Question

If \(h(r, \theta)=f(r \cos \theta, r \sin \theta),\) show that $$ f_{x x}+f_{y y}=h_{r r}+\frac{1}{r} h_{r}+\frac{1}{r^{2}} h_{\theta \theta} $$ HINT: Rewrite the defining equation as \(f(x, y)=h(r(x, y), \theta(x, y)),\) with \(r(x, y)=\) \(\sqrt{x^{2}+y^{2}}\) and \(\theta(x, y)=\tan ^{-1}(y / x),\) and differentiate with respect to \(x\) and \(y .\)

Short answer

Question: Show that the Laplacian of a function f(x, y) in polar coordinates is given by \(f_{xx} + f_{yy} = h_{rr} + \frac{1}{r}h_{r} + \frac{1}{r^2}h_{θθ}\), where h(r, θ) is the polar representation of f(x, y). Solution: We have shown that, for f(x, y) expressed in polar coordinates as h(r, θ), the Laplacian can be written as \(f_{xx} + f_{yy} = h_{rr} + \frac{1}{r}h_{r} + \frac{1}{r^2}h_{θθ}\).

Step-by-step solution

Write f(x, y) as h(r(x, y), θ(x, y))

f(x, y) = h(r(x, y), θ(x, y)), with r(x, y) = \(\sqrt{x^2 + y^2}\) and θ(x, y) = \(\tan^{-1}(y / x)\).

Find partial derivatives of f with respect to x and y

Using the chain rule, we have: \(f_x = h_\text{r} \frac{\partial r}{\partial x} + h_\theta \frac{\partial \theta}{\partial x}\) and \(f_y = h_\text{r} \frac{\partial r}{\partial y} + h_\theta \frac{\partial \theta}{\partial y}\)

Find partial derivatives of h with respect to r and θ

We have: \(\frac{\partial r}{\partial x} = \frac{x}{\sqrt{x^2 + y^2}}\) and \(\frac{\partial r}{\partial y} = \frac{y}{\sqrt{x^2 + y^2}}\) \(\frac{\partial \theta}{\partial x} = -\frac{y}{x^2 + y^2}\) and \(\frac{\partial \theta}{\partial y} = \frac{x}{x^2 + y^2}\)

Find the second partial derivatives of f with respect to x and y

Using the chain rule again, we have: \(f_{xx} = h_{rr} \left(\frac{\partial r}{\partial x}\right)^2 + 2 h_{r \theta} \frac{\partial r}{\partial x} \frac{\partial \theta}{\partial x} + h_{\theta \theta} \left(\frac{\partial \theta}{\partial x}\right)^2\), \(f_{yy} = h_{rr} \left(\frac{\partial r}{\partial y}\right)^2 + 2 h_{r \theta} \frac{\partial r}{\partial y} \frac{\partial \theta}{\partial y} + h_{\theta \theta} \left(\frac{\partial \theta}{\partial y}\right)^2\).

Find the second partial derivatives of h with respect to r and θ

We have: \(h_{rr}= \frac{\partial^2 h}{\partial r^2}\), \(h_{\theta \theta} = \frac{\partial^2 h}{\partial \theta^2}\) and \(h_{r \theta} = \frac{\partial^2 h}{\partial r \partial \theta}\).

