Simple partial differentiation $x = r\cos\theta$ and $y = r\sin\theta$

Question

If

\begin{align} x &= r\cos\theta,\\ y &= r\sin\theta, \end{align}

find

$$\dfrac{\partial^2\theta}{\partial{x}\partial{y}}.$$

How can I find this partial derivative?

I need to prove that

$$ \frac{\partial^2\theta}{\partial{x}\partial{y}} = -\frac{\cos2\theta}{r^2}.$$

Answer 1

I'll assume that $x$ and $y$ are independent. All we need do is carefully differentiate both sides of relations making use of the chain rule and the product rule. (Inverse trig functions are not continuous as $\theta$ crosses major axes on the plane, so their use in this kind of differentiation problem should be avoided, unless say you have extra information that $(x,y)$ will be confined to quadrant I.)

Since $\tan(\theta)={y\over x}$, we can use this as a starting point with no $r$ involved. Applying ${\partial\over\partial y}$ and ${\partial\over\partial x}$: $$ \begin{align} \sec^2(\theta)\frac{\partial \theta}{\partial y}&=\frac{1}{x}\\ \sec^2(\theta)\frac{\partial \theta}{\partial x}&=-\frac{y}{x^2}\\ \end{align} $$ Differentiating the first equation with respect to $x$ gives: $$ \begin{align} 2\,\sec^2(\theta)\tan(\theta)\frac{\partial \theta}{\partial y}\frac{\partial \theta}{\partial x}+\sec^2(\theta)\frac{\partial^2 \theta}{\partial x\partial y}&=-\frac{1}{x^2}\\ \end{align} $$ Substituting in from the first equation, $$ \begin{align} \frac{2}{x}\tan(\theta)\frac{\partial \theta}{\partial x}+\sec^2(\theta)\frac{\partial^2 \theta}{\partial x\partial y}&=-\frac{1}{x^2}\\ \end{align} $$ Multiplying be $\sec^2(\theta)$ and subbing in using the second equation: $$ \begin{align} -\frac{2y}{x^3}\tan(\theta)+\sec^4(\theta)\frac{\partial^2 \theta}{\partial x\partial y}&=-\frac{1}{x^2}\sec^2(\theta)\\ \end{align} $$ So we have $$ \begin{align} \frac{\partial^2 \theta}{\partial x\partial y}&=-\frac{1}{x^2}\cos^2(\theta)+\frac{2y}{x^3}\tan(\theta)\cos^4(\theta)\\ \end{align}$$

Now there are many alternative expressions for this, if you sub in the original relations between $x$, $y$, $\theta$, and $r$. Here is one: $$ \begin{align} \frac{\partial^2 \theta}{\partial x\partial y}&=-\frac{1}{r^2}+\frac{2y}{x^3}\frac{y}{x}\frac{x^4}{r^4}\\ &=-\frac{1}{r^2}+\frac{2y^2}{r^4}\\ &=\frac{1}{r^2}\left(-1+\frac{2y^2}{r^2}\right)\\ &=\frac{1}{r^2}\left(-1+2\sin^2(\theta)\right)\\ \end{align}$$

And then a double-angle formula for cosine gives the desired result.

Answer 2

Use $$ \frac{y}{x} = \tan \theta \Rightarrow \theta = \tan^{-1}\left(\frac{y}{x}\right) + 2 \pi n, $$ where $n$ is an integer, $$ \frac{d}{dx} \tan^{-1} x = \frac{1}{1+x^2}, $$ and $$ \frac{\partial}{\partial u} f\left(g\left(u,v\right)\right) = \frac{df}{dg} \frac{\partial g}{\partial u}. $$

Answer 3

Using $\theta = \arcsin\left(\frac{y}{r}\right)$ and $\arcsin'(t) = \frac{1}{\sqrt{1-t^2}}$ we get $$ \frac{\partial \theta}{\partial y} = \frac{1}{r} \frac{1}{\sqrt{1- \left(\frac{y}{r}\right)^2}} = \frac{1}{x}$$ using $r^2 = x^2 + y^2$. Hence $$\frac{ \partial \theta}{\partial x\partial y} = -\frac{1}{x^2} = -\frac{\cos^2(\theta)}{r^2} $$

Answer 4

Note that the following argument never involves division by any variable other than $r$, nor does it use inverse trig functions. This makes the argument valid everywhere except at the origin, which is OK since the partial derivative is clearly undefined there anyway, since at the origin $\theta$ can be any angle. These features distinguish this answer from other answers, including my own earlier answer.

