Maths WorkshopMaths problems with full solutions

Vector calculus

34 problems · Line integrals, surface integrals and the classical theorems.

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Ex.591The divergence

#divergence #vector field #computation

Vector calculus

Ex.592Closed paths in conservative fields

#closed path #conservative #zero integral

Vector calculus

Ex.593Line integral of a scalar field

#line integral #scalar field #parametrization

Vector calculus

Ex.794Line integral of a differential form

#line integral #differential form #parametrization

Vector calculus

Ex.1198The curl

#curl #vector field #rotation

Vector calculus

Ex.1201Flux through a flat surface

#flux #plane surface #dot product

Vector calculus

Ex.1202An application of Green's theorem

#Green #application #area

Vector calculus

Ex.1203Area using Green's theorem

#area #Green #closed curve

Vector calculus

Ex.1204Green with symmetry

#Green #symmetry #shortcut

Vector calculus

Ex.1210Conservative vector field

#conservative #gradient #path independence

Vector calculus

Ex.1211Fundamental theorem for line integrals

#fundamental theorem #line integral #potential

Vector calculus

Ex.1212Line integral along a circle

#line integral #circle #parametrization

Vector calculus

Ex.1213Line integral on a piecewise path

#piecewise path #line integral #work

Vector calculus

Ex.1214Work done by a force field

#work #line integral #vector field

Vector calculus

Ex.1215Testing whether a field is conservative

#conservative #curl #test

Vector calculus

Ex.1518Divergence theorem (Ostrogradsky)

#divergence theorem #flux #surface

Vector calculus

Ex.1519Green-Riemann formula

#Green-Riemann #line integral #double integral

Vector calculus

Ex.1520Stokes' theorem

#Stokes #curl #surface integral

Vector calculus

Ex.1903Gauss divergence theorem for a cylinder

#divergence #cylinder #flux

Vector calculus

Ex.1904The divergence theorem

#divergence theorem #flux #surface

Vector calculus

Ex.1905Flux through a cube

#divergence #cube #flux

Vector calculus

Ex.1908Divergence theorem with variable divergence

#divergence #sphere #variable density

Vector calculus

Ex.1910Recovering a field from its curl (2D)

#Green #curl #reconstruction

Vector calculus

Ex.1911Green's reciprocity

#Green #reciprocity #symmetry

Vector calculus

Ex.1912Green's theorem on an annulus

#Green #annulus #inner boundary

Vector calculus

Ex.1913Green's theorem

#Green theorem #curl #region

Vector calculus

Ex.1914Green's theorem with a general field

#Green #general field #region

Vector calculus

Ex.1917Area of a region using line integral

#line integral #area #polar

Vector calculus

Ex.1918Conservative fields in 3D

#conservative #3D #potential

Vector calculus

Ex.1919Finding a potential function

#potential #conservative #integration

Vector calculus

Ex.1925Stokes with a hemisphere

#Stokes #hemisphere #closed curve

Vector calculus

Ex.1926Stokes' theorem

#Stokes theorem #curl #surface

Vector calculus

Ex.2223Stokes on a cone

#Stokes #cone #surface

Vector calculus

Ex.2224Stokes on a torus (statement)

#Stokes #torus #closed surface

Vector calculus