Search results for “calculus”

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Calculus of Variations Differential Geometry

6 results

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.

380

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.

260

Vector calculus. Line integrals of the first kind

2026Problem sheetSergey

Integration with respect to arc length: parametrising a curve, the length element $ds = |\mathbf{r}'(t)|\,dt$ , arc length, curves in polar coordinates, the mass and centre of mass of a wire, symmetry instead of computation.

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.

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.

310

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.

340