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Calculus 3: Flux

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All Calculus 33D SpaceVector FunctionsDot and cross productEquations of lines and planesParametric curves, conic sectionsTangent vectors and arc lengthCylinders and quadric surfacesIntegrals of vector functionsArc length and curvatureMultivariable functionsSurface parameterizationPartial derivativesLinearization, chain rule, gradientTangent planes and linear approximationsOptimizationLagrange multipliersDouble integralsTriple integralsChanging coordinates for integrationSurface areaVector fields, divergence, and curlLine integralsGreen's TheoremFluxStokes' TheoremDivergence TheoremComplex numbers
Flux Through a Curved Surface

For a vector field F=⟨x2,xy,z⟩F = \langle x^2, xy, z \rangle and the surface given by z=x+y2z = x + y^2, find the flux of the field through this surface over the range xx from 0 to 1 and yy from 0 to 1.

FluxMediumVideo
Flux of a Vector Field across z 1 x2 y2

Compute the flux of a vector field across the surface z=1−x2−y2,z = 1 - x^2 - y^2, with z≥0,z \geq 0, where the vector field is f(x,y,z)=(x,y,z).\mathbf{f}(x, y, z) = (x, y, z). The surface is oriented with outward normals from the perspective of starting at the origin and moving out to the surface.

FluxMediumVideo