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My question is from the book "Tensor Analysis" by Barry Spain. I am asked to show that how a vector transforms from rectangular Cartesian coordinates to polar coordinates. I have attached the question in jpeg format. I have came up with a solution but the angular component in my solution is r^2 times the angular component given in the book. I have checked some books about this subject and found out that both the solution given in the attachment and I have found exist. I am pretty confused about this and I assume that this book is wrong. I will be grateful if someone can provide some insight.

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# Transformation of a Vector

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