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(a) Find the real and positive constants n and y such that the following velocity field V is conservative V(x, y, z)= [2ŋx sin(z)] i+ [√y z²e¯ ]] + [x² cos(z) - 2ze->] k (b) Consider a force field F(x, y, z)=(x, y, z) where is the conservative form of V from part (a). Find o such that F= Vo. (c) Can the divergence of F be zero on the plane z=0? Justify your answer using the divergence of F on this plane. Classify the points on z=0 as source, sink or neither.

(a) Find the real and positive constants n and y such that the following velocity field V is conservative V(x, y, z)= [2ŋx sin(z)] i+ [√y z²e¯ ]] + [x² cos(z) - 2ze->] k (b) Consider a force field F(x, y, z)=(x, y, z) where is the conservative form of V from part (a). Find o such that F= Vo. (c) Can the divergence of F be zero on the plane z=0? Justify your answer using the divergence of F on this plane. Classify the points on z=0 as source, sink or neither.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Find the real and positive constants n and y such that the following velocity field V is conservative V(x, y, z) = [2ŋx sin(z)] i+[√y z²e¯ ]] + [x² cos(z) - 2ze->] k (b) Consider a force field F(x, y, z)=(x, y, z) where is the conservative form of V from part (a). Find o such that F = Vo. (c) Can the divergence of F be zero on the plane z=0? Justify your answer using the divergence of F on this plane. Classify the points on z=0 as source, sink or neither.
Transcribed Image Text:(a) Find the real and positive constants n and y such that the following velocity field V is conservative V(x, y, z) = [2ŋx sin(z)] i+[√y z²e¯ ]] + [x² cos(z) - 2ze->] k (b) Consider a force field F(x, y, z)=(x, y, z) where is the conservative form of V from part (a). Find o such that F = Vo. (c) Can the divergence of F be zero on the plane z=0? Justify your answer using the divergence of F on this plane. Classify the points on z=0 as source, sink or neither.
1. Change of Variable of Integration in 2D [ f(x,y) drdy = f(z(u, v),y(u, 0)). («, ») ducho 2. Transformation to Polar Coordinates The useful formulas 3. Change of Variable of Integration in 3D [, (2, 2) dadydz = [[F(w, x, w)|J(u, v, w)| dududw 4. Transformation to Cylindrical Coordinates z=rcos, y=rsin, Jr.)=r 6. Line Integrals 5. Transformation to Spherical Coordinates x=rcos 0, y = rsin, ===, J(r,0,2)=r x=rcos@sind, y=rsin@sind, 2=rcoso, Jr.,0,0)=²sin 7. Work Integrals [1(x, y, z) ds = [ f(x(t), y(t),= (t)) √√x²(1)² + y′(t)² + 2′(t)²³ dt 8. Surface Integrals [F(x, y, z) - dx = [° R² + R$/ + d dr dt [ 92,9,2) ds = [[ 9(2.9.1(2.9)) √ 12 + 12 + 1 dady 9. Flux Integrals For a surface with upward unit normal, J.P. 11. Stokes' Theorem = [[₁-Fife - Faly + Pa dyda F-nds= 10. Gauss' (Divergence) Theorem JIL V. FdV = dv = [[₁, F [[ F. ÂdS (V x F). ÂdS= s = [ F.. F.dr
Transcribed Image Text:1. Change of Variable of Integration in 2D [ f(x,y) drdy = f(z(u, v),y(u, 0)). («, ») ducho 2. Transformation to Polar Coordinates The useful formulas 3. Change of Variable of Integration in 3D [, (2, 2) dadydz = [[F(w, x, w)|J(u, v, w)| dududw 4. Transformation to Cylindrical Coordinates z=rcos, y=rsin, Jr.)=r 6. Line Integrals 5. Transformation to Spherical Coordinates x=rcos 0, y = rsin, ===, J(r,0,2)=r x=rcos@sind, y=rsin@sind, 2=rcoso, Jr.,0,0)=²sin 7. Work Integrals [1(x, y, z) ds = [ f(x(t), y(t),= (t)) √√x²(1)² + y′(t)² + 2′(t)²³ dt 8. Surface Integrals [F(x, y, z) - dx = [° R² + R$/ + d dr dt [ 92,9,2) ds = [[ 9(2.9.1(2.9)) √ 12 + 12 + 1 dady 9. Flux Integrals For a surface with upward unit normal, J.P. 11. Stokes' Theorem = [[₁-Fife - Faly + Pa dyda F-nds= 10. Gauss' (Divergence) Theorem JIL V. FdV = dv = [[₁, F [[ F. ÂdS (V x F). ÂdS= s = [ F.. F.dr
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