Calculus 4: Vector Calculus on Teddy Rook Bookhttps://teddyrookbook.com/math/calculus-4/Recent content in Calculus 4: Vector Calculus on Teddy Rook BookTeddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenVector Fieldshttps://teddyrookbook.com/math/calculus-4/vector-fields/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/vector-fields/A vector field attaches an arrow to every point. Calculus 4: Vector Calculus · Math LadderGradient Fields and Potentialshttps://teddyrookbook.com/math/calculus-4/gradient-fields/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/gradient-fields/F = ∇f for a potential function f. Calculus 4: Vector Calculus · Math LadderDivergence and Curlhttps://teddyrookbook.com/math/calculus-4/divergence-curl/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/divergence-curl/∇ · F measures spreading out; ∇ × F measures rotation. Calculus 4: Vector Calculus · Math LadderWhat Curl and Divergence Measurehttps://teddyrookbook.com/math/calculus-4/curl-divergence-meaning/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/curl-divergence-meaning/Read sources, sinks and spin from a picture of a field. Calculus 4: Vector Calculus · Math LadderParametrizing Curves for Integrationhttps://teddyrookbook.com/math/calculus-4/parametrize-curves/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/parametrize-curves/Write segments, circles and graphs as r(t) with an orientation. Calculus 4: Vector Calculus · Math LadderLine Integrals of Functionshttps://teddyrookbook.com/math/calculus-4/scalar-line-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/scalar-line-integrals/∫_C f ds adds f along a curve, weighted by arc length. Calculus 4: Vector Calculus · Math LadderLine Integrals of Vector Fieldshttps://teddyrookbook.com/math/calculus-4/work-line-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/work-line-integrals/∫_C F · dr measures work along a path. Calculus 4: Vector Calculus · Math LadderLine Integrals in the Form ∫P dx + Q dyhttps://teddyrookbook.com/math/calculus-4/differential-form/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/differential-form/Write and compute work with differentials. Calculus 4: Vector Calculus · Math LadderCirculation and Flux across Plane Curveshttps://teddyrookbook.com/math/calculus-4/circulation-flux-curves/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/circulation-flux-curves/Flux across a curve uses the outward normal. Calculus 4: Vector Calculus · Math LadderThe Fundamental Theorem for Line Integralshttps://teddyrookbook.com/math/calculus-4/ftc-line-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/ftc-line-integrals/∫_C ∇f · dr = f(end) − f(start). Calculus 4: Vector Calculus · Math LadderTesting for Conservative Fieldshttps://teddyrookbook.com/math/calculus-4/conservative-test/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/conservative-test/On a simply connected region, curl F = 0 means F is conservative. Calculus 4: Vector Calculus · Math LadderFinding a Potential Functionhttps://teddyrookbook.com/math/calculus-4/find-potential/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/find-potential/Integrate one component and match the others. Calculus 4: Vector Calculus · Math LadderPath Independence and Closed Loopshttps://teddyrookbook.com/math/calculus-4/path-independence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/path-independence/For conservative fields, work depends only on the endpoints. Calculus 4: Vector Calculus · Math LadderGreen's Theorem: Circulation Formhttps://teddyrookbook.com/math/calculus-4/greens-circulation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/greens-circulation/∮ P dx + Q dy = ∬ (Qx − Py) dA. Calculus 4: Vector Calculus · Math LadderGreen's Theorem: Flux Formhttps://teddyrookbook.com/math/calculus-4/greens-flux/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/greens-flux/Outward flux equals the integral of the divergence. Calculus 4: Vector Calculus · Math LadderArea with Green's Theoremhttps://teddyrookbook.com/math/calculus-4/greens-area/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/greens-area/Find an area with a line integral, such as ½∮(x dy − y dx). Calculus 4: Vector Calculus · Math LadderGreen's Theorem for Regions with Holeshttps://teddyrookbook.com/math/calculus-4/greens-holes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/greens-holes/Orient inner boundaries the opposite way. Calculus 4: Vector Calculus · Math LadderParametric Surfaceshttps://teddyrookbook.com/math/calculus-4/parametric-surfaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/parametric-surfaces/r(u, v) describes a surface with two parameters. Calculus 4: Vector Calculus · Math LadderSurface Area of a Parametric Surfacehttps://teddyrookbook.com/math/calculus-4/parametric-surface-area/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/parametric-surface-area/dS = |r_u × r_v| du dv. Calculus 4: Vector Calculus · Math LadderSurface Integrals of Functionshttps://teddyrookbook.com/math/calculus-4/scalar-surface-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/scalar-surface-integrals/∬_S f dS adds f over a surface. Calculus 4: Vector Calculus · Math LadderOriented Surfaceshttps://teddyrookbook.com/math/calculus-4/oriented-surfaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/oriented-surfaces/Choose a consistent unit normal, like outward on a closed surface. Calculus 4: Vector Calculus · Math LadderFlux Integralshttps://teddyrookbook.com/math/calculus-4/flux-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/flux-integrals/∬_S F · dS measures flow through a surface. Calculus 4: Vector Calculus · Math LadderStokes' Theoremhttps://teddyrookbook.com/math/calculus-4/stokes-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/stokes-theorem/Circulation around a boundary equals the flux of the curl through the surface. Calculus 4: Vector Calculus · Math LadderUsing Stokes' Theorem to Switch Surfaceshttps://teddyrookbook.com/math/calculus-4/stokes-surfaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/stokes-surfaces/Any surface with the same boundary gives the same flux of the curl. Calculus 4: Vector Calculus · Math LadderThe Divergence Theoremhttps://teddyrookbook.com/math/calculus-4/divergence-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/divergence-theorem/Outward flux through a closed surface equals the integral of the divergence inside. Calculus 4: Vector Calculus · Math LadderUsing the Divergence Theoremhttps://teddyrookbook.com/math/calculus-4/divergence-applications/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/divergence-applications/Close an open surface or swap a hard flux for an easy volume integral. Calculus 4: Vector Calculus · Math LadderOne Idea, Four Theoremshttps://teddyrookbook.com/math/calculus-4/big-picture/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/big-picture/FTC, Green, Stokes and Gauss all relate a region to its boundary. Calculus 4: Vector Calculus · Math LadderFluid Flow, Sources and Circulationhttps://teddyrookbook.com/math/calculus-4/fluid-flow/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/fluid-flow/Interpret flux and curl for moving fluids. Calculus 4: Vector Calculus · Math LadderGauss's Law and Electric Fluxhttps://teddyrookbook.com/math/calculus-4/gauss-law/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/gauss-law/Flux through a closed surface counts the charge inside. Calculus 4: Vector Calculus · Math LadderMaxwell's Equations in Two Formshttps://teddyrookbook.com/math/calculus-4/maxwell-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/maxwell-equations/Move between integral and differential forms using the theorems. Calculus 4: Vector Calculus · Math LadderMass and Centers of Mass of Thin Shellshttps://teddyrookbook.com/math/calculus-4/shell-mass/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-4/shell-mass/Integrate density over a surface. Calculus 4: Vector Calculus · Math Ladder