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Transformation of a Vector

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.

Re: Transformation of a Vector

Any ideas? I can post my solution method if it is necessary.

Re: Transformation of a Vector

Hi Vesnog! :)

See here for a derivation.

Re: Transformation of a Vector

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.

1 Attachment(s)

Re: Transformation of a Vector

The solution for this transformation is also given as in my book on a book available through Google. I am attaching a screenshot to pinpoint my point.

Re: Transformation of a Vector

Ah, now I understand what you mean.

But it is not a factor of $\displaystyle r^2$, but a factor of $\displaystyle r$.

When we use polar coordinates $\displaystyle (r,\theta)$, a small change is expressed as $\displaystyle \mathbf{dr} = dr \mathbf{\hat r} + r d\theta \boldsymbol{\hat \theta}$.

This change is identified by $\displaystyle (dr, d\theta)$.

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.

Re: Transformation of a Vector

Thanks for your replies. First of all that should be a factor of r sorry for that, I am glad you got my point; however, I still could not understand your last statement(sentence).

Re: Transformation of a Vector

It's an arbitrary decision to take a factor r out.

It does not have any particular meaning or significance.

Re: Transformation of a Vector

It may be significant because of dimensional analysis by the way the book should explicitly give a reason why it ignores r in my opinion. There may be a more subtle reason behind this.