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 Homework Statement

3.1 (Kibble's Classical Mechanics) Find which of the following forces are conservative, and for those that
are find the corresponding potential energy function (a and b are constants)
 Homework Equations

Fx = ax + by², Fy = az + 2bxy, Fz = ay + bz²
Fr = 2ar sin θ sin ϕ, Fθ = ar cosθ sin ϕ, Fϕ = ar cosϕ
The first force components:
Fx = ax + by², Fy = az + 2bxy, Fz = ay + bz²
I calculated the integral V=∫Fdr, using dr=(dx,dy,dz)
The result I found was
(1/2(ax²)+2azy+2bxy²+1/3bz³)
The answer in the book (Kibble's Classical Mechanics): (1/2(ax²)+azy+bxy²+1/3bz³)
The second force:
Fr = 2ar sin θ sin ϕ, Fθ = ar cosθ sin ϕ, Fϕ = ar cosϕ
I calculated the integral V=∫Fdr, using dr=(dr,rdθ,rsinθdϕ)
The result I found was
arsinθsinϕ(r+1)
The answer in the book (Kibble's Classical Mechanics): ar²sinθsinϕ
Me and my colleagues can't find where we have mistaken. Can you help us? Our solutions are incorrect?
Fx = ax + by², Fy = az + 2bxy, Fz = ay + bz²
I calculated the integral V=∫Fdr, using dr=(dx,dy,dz)
The result I found was
(1/2(ax²)+2azy+2bxy²+1/3bz³)
The answer in the book (Kibble's Classical Mechanics): (1/2(ax²)+azy+bxy²+1/3bz³)
The second force:
Fr = 2ar sin θ sin ϕ, Fθ = ar cosθ sin ϕ, Fϕ = ar cosϕ
I calculated the integral V=∫Fdr, using dr=(dr,rdθ,rsinθdϕ)
The result I found was
arsinθsinϕ(r+1)
The answer in the book (Kibble's Classical Mechanics): ar²sinθsinϕ
Me and my colleagues can't find where we have mistaken. Can you help us? Our solutions are incorrect?