First we can take the first relation and apply $\partial\over\partial x$ and then we can take the first relation and apply $\partial\over\partial y$. Lastly, we can apply both. $$ \begin{align} x & = r\cos(\theta)& (1)\\ 1 & = r_x\cos(\theta)-r\sin(\theta)\,\theta_x & (2)\\ 0 & = r_y\cos(\theta)-r\sin(\theta)\,\theta_y & (3)\\ 0 & = r_{xy}\cos(\theta)-r_x\sin(\theta)\theta_y-r_y\sin(\theta)\,\theta_x-r\cos(\theta)\,\theta_x\theta_y-r\sin(\theta)\,\theta_{xy}&(4) \end{align} $$

If we treat the second relation the same way: $$ \begin{align} y & = r\sin(\theta)& (5)\\ 0 & = r_x\sin(\theta)+r\cos(\theta)\,\theta_x&(6)\\ 1 & = r_y\sin(\theta)+r\cos(\theta)\,\theta_y&(7)\\ 0 & = r_{xy}\sin(\theta)+r_x\cos(\theta)\theta_y+r_y\cos(\theta)\,\theta_x-r\sin(\theta)\,\theta_x\theta_y+r\cos(\theta)\,\theta_{xy}&(8) \end{align} $$

Equation (6) helps us simplify equation (4), and then equation (3) helps us simplify equation (8): $$ \begin{align} 0 & = r_{xy}\cos(\theta)-r_y\sin(\theta)\,\theta_x-r\sin(\theta)\,\theta_{xy}&(9)\\ 0 & = r_{xy}\sin(\theta)+r_x\cos(\theta)\theta_y+r\cos(\theta)\,\theta_{xy}&(10) \end{align} $$ Multiplying the above by $-\sin(\theta)$ and $\cos(\theta)$ respectively, and adding equations lets us eliminate the second derivatives of $r$: $$ \begin{align} 0 & = r_y\sin^2(\theta)\,\theta_x+r_x\cos^2(\theta)\,\theta_y+r\,\theta_{xy}&(11)\\ \end{align} $$ Multiplying by $r$, then subbing in equations (7) and (2), and then expanding and simplifying: $$ \begin{align} 0 & = r_y\sin(\theta)r\sin(\theta)\,\theta_x+r_x\cos(\theta)r\cos(\theta)\,\theta_y+r^2\,\theta_{xy}&(12)\\ 0 & = r_y\sin(\theta)(r_x\cos(\theta)-1)+r_x\cos(\theta)(1-r_y\sin(\theta))+r^2\,\theta_{xy}&(13)\\ 0 & = -r_y\sin(\theta)+r_x\cos(\theta)+r^2\,\theta_{xy}&(14)\\ \end{align} $$ The partials $r_x$ and $r_y$ are easy to establish in terms of $x$ and $y$, unlike $\theta_x$ and $\theta_y$, which don't have clear formulas because of the issues regarding discontiunity of inverse trig functions. Basically, $r^2=x^2+y^2\implies2rr_x=2x$ and $2rr_y=2y\implies r_x=x/r$ and $r_y=y/r$. $$ \begin{align} 0 & = -\frac{y}{r}\sin(\theta)+\frac{x}{r}\cos(\theta)+r^2\,\theta_{xy}&(15)\\ 0 & = -\sin^2(\theta)+\cos^2(\theta)+r^2\,\theta_{xy}&(16)\\ 0 & = \cos(2\theta)+r^2\,\theta_{xy}&(17)\\ \end{align} $$ Which implies the desired result $$\theta_{xy}=-\frac{\cos(2\theta)}{r^2}$$

Answer 5

Here's another method in which we avoid any risk of accidentally dividing by $0$, I'm assuming here that $x,y$ are not dependent--that is, neither is a function with the other as a variable, and so $\frac{\partial x}{\partial y}=\frac{\partial y}{\partial x}=0$. Let me know if that's incorrect, and I'll adapt my answer accordingly.

