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Question: (d) Find the work done by F in moving a particle along any closed path C. (e) Consider two paths C₁ and C₂. Suppose the work done by the particle in moving through F along path C₁ from r(0) to r(2) is 10. Find the work done by F in moving a particle along path C₂ from r(0) to r(2).

Question: (d) Find the work done by F in moving a particle along any closed path C. (e) Consider two paths C₁ and C₂. Suppose the work done by the particle in moving through F along path C₁ from r(0) to r(2) is 10. Find the work done by F in moving a particle along path C₂ from r(0) to r(2).

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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(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. (a) Find the real and positive constants 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-¹] k (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. Question: (d) Find the work done by F in moving a particle along any closed path C. (e) Consider two paths C₁ and C₂. Suppose the work done by the particle in moving through F along path C₁ from r(0) to r(2) is 10. Find the work done by F in moving a particle along path C₂ from r(0) to r(2). The solutions of (a) (b) (c)are given as references, please solve for (d) and (e) We have already proved that: n = and y = 1 (a) (b) can let the velocity field V is conservative. F = Vô where p(x,y,z) = ≤sin(xz) − z?² e¯y – (c) div(F) = -2e < 0 therefore the points on the plane z = 0 are all sink.
Transcribed Image Text:(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. (a) Find the real and positive constants 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-¹] k (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. Question: (d) Find the work done by F in moving a particle along any closed path C. (e) Consider two paths C₁ and C₂. Suppose the work done by the particle in moving through F along path C₁ from r(0) to r(2) is 10. Find the work done by F in moving a particle along path C₂ from r(0) to r(2). The solutions of (a) (b) (c)are given as references, please solve for (d) and (e) We have already proved that: n = and y = 1 (a) (b) can let the velocity field V is conservative. F = Vô where p(x,y,z) = ≤sin(xz) − z?² e¯y – (c) div(F) = -2e < 0 therefore the points on the plane z = 0 are all sink.
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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