What is Double integral? Triple integrals? Line & Surface integral? Volume integral?
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- Опубліковано 7 сер 2024
- #some2
After watching this video you will understand that ... A line integral is the generalization of simple integral. A surface integral is generalization of double integral. A volume integral is generalization of triple integral.
0:00 Intro
0:18 Simple Integral
0:46 Double Integral
1:06 Line Integral
2:02 Double and Surface Integrals
3:01 Parametric Surface
4:48 Triple and Volume Integrals
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Amazing video
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This is great. Too bad we don't have holograms in schools.
Yes
0:38 0:38 0:38 0:38 0:38
India 0:32
Simple integral: ∫dx
Double integral: ∬dA = ∬dydx ∬ rdrdθ
Triple integral: ∭dV = ∭dzdydx = ∭rdzdrdθ = ∭ρ^2*sin(φ)dρdφdθ
Line integral: ∮ F ∙ dr = ∮ (F ∙ n)ds = ∬(Ny-Mx)dA = ∯ curl F ∙ dS = ∫y(x)√(1+(y')^2)dx = ∫r(t)√((x')^2+(y')^2)dt = ∫r(θ)√(r^2+(r')^2)dθ
Surface integral: ∯ F ∙ dS = ∯(F ∙ N)dS = ∬z(x, y)√(1+zx^2+zy^2)dydx = ∬r(s, t)√(rs ⨉ rt)dsdt = ∬g(x, y, z) * ∇g/(∇g ∙ n)dA = ∬r(θ, z)√(r^2+rθ^2+(r*rz)^2)dθdz = ∬ρ(φ, θ)√((ρ^2*sin(φ))^2+(ρ*sin(φ)*ρφ)^2+(ρ*ρθ)^2)dφdθ
Volume integral: ∰ F ∙ dS = ∰(F ∙ N)dS = ⨌dV = ⨌dwdzdydx = ⨌rdwdzdrdθ = ⨌r1r2dr2dθ2dr1dθ1 = ⨌ρ^2*sin(φ)dwdρdφdθ = ⨌р^3*sin^2(φ)*sin(ψ)dрdφdψdθ
Using Green's Theorem, you can convert a line to a surface integral.
Using the Divergence Theorem, you can convert from a surface integral into an interior integral.
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