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I need some help understanding the solution to a problem.

Equations:

x = r.cos(θ)

y = r.sin(θ)

r = x^{2}+ y^{2}

theta = arctan(y/x)

Question:

Determine the Jacobian Matrix for (x,y)^{T}and for (r, θ)^{T}

SOLUTION:

I understand and can compute by myself the Jacobian for (x,y)^{T}, but the solution to for J(r, θ) i dont understand.

J(r,θ) = ( (@r/@x, @r/@y) , (@θ/@x, @θ,@y) )^{T}

Ok so that makes sense...

Then they gave me this...

(@r/@x, @r/@y) =( x / sqrt( x^{2}+ y^{2}) , y / sqrt(x^{2}+ y^{2}) )

Why isn't it just 2x and 2y respectively? why does it resemble something similar to a magnitude?

thanks for your help.

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# Jacobian Matrix for Polar Coordinates

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