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I will us e the symbol A in place of Ф.

x = r cos A and y = r sin A

dx = - r sin A dA + Cos A dr

dx / x² = [- r Sin A dA + Cos A dr ] / (r² cos²A)

dy / y² = [ r cos A dA + sin A dr ] / (r² Sin² A)

dx / x² + dy /y²

= [ - r Sin³ A dA + Cos A Sin² A dr + r Cos³A dA + Sin A Cos²A dr]

/ (r² Sin²A Cos²A)

= [ r (CosA - SinA) (Cos²A+CosA SinA + Sin²A) dA + (CosA Sin²A + SinA Cos²A) dr ] / [r² Sin²A Cos²A ]

= [ r (cosA - Sin A)(1+cosA SinA) dA + CosA SinA (sinA + CosA) dr ]

/ [r² Sin²A Cos²A]

= [ (x - y) (1 + xy/r²) dA + xy/r³ (x + y) dr ] / x²y²

x = r cos A and y = r sin A

dx = - r sin A dA + Cos A dr

dx / x² = [- r Sin A dA + Cos A dr ] / (r² cos²A)

dy / y² = [ r cos A dA + sin A dr ] / (r² Sin² A)

dx / x² + dy /y²

= [ - r Sin³ A dA + Cos A Sin² A dr + r Cos³A dA + Sin A Cos²A dr]

/ (r² Sin²A Cos²A)

= [ r (CosA - SinA) (Cos²A+CosA SinA + Sin²A) dA + (CosA Sin²A + SinA Cos²A) dr ] / [r² Sin²A Cos²A ]

= [ r (cosA - Sin A)(1+cosA SinA) dA + CosA SinA (sinA + CosA) dr ]

/ [r² Sin²A Cos²A]

= [ (x - y) (1 + xy/r²) dA + xy/r³ (x + y) dr ] / x²y²