Course category:2nd year
At the end of the module you should be able to
Introduction to vector calculus. Elementary topology of Euclidean space.
Limits and continuity. Basic properties of vector limits.
Multivariable differentiation. Linear maps. Composition of continuous functions. Linear approximation and the multivariable derivative. Basic properties of the derivative. The Chain Rule. Paths, directional derivatives, partial derivatives and the Jacobian matrix. The differentiability test.
Line integrals. Line integrals and mechanical work. Fundamental theorems of line integration. Poincare's Lemma, curl, work and energy. Line integration with respect to arc length.
Green's Theorem. Harmonic conjugates, Green's Theorem. Change of variables formula for double integrals (revision from Calculus). Examples of Green's Theorem, application to area.
Divergence Theorem in two dimensions.
Symmetry principles for double integrals.
Review of vector algebra.
Index notation. Kronecker delta and epsilon, and their properties.
Vector differential operators in 3 dimensional Euclidean space. Gradient, divergence, curl, directional derivatives and the Laplacian; differentiation of products; chain rules. (Partly a review of objects introduced in Calculus.) Div, grad and Laplacian in curvilinear coordinates; solution by separation of variables; physical examples (eg. vibrations of a drumskin).
Parametric description of surfaces in space, and their orientation.
Surface and volume integrals. Integration over a parametric surface and volume integrals, including Cartesian, spherical polar, and cylindrical polar coordinates.
Stokes and Gauss theorems, and their applications.
Intrinsic definitions of gradient, divergence and curl. (if time permits)
Poincare's lemma. Scalar and vector potentials.(if time permits)
Three hour examination summer term.
a mature study of fundamental ideas in advanced calculus.
Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.
Department of Mathematics, University of York, Heslington, York, UK. YO10 5DD