--- title: "Verify Gauss' divergence theorem F=4xi−2y²j+z²k" subject: "Mathematics III" chapter: 2 key: "semester_3/mathematics_iii:verify-gauss-divergence-theorem-f-4xi" source: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-gauss-divergence-theorem-f-4xi site: BCT Engineering --- # Verify Gauss' divergence theorem F=4xi−2y²j+z²k Mathematics III · Chapter 2: Line, Surface and Volume Integrals - Asked in: 2067 Magh - Video: [Gauss divergence - F=4xi−2y^2j+z^2k and the region bounded by x^2+y^2=4, z= 0 and z=3](https://www.youtube.com/watch?v=p9jM8ZmzCdM) — Bright Concept - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-gauss-divergence-theorem-f-4xi - Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2 ## Question Verify Gauss' theorem for F = 4x i − 2y² j + z² k taken over the region bounded by x² + y² = 4, z = 0 and z = 3. ## Short answer div F = 4−4y+2z. Volume integral in cylindrical coords over r≤2,0≤z≤3 gives 84π. Surface integral: top (z=3) flux 36π, bottom 0, curved 48π; total 84π. Gauss' theorem holds. ## Answer **Gauss' theorem:** ∬_S F·n dS = ∭_V div F dV. F = 4x i − 2y² j + z² k over the cylinder x² + y² ≤ 4, 0 ≤ z ≤ 3. #### Volume integral ```text div F = 4 − 4y + 2z ∭ (4 − 4y + 2z) dV with cylindrical coordinates (r ≤ 2, 0 ≤ z ≤ 3): 4 ∭ dV = 4 · π(4)(3) = 48π −4 ∭ y dV = 0 (y = r sin θ, integral over θ is 0) 2 ∭ z dV = 2 · (π·4) · ∫₀³ z dz = 2 · 4π · 9/2 = 36π Total = 84π ``` #### Surface integral - **Top** z = 3, n = k: F·n = z² = 9. Flux = 9 · 4π = 36π. - **Bottom** z = 0, n = −k: F·n = −z² = 0. Flux = 0. - **Curved surface**: n = (x i + y j)/2, F·n = (4x² − 2y³)/2. With x = 2cos θ, y = 2sin θ, dS = 2 dθ dz: ```text ∫₀³ ∫₀^2π (4x² − 2y³) dθ dz = 3 ∫₀^2π (16cos²θ − 16 sin³θ) dθ = 3 (16π − 0) = 48π ``` Total = 36π + 0 + 48π = 84π, equal to the volume integral. So **Gauss' theorem is verified.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. div F for F = 4x i - 2y^2 j + z^2 k is: A. 4 + 4y + 2z B. 4 - 2y + 2z C. 4 - 4y + 2z D. 4 - 4y + z Answer: C. d/dx(4x)=4, d/dy(-2y^2)=-4y, d/dz(z^2)=2z. (Key point: Divergence) 2. The volume integral equals: A. 48 pi B. 36 pi C. 120 pi D. 84 pi Answer: D. 48pi - 0 + 36pi. (Key point: Volume integral) 3. The term -4 y dV integrates to 0 because: A. y = r sin theta integrates to 0 over 0 to 2pi B. y is always zero C. the cylinder has no volume in y D. z cancels it Answer: A. Symmetry in theta. (Key point: y term) 4. Top + bottom + curved fluxes are: A. 36pi + 36pi + 12pi B. 36pi + 0 + 48pi C. 9pi + 0 + 48pi D. 36pi + 0 + 36pi Answer: B. Top 9*4pi=36pi, bottom 0, curved 48pi, total 84pi. 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