables; differential operators; line, surface and volume integrals; curvilinear coordinates; integral theorems;
- Demonstrate the ability to work with limits and continuity; obtain extrema of functions of several variables; calculate line, surface and volume integrals; work in curvilinear coordinates; apply integral theorems;
- Define fundamental concepts of vector calculus; the relations between line, surface and volume integrals.
- Apply and interpret theorems of calculus in a rigorous way.
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 31 January 2024