In multivariable calculus, why do we normalize $\frac{\partial}{\partial \theta}$ in polar coordinates?

Question

I'm TA'ing multivariable this semester, and I just noticed that we always tend to normalize all our basis vectors when using polar coordinates. This is in stark contrast what I'm used to in differential geometry, as we'd prefer that our coordinate basis to transform by the law \begin{align*} \frac{\partial}{\partial x}&=\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta}\\ &=\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}, \end{align*} and similarly with $\frac{\partial}{\partial y}$. This $\frac{1}{r}$ factor makes up for the fact that if we travel in the angular direction, we cover more ground the further we are from the origin. So for example, the gradient in these "geometric" polar coordinates would take on the form $$\nabla f |_{(r,\theta)}=(\frac{\partial}{\partial r},\frac{1}{r^2}\frac{\partial}{\partial \theta})$$ which agrees with the usual way of defining gradients locally by $\nabla f=g^{ij}(\partial_if)\partial_j$. This in opposition to the more common $\frac{1}{r}$ factor which comes using the normalized polar coordinate system. So why are we normalizing these coordinates? If you insist on working in an orthonormal frame, why not call it a polar frame instead of polar coordinates to avoid bad practices in the future?

Edit: Let me put in an explicit computation in with the "geometric" (which I learned is called holonomic) basis. Consider $$f(x,y)=\frac{x}{x^2+y^2},$$ so that in polar coordinates, $$f(r,\theta)=\frac{\cos\theta}{r}.$$ One sees: \begin{align*} \nabla f&=\frac{\partial f}{\partial x}\bigg\vert_{(r,\theta)}\frac{\partial}{\partial x}+\frac{\partial f}{\partial y}\bigg\vert_{(r,\theta)}\frac{\partial}{\partial y}\\ &=\frac{\sin^2\theta-\cos^2\theta}{r^2}\left(\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)-\frac{2\cos\theta\sin\theta}{r^2}\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)\\ &=\frac{\sin^2\theta\cos\theta-\cos^3\theta-2\cos\theta\sin^2\theta}{r^2}\frac{\partial}{\partial r}+\frac{-\sin^3\theta+\cos^2\theta\sin\theta-2\cos^2\theta\sin\theta}{r^3}\frac{\partial}{\partial \theta}\\ &=-\frac{\cos\theta}{r^2}\frac{\partial}{\partial r}-\frac{\sin\theta}{r^3}\frac{\partial}{\partial \theta}\\ &=\frac{\partial f}{\partial r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial f}{\partial \theta}\frac{\partial}{\partial \theta}. \end{align*}

Answer 1

My naïve guess is simply that people like orthonormal bases so that they can apply a Pythagoras-like formula to get lengths $$\|a e_r + b e_\theta \| =\sqrt{a^2+b^2}$$ and so they avoid introducing the components of the metric tensor. Projection formulas are also easier, like the component of $v$ in the angular direction is $(v\cdot e_\theta)e_\theta$. This last point can be remedied by using the dual basis of the dual space, but again most people avoid talking about linear functionals by using their Riesz representatives.

Finally, the vector calculus formalism is not designed to work nicely in arbitrary coordinates (an example of this is how the vector Laplacian has to be defined using the curl of the curl). People using vector calculus tend to not care about coordinate invariance of their expressions, so they are happy (or at least won't complain as much as a differential geometer would) having different expressions in different coordinate systems.

Answer 2

I am grateful to @JackozeeHakkiuz who made the comment that your question has to do with holonomic / non holonomic coordinates. This PSE post also treats the subject with the example of 2D polar coordinates:

• $\{\partial_r,\partial_\theta\}=\{\hat{\boldsymbol{e}}_r,r\hat{\boldsymbol{e}}_\theta\}$ is the holonomic basis which is orthogonal but not orthonormal.

• In contrast, $\{\partial_r,\frac{1}{r}\partial_\theta\}= \{\hat{\boldsymbol{e}}_r,\hat{\boldsymbol{e}}_\theta\}$ is not holonomic but orthonormal.

• Clearly, the holonomic operators commute while the non holonomic ones don't.

• The metric that makes the above mentioned orthogonality / non-orthogonality happend is (in both cases) the familiar $ds^2=dr^2+r^2\,d\theta^2$ which we can write as the $2$-tensor $$ g=\begin{pmatrix}1&0\\0&r^2\end{pmatrix}\,. $$ Note that $$ g(\hat{\boldsymbol{e}}_\theta,\hat{\boldsymbol{e}}_\theta)= g\Big(\frac{1}{r}\partial_\theta,\frac{1}{r}\partial_\theta\Big)= r^2\frac{1}{r^2}=1\,. $$

• If we change to a metric that looks Euclidean $$ \eta=\begin{pmatrix}1&0\\0&1\end{pmatrix}\, $$ then the holonomic basis becomes (miraculously) orthonormal. However, the scaling is now built into the basis vector $r\hat{\boldsymbol{e}}_\theta$ which -as we know- is not normalized in the traditional sense using $g\,.$ To me this seems like an easy way to understand what tetrads (moving coordinate systems) are.

• The factor $\displaystyle \frac{1}{r^2}$ caused a lot of confusion in the comments (including mine) but @JackozeeHakkiuz finally cleared this up: For a differentiable function $f$, the $\mathbb R^2$-valued function $$ \Big(\partial_rf,\frac{1}{r}\partial_\theta f\Big) $$ are the components of its gradient (in orthogonal polar coordinates). Using the basis vector fields $\{\partial_r,\frac{1}{r}\partial_\theta\}$ we get the gradient vector field as $$ (\partial_rf)\,\partial_r + \Big(\frac{1}{r}\partial_\theta f\Big)\frac{1}{r}\partial_\theta=(\partial_rf)\,\partial_r + \Big(\frac{1}{r^2}\partial_\theta f\Big)\,\partial_\theta\,. $$ In other words, in the holonomic basis $\{\partial_r,\partial_\theta\}$ the second component carries the factor $\displaystyle\frac{1}{r^2}\,.$

• The linked PSE post also gives the example of a closed loop out of which I created the following picture. It shows that if we perform a "coordinate loop" starting in $(r,\theta)=(1,0)$ and incrementing these coordinates by the same amounts $\Delta r=\pm 1$ resp. $\Delta\theta=\pm\frac{\pi}{2}$ the loop will be closed provided at $r=2$ the basis vector is scaled up to the holonomic $r\hat{\boldsymbol{e}}_\theta$. The closedness of the loop is clearly related to the vanishing of the commutator $$ [\hat{\boldsymbol{e}}_r,r\hat{\boldsymbol{e}}_\theta]=[\partial_r,\partial_\theta]\,. $$ That order independence (closedness of the loop) must also be the reason why the holonomic basis is called a coordinate basis.