We may first observe that $r^2=x^2+y^2$, so since $\frac{\partial x}{\partial y}=0$ and $y=r\sin\theta$, then $$2r\frac{\partial r}{\partial y}=2x\frac{\partial x}{\partial y}+2y=2y=2r\sin\theta.$$ Now, there are two ways this can happen: either (i) $r=0$ (in which case your given expression for the mixed partial cannot hold) or (ii) $r\neq 0$ and $\frac{\partial r}{\partial y}=\sin\theta$. Similarly, either (i) $r=0$ or (ii) $r\neq 0$ and $\frac{\partial r}{\partial x}=\cos\theta$.

Let's see what happens in the case that $r\neq 0$, $\frac{\partial r}{\partial y}=\sin\theta$, and $\frac{\partial r}{\partial x}=\cos\theta$. Since $x=r\cos\theta$, $\frac{\partial x}{\partial y}=0$, and $\frac{\partial r}{\partial y}=\sin\theta$, then by product and chain rules, we have $$0=\cos\theta\frac{\partial r}{\partial y}-r\sin\theta\frac{\partial\theta}{\partial y}=\cos\theta\sin\theta-r\sin\theta\frac{\partial\theta}{\partial y}.\tag{1}$$ Similarly, since $y=r\sin\theta$ and $\frac{\partial r}{\partial y}=\sin\theta$, then $$1=\sin^2\theta+r\cos\theta\frac{\partial\theta}{\partial y}.\tag{2}$$ With some rearrangement, dividing through by $r\neq 0$, and an application of the Pythagorean Identity, we have $$\frac1r\cos\theta\sin\theta=\sin\theta\frac{\partial\theta}{\partial y},\tag{$1'$}$$ $$\frac1r\cos^2\theta=\cos\theta\frac{\partial\theta}{\partial y}.\tag{$2'$}$$ Now, since $\frac{\partial r}{\partial x}=\cos\theta$, then differentiating $(2')$ with respect to $x$, we have again by chain and product rules, and by substitution using $(1')$, that

$\begin{eqnarray*} -2\cos\theta\sin\theta\frac{\partial\theta}{\partial x} & = & \cos\theta\frac{\partial\theta}{\partial y}\frac{\partial r}{\partial x}-r\sin\theta\frac{\partial\theta}{\partial y}\frac{\partial\theta}{\partial x}+r\cos\theta\frac{\partial^2\theta}{\partial x\partial y}\\ & = & \cos^2\theta\frac{\partial\theta}{\partial y}-r\sin\theta\frac{\partial\theta}{\partial y}\frac{\partial\theta}{\partial x}+r\cos\theta\frac{\partial^2\theta}{\partial x\partial y}\\ & = & \cos^2\theta\frac{\partial\theta}{\partial y}-\cos\theta\sin\theta\frac{\partial\theta}{\partial x}+r\cos\theta\frac{\partial^2\theta}{\partial x\partial y}, \end{eqnarray*}$

and so $$0=\left(\cos\theta\frac{\partial\theta}{\partial y}+\sin\theta\frac{\partial\theta}{\partial x}+r\frac{\partial^2\theta}{\partial x\partial y}\right)\cos\theta.\tag{3}$$

Since $\frac{\partial r}{\partial x}=\cos\theta$, then differentiating $(1')$ with respect to $x$, we have again by chain and product rules, and by substitution using $(2')$, that

$\begin{eqnarray*} (\cos^2\theta-\sin^2\theta)\frac{\partial\theta}{\partial x} & = & \sin\theta\frac{\partial\theta}{\partial y}\frac{\partial r}{\partial x}+r\cos\theta\frac{\partial\theta}{\partial y}\frac{\partial\theta}{\partial x}+r\sin\theta\frac{\partial^2\theta}{\partial x\partial y}\\ & = & \cos\theta\sin\theta\frac{\partial\theta}{\partial y}+r\cos\theta\frac{\partial\theta}{\partial y}\frac{\partial\theta}{\partial x}+r\sin\theta\frac{\partial^2\theta}{\partial x\partial y}\\ & = & \cos\theta\sin\theta\frac{\partial\theta}{\partial y}+\cos^2\theta\frac{\partial\theta}{\partial x}+r\sin\theta\frac{\partial^2\theta}{\partial x\partial y}, \end{eqnarray*}$

and so $$0=\left(\cos\theta\frac{\partial\theta}{\partial y}+\sin\theta\frac{\partial\theta}{\partial x}+r\frac{\partial^2\theta}{\partial x\partial y}\right)\sin\theta.\tag{4}$$

