Any ideas? I can post my solution method if it is necessary.
Greetings,
My question is from the book "Tensor Analysis" by Barry Spain. I am asked to show that what the components of a vector become upon transforming 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 other books on this subject and found out that both the solution given in the attachment and the one I found exist. I am pretty confused about this, and I assume that this book is wrong or I am doing a terrible mistake. I will be grateful if someone can provide some insight.
Note: I found it appropriate to post this in differential geometry section since it is related with tensors somehow.
Okay my derivation was similar and it is included in many textbooks but my problem is that there is an additional factor of r^2 in the angular part of the vector in my solution and in Wikipedia's solution. My question is this, if it is not clear I can explain it again.
Ah, now I understand what you mean.
But it is not a factor of , but a factor of .
When we use polar coordinates , a small change is expressed as .
This change is identified by .
Note that the vector formula has an extra r in it where the change in angle is identified, but when we identify them as a coordinate pair, the r is left out.