Fun example: empty colimit does not commute with empty limit
2026NotesBogdan Zavyalov
Why the definition of a filtered category insists on non-emptiness: the empty colimit fails to commute with the empty limit, and that is the only failure.
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310
Search results for “limits”
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11 results
Vector calculus. Surface integrals of the first kind
2026Problem sheetSergey
Integration with respect to surface area: the area element for an explicit and a parametric surface, pieces of planes, the sphere in spherical coordinates.
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340
Vector calculus. Surface integrals of the second kind. Flux
2026Problem sheetSergey
The flux of a vector field through an oriented surface: the normal and the choice of side, the explicit and parametric forms, the lateral surface of a cylinder.
0
380
Vector calculus. Line integrals of the first kind
2026Problem sheetSergey
Integration with respect to arc length: parametrising a curve, the length element , arc length, curves in polar coordinates, the mass and centre of mass of a wire, symmetry instead of computation.
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530
Vector calculus. Line integrals of the second kind. Work of a field
2026Problem sheetSergey
Integrating a vector field along a directed curve: parametrising a segment and a graph, the work of a force, path dependence, a potential field and its potential, a curl-free field in a domain that is not simply connected.
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400
Vector calculus. Stokes' theorem. Circulation
2026Problem sheetSergey
Circulation of a field as the flux of its curl: computing the curl, matching the orientation of the contour and the normal, circulation computed two ways.
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310
Vector calculus. Green's theorem
2026Problem sheetSergey
A contour integral as a double integral: the hypotheses of Green's theorem, orientation of the contour, area as a contour integral, closing an open arc, an annulus, a singular point inside the contour.
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260
Cramer's Rule and Gaussian Elimination
2026TheorySergey
Two methods for solving a SLAE — Cramer's rule via determinants and the universal Gaussian elimination method, with examples.
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250