Change from rectangular to spherical coordinates. (Let ρ≥0, 0≤θ≤2π, and 0≤ϕ≤π.) a) (0, −8, 0)(ρ,θ,ϕ)=? b) (−1, 1, −2)(ρ,θ,ϕ)=?

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Change from rectangular to spherical coordinates. (Let ρ≥0, 0≤θ≤2π, and 0≤ϕ≤π.) a) (0, −8, 0)(ρ,θ,ϕ)=? b) (−1, 1, −2)(ρ,θ,ϕ)=?

Maciej Morrow · Accepted Answer

Step 1a) Consider the rectangular coordinate&amp;nbsp;(0,&amp;nbsp;−8,&amp;nbsp;0)The objective is to convert the rectangular coordinate to spherical coordinate&amp;nbsp;(ρ,θ,ϕ)Here,&amp;nbsp;ρ≥0,&amp;nbsp;0≤θ≤2π,&amp;nbsp;0≤ϕ≤πConsider the following statement:The point P in&amp;nbsp;R3&amp;nbsp;with rectangular coordinates&amp;nbsp;(x,y,z)&amp;nbsp;and spheerical coordinates&amp;nbsp;(ρ,θϕ)The relation between rectangular coordinates&amp;nbsp;(x,y,z)&amp;nbsp;and spherical coordinates&amp;nbsp;(ρ,θϕ)&amp;nbsp;is,x=ρsin⁡ϕcos⁡θy=ρsin⁡ϕsin⁡θz=ρcos⁡ϕStep 2The rectangular coordinate is&amp;nbsp;(x,y,z)=(0,&amp;nbsp;−8,&amp;nbsp;0)That is,&amp;nbsp;x=0,&amp;nbsp;y=−8,&amp;nbsp;z=0.From the statement, it can be inferred thatρ=x2+y2+z2=02+(−8)2+02{x=0,&amp;nbsp;y=−8,z=0}=64=8From the statement, it can be inferred thatz=ρcos⁡ϕ0=8cos⁡ϕ{z=0,ρ=8}cos⁡ϕ=0ϕ=π2{0≤ϕ≤π}Step 3From the statement, it can be inferred thaty=ρsin⁡ϕsin⁡θsin⁡θ=yρsin⁡ϕReplace&amp;nbsp;y=−8,ρ=8,ϕ=π2sin⁡θ=−88sin⁡π2=−1{sin⁡π2=1}θ=3π2{0≤θ≤π2}Therefore, the spherical coordinate of the point&amp;nbsp;(0,&amp;nbsp;−8,&amp;nbsp;0)&amp;nbsp;is&amp;nbsp;(8,&amp;nbsp;3π2,&amp;nbsp;π2)Step 4Consider the rectangular coordinate&amp;nbsp;(−1,&amp;nbsp;1,&amp;nbsp;−2)The objective is to convert the rectangular coordinate to spherical coordinate&amp;nbsp;(ρ,θ,ϕ).Consider the following statement:The point P in&amp;nbsp;R3&amp;nbsp;with rectangular coordinates&amp;nbsp;(x,y,z)&amp;nbsp;and spherical coordinates&amp;nbsp;(ρ,θ,ϕ).The relation between rectangular coordinates&amp;nbsp;(x,y,z)&amp;nbsp;and spherical coordinates&amp;nbsp;(ρ,θ,ϕ)&amp;nbsp;is,x=ρsin⁡ϕcos⁡θy=ρsin⁡ϕsin⁡θz=ρcos⁡ϕThe rectangular coordinate is&amp;nbsp;(x,y,z)=(−1,&amp;nbsp;1,&amp;nbsp;−2)That is,&amp;nbsp;x=−1,&amp;nbsp;y=1,&amp;nbsp;z=−2.Thus, from the statement,ρ=x2+y2+z2=(−1)2+(1)2+(−2)2{x=−1,&amp;nbsp;y=1,&amp;nbsp;z=−2}=1+1+4=6From the statement, it can be inferred thatz=ρcos⁡ϕ−2=6cos⁡ϕ&amp;nbsp;

Change from rectangular to spherical coordinates. (Let \rho\geq0,\ 0\leq\theta\leq\2\pi, and 0\leq\phi\leq\pi.)a) (0,\ -8,\ 0)(\rho,\theta, \phi)=?b) (-1,\ 1,\ -2)(\rho, \theta, \phi)=?

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