Intermediate variable chain rule question.

[v0id19]

Homework Statement

Suppose that w=f(x,y), x=r*cos(θ), y=r*sin(θ). Show that:
$$(\frac{\partial w}{\partial x})^2 + (\frac{\partial w}{\partial y})^2 =(\frac{\partial w}{\partial r})^2 + \frac{1}{r^2} (\frac{\partial w}{\partial \theta})^2$$

Homework Equations

the multivariable chain rule

The Attempt at a Solution

we just were taught this yesterday, but my prof didn't exactly do a good job, and I'm doing a good job at figuring it out and understanding it, but all I'm able to do is use the chain rule with respect to the independent variables (r and θ in this case), and i can't figure out how to use it for the intermediate variables x and y and my book doesn't have any examples of this. I'm sure it's something really obvious that I'm missing, but i just haven't had that lightbulb moment...

 

[Billy Bob]

w depends on x and y

x depends on r and theta; y depends on r and theta

If r changes, then x and y are influenced, and each of them influences w, so the formula for $$\frac{\partial w}{\partial r}$$ will have to include x and y.

This helps you remember that the formula is
$$\frac{\partial w}{\partial r}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial r}$$

The intuition and formula for $$\frac{\partial w}{\partial \theta}$$ are similar.

--


Now turning to your problem, on the left hand side, there is nothing to do. $$\frac{\partial w}{\partial x}$$ and $$\frac{\partial w}{\partial y}$$ do not simplify.

On the right hand side, use
$$\frac{\partial w}{\partial r}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial r}$$
and the analogous formula for $$\frac{\partial w}{\partial \theta}$$, find the partials with respect to r and theta, and simplify.

 

[LCKurtz]

Part of the confusion in this type of problem stems from the abuse of notation whereby the same notation, in this case w is used for the function whether it is expressed in terms of x and y or r and theta. To more properly phrase the question it should be given as follows:

If $$w(x,y) = W(r,\theta)$$, where $$x = r\cos(\theta)\, y = r \sin(\theta)$$, show that

$$w_{x}^2 + w_{y}^2 = W_r^2+\frac 1 {r^2}W_{\theta}^2$$

Now start with the right side and use the chain rule in this form:

$$W_r = w_r = w_x x_r + w_y y_r$$

$$W_{\theta} = w_{\theta} = w_x x_{\theta} + w_y y_{\theta}$$

The partials $$x_r,\ y_r,\ x_{\theta},\ y_{\theta}$$ are easy to calculate from your equations. Manipulate that a bit and you should get the required equation.

Once you have done this a couple of times, it likely won't confuse you to use the lower case w for W. Although they aren't the same function, many texts do this.