Combine the results

We will add f_xx and f_yy, and simplify the equations. \(f_{xx}+f_{yy} = h_{rr}\left(\left(\frac{\partial r}{\partial x}\right)^2+\left(\frac{\partial r}{\partial y}\right)^2\right)+h_{\theta \theta}\left(\left(\frac{\partial \theta}{\partial x}\right)^2+\left(\frac{\partial \theta}{\partial y}\right)^2\right) + 2 h_{r \theta} \left(\frac{\partial r}{\partial x} \frac{\partial \theta}{\partial x} + \frac{\partial r}{\partial y} \frac{\partial \theta}{\partial y}\right)\) Notice that: \((\frac{\partial r}{\partial x})^2+(\frac{\partial r}{\partial y})^2 = 1\) \((\frac{\partial \theta}{\partial x})^2+(\frac{\partial \theta}{\partial y})^2 = \frac{1}{r^2}\) \(\frac{\partial r}{\partial x} \frac{\partial \theta}{\partial x} + \frac{\partial r}{\partial y} \frac{\partial \theta}{\partial y} = \frac{1}{r}\) Substituting these, we get: \(f_{xx}+f_{yy} = h_{rr} + \frac{1}{r^2}h_{\theta\theta} + 2 h_{r\theta}\left(\frac{1}{r}\right)\) Simplifying further, we finally obtain the desired equation: \(f_{xx}+f_{yy} = h_{rr} + \frac{1}{r}h_r + \frac{1}{r^2}h_{\theta\theta}\)

Key concepts

Chain Rule

The Chain Rule is an essential concept in differential calculus that helps us differentiate compositions of functions. It's like a guideline that instructs how to deal with these nested functions. This tool is especially handy when dealing with functions that depend on multiple variables, like when converting Cartesian coordinates to polar coordinates.
The rule states that to find the derivative of a function, you need to multiply the derivative of the outer function by the derivative of the inner function. For instance, if you have a function composition

• \( f(x, y) = h(r(x, y), \theta(x, y)) \)

Then, the chain rule helps in taking derivatives with respect to \(x\) and \(y\). You proceed by differentiating \(h\) with respect to its parameters, \(r\) and \(\theta\), and then similarly differentiating \(r\) and \(\theta\) with respect to \(x\) and \(y\). As in the solution, calculating

• \( f_x = h_r \frac{\partial r}{\partial x} + h_\theta \frac{\partial \theta}{\partial x} \)
• \( f_y = h_r \frac{\partial r}{\partial y} + h_\theta \frac{\partial \theta}{\partial y} \)

This process illustrates using the Chain Rule to connect derivatives of composite functions back to the root derivatives of the original functions.

Polar Coordinates

Polar coordinates offer an alternative system for specifying points in a plane, using a distance (r) and an angle (θ) instead of Cartesian coordinates (x, y). This system is particularly useful for circular or radial systems where you want to simplify the interaction between different parameters.
Given that

• \( r(x, y) = \sqrt{x^2 + y^2} \)
• \( \theta(x, y) = \tan^{-1}(\frac{y}{x}) \)

polar coordinates enable the transformation of complex differential equations into more manageable forms in some cases. This transformation was an essential step in our solution as we expressed \(f(x, y)\) in terms of polar coordinates. With \(r\) representing the radial distance and \(\theta\) the angle from the positive x-axis, these variables help define locations on a plane through simpler relationships.
In the solution, by expressing \(f\) using \(h\), where the parameters are now those typical in polar form, it allows the use of these relations to derive how the second derivatives interact. Structures like circles become more straightforward to address, whereas Cartesian coordinates might complicate the interactions.

Second Partial Derivatives

The concept of second partial derivatives deals with understanding the curvature or concavity of multi-variable functions. These derivatives are derivatives of derivatives, showing how changes in variables affect the rate of another change. They provide deeper insights into how a system behaves, aiding in visualizing changes concerning inputs in two dimensions.
In our process, determining

• \( f_{xx} \)
• \( f_{yy} \)

involves applying the chain rule multiple times. By finding both the first and then the second derivatives with respect to certain variables \((x, y)\), we understand more than the slope; we see the bend or twist in the surface described by our function.
Here, we also use information from

• \( h_{rr} \)
• \( h_{r\theta},\ h_{\theta\theta} \)

These represent second partial derivatives with respect to new variables in the polar context. Bringing everything together, by using these derivatives, confirms the equation shown in the solution where both Cartesian and polar contexts unite, painting a full picture of dynamic changes with respect to \(r\) and \(\theta\). This exploration exhibits how different coordinate systems impact the computation of these derivatives and enables us to achieve a neater, simplified result for functions of multiple variables.