Vector Transformation in Cartesian and Polar Coordinates

[Septim]

Greetings,

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.

 

[Septim]

Any ideas ?

 

[mathman]

My guess: normalization question. Both answers may be right.

 

[Ferramentarius]

Here is a solution based on using vector u_r = [cos(A) , sin(A)] and vector u_A = [-sin(A), cos(A)] normal to it. Edit: It appears the answer is not exactly the same.

 

[Septim]

Here is a solution based on using vector u_r = [cos(A) , sin(A)] and vector u_A = [-sin(A), cos(A)] normal to it. Edit: It appears the answer is not exactly the same.

I just saw your post much later but I did not understand your argument.

P.S The link is not accessible.

 

[Septim]

The link was accessible and I saw the solution which is similar to mine though less detail is provided. The two answers differ by a factor of r and I think the solution in the link you suggest is the correct one, since it has the dimensions of acceleration and this is acceleration am I correct?