Now, $\cos\theta$ and $\sin\theta$ are never simultaneously $0$, so by $(3)$ and $(4)$, it follows that $$\cos\theta\frac{\partial\theta}{\partial y}+\sin\theta\frac{\partial\theta}{\partial x}+r\frac{\partial^2\theta}{\partial x\partial y}=0$$ in any case, so since $r\neq 0$, then $$\frac{\partial^2\theta}{\partial x\partial y}=-\frac1r\cos\theta\frac{\partial\theta}{\partial y}-\frac1r\sin\theta\frac{\partial\theta}{\partial x},$$ whence substitution using $(2')$ yields $$\frac{\partial^2\theta}{\partial x\partial y}=-\frac{\cos^2\theta}{r^2}-\frac1r\sin\theta\frac{\partial\theta}{\partial x}.\tag{5}$$ Since $x=r\cos\theta$ and $\frac{\partial r}{\partial x}=\cos\theta$, then in a similar fashion to our development of $(2')$ we find that $$\frac1r\sin^2\theta=-\sin\theta\frac{\partial\theta}{\partial x},\tag{6}$$ and subbing this into $(5)$ gives us $$\frac{\partial^2\theta}{\partial x\partial y}=-\frac{\cos^2\theta-\sin^2\theta}{r^2}.\tag{$5'$}$$ Finally, a double angle identity for cosine gives us the desired result.

Answer 6

Be $f(x,y)$ a real function and be

$$\bar{f}(r,\theta)=f(r\cos\theta,r\sin\theta)=f(x,y)$$

its corresponding function after the change of variables. We have by the chain rule

$$ \left\{ \begin{align} \frac{\partial{\bar{f}}}{\partial{r}}&= \cos\theta\frac{\partial{f}}{\partial{x}} +\sin\theta\frac{\partial{f}}{\partial{y}},\\ \frac{\partial{\bar{f}}}{\partial{\theta}}&= -r\sin\theta\frac{\partial{f}}{\partial{x}} +r\cos\theta\frac{\partial{f}}{\partial{y}}. \end{align} \right. $$

Solving this system with respect to $\partial{f}/\partial{x}$, $\partial{f}/\partial{y}$, we have

$$ \left\{ \begin{align} \frac{\partial{f}}{\partial{x}}&= \cos\theta\frac{\partial{\bar{f}}}{\partial{r}} -\frac{\sin\theta}{r}\frac{\partial{\bar{f}}}{\partial{\theta}},\\ \frac{\partial{f}}{\partial{y}}&= \sin\theta\frac{\partial{\bar{f}}}{\partial{r}} +\frac{\cos\theta}{r}\frac{\partial{\bar{f}}}{\partial{\theta}}. \end{align} \right. $$

Applying $\partial/\partial{y}$ to $\partial{f}/\partial{x}$ we have

\begin{align} &\frac{\partial^2{f}}{\partial{x}\partial{y}} =\frac{\partial}{\partial{y}}\frac{\partial{f}}{\partial{x}} =\left(\sin\theta\frac{\partial{}}{\partial{r}} +\frac{\cos\theta}{r}\frac{\partial{}}{\partial{\theta}}\right) \left(\cos\theta\frac{\partial{\bar{f}}}{\partial{r}} -\frac{\sin\theta}{r}\frac{\partial{\bar{f}}}{\partial{\theta}}\right)\\ &\qquad=\frac{\sin2\theta}{2} \frac{\partial^2{\bar{f}}}{\partial{r^2}} +\frac{\cos2\theta}{r} \frac{\partial^2{\bar{f}}}{\partial{r}\partial\theta} -\frac{\sin2\theta}{2r^2} \frac{\partial^2{\bar{f}}}{\partial{\theta^2}} -\frac{\sin2\theta}{2r} \frac{\partial{\bar{f}}}{\partial{r}} -\frac{\cos2\theta}{r^2} \frac{\partial{\bar{f}}}{\partial{\theta}}. \end{align}

If we take $\bar{f}=\theta$ and take into account

$$ \frac{\partial^2\theta}{\partial{r^2}} =\frac{\partial^2\theta}{\partial{r}\partial\theta} =\frac{\partial^2\theta}{\partial\theta^2} =\frac{\partial\theta}{\partial{r}}=0, \qquad\frac{\partial\theta}{\partial\theta}=1, $$

only the last term remains, and we have

$$ \frac{\partial^2\theta}{\partial{x}\partial{y}}=-\frac{\cos2\theta}{r^2}. $$