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x-cos-sin-y-sin-cos-z-sin-r-x-e-x-y-e-y-z-e-z-r-r-r-

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Question Number 4922 by 123456 last updated on 22/Mar/16
{ ((x(ρ,θ,ψ)=ρ cos θ+ψ sin θ)),((y(ρ,θ,ψ)=ρ sin θ+ψ cos θ)),((z(ρ,θ,ψ)=ψ sin θ)) :} r(ρ,θ,ψ)=x(ρ,θ,ψ) e_x +y(ρ,θ,ψ) e_y +z(ρ,θ,ψ) e_z (∂r/∂ρ)=? (∂r/∂θ)=? (∂r/∂ψ)=?
{x(ρ,θ,ψ)=ρcosθ+ψsinθy(ρ,θ,ψ)=ρsinθ+ψcosθz(ρ,θ,ψ)=ψsinθr(ρ,θ,ψ)=x(ρ,θ,ψ)ex+y(ρ,θ,ψ)ey+z(ρ,θ,ψ)ez∂r∂ρ=?∂r∂θ=?∂r∂ψ=?
Commented by prakash jain last updated on 22/Mar/16
(∂x/∂ρ)=cos θ,(∂y/∂ρ)=sin θ,(∂z/∂ρ)=0 (∂x/∂θ)=−ρsin θ+ψcos θ,(∂y/∂θ)=ρcos θ−ψsin θ,(∂z/∂θ)=ψcos θ (∂x/∂ψ)=sin θ,(∂y/∂θ)=ψcos θ,(∂z/∂θ)=sin θ
∂x∂ρ=cosθ,∂y∂ρ=sinθ,∂z∂ρ=0∂x∂θ=−ρsinθ+ψcosθ,∂y∂θ=ρcosθ−ψsinθ,∂z∂θ=ψcosθ∂x∂ψ=sinθ,∂y∂θ=ψcosθ,∂z∂θ=sinθ
Commented by prakash jain last updated on 24/Mar/16
I thin e_x ,e_y are unit vectors.
Ithinex,eyareunitvectors.
Commented by 123456 last updated on 24/Mar/16
unit vetor in direction of x,y and z
unitvetorindirectionofx,yandz
Commented by Yozzii last updated on 23/Mar/16
What exactly are e_x ,e_y and e_z ? Are they vectors such that r= (((x(ρ,φ,ψ))),((y(ρ,φ,ψ))),((z(ρ,φ,ψ))) ) ?
Whatexactlyareex,eyandez?Aretheyvectorssuchthatr=(x(ρ,ϕ,ψ)y(ρ,ϕ,ψ)z(ρ,ϕ,ψ))?
Commented by Yozzii last updated on 24/Mar/16
Then, you can combine Prakash′s result and the vector form of r= ((a),(b),(c) ) to get (∂r/∂x_i )= (((∂a/∂x_i )),((∂b/∂x_i )),((∂c/∂x_i )) ) where x_i is the i−th variable for a,b and c being functions of i∈N variables.
Then,youcancombinePrakash′sresultandthevectorformofr=(abc)toget∂r∂xi=(∂a∂xi∂b∂xi∂c∂xi)wherexiisthei−thvariablefora,bandcbeingfunctionsofi∈Nvariables.

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