The problem of automatic computation of cuts for magnetic scalar potentials in 3-dimensional finite element meshes is often regarded as too difficult to handle in a general way. This thesis applies basic techniques from algebraic topology to formulate, analyze, and implement a general, efficient, simple cuts algorithm and gives a concrete formulation of an algorithm for using cuts and a magnetic scalar potential coupled to a stream function for computing eddy currents on thin conducting surfaces. Where they do not rely on standard finite element theory the algorithms are based on integer data structures and arithmetic, so that questions of analysis in "numerical analysis" are moot and the algorithms scale with problem size. The method of computing and using cuts is a general approach for 3-d low-frequency engineering magnetics problems which is immune to rounding and cancellation errors and is an order of magnitude faster than typical vector methods.; Functorial data structures formulated for the cuts algorithm have a wide range of applicability for topological computations on 3-dimensional finite element meshes, and are a means of interpreting the formal link between finite elements, cohomology theory, and lumped circuit parameters of electrical engineering.
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