Problem 53 - Calculus Early Transcendentals

Question

\(49-54\) Assume that all the given functions have continuous second-order partial derivatives. If \(z=f(x, y),\) where \(x=r \cos \theta\) and \(y=r \sin \theta,\) show that $$\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=\frac{\partial^{2} z}{\partial r^{2}}+\frac{1}{r^{2}} \frac{\partial^{2} z}{\partial \theta^{2}}+\frac{1}{r} \frac{\partial z}{\partial r}$$

Step-by-Step Solution

Answer

Both sides of the equation represent the Laplacian in different coordinate systems.

Step 1: Setup the problem

We begin by identifying the necessary transformations between Cartesian coordinates \(x, y\) and polar coordinates \(r, \theta\). These transformations are given by \(x = r \cos \theta\) and \(y = r \sin \theta\). We know that \(z = f(x, y)\) and aim to demonstrate the Laplacian relation in polar coordinates.

Step 2: Calculate first-order partial derivatives

Using the chain rule, we find \(\frac{\partial z}{\partial x} = \frac{\partial f}{\partial x}\) and \((\frac{\partial z}{\partial y} = \frac{\partial f}{\partial y}\). The transformation to polar coordinates doesn't affect these first derivatives directly, so simply calculate as usual.

Step 3: Calculate second-order partial derivatives in Cartesian coordinates

Calculate \(\frac{\partial^2 z}{\partial x^2}\) and \(\frac{\partial^2 z}{\partial y^2}\) using the product and chain rules. These represent how the function changes in space in both Cartesian directions. Assume the function has continuous derivatives as stated.

Step 4: Express derivatives in polar coordinates

Using the transformations, rewrite the function \(z = f(x=r\cos\theta, y=r\sin\theta)\). Apply the chain rule to express all derivatives in terms of \(r\) and \(\theta\).

Step 5: Compute the Laplacian in Polar Coordinates

Derive expressions for each term: \(\frac{\partial^2 z}{\partial r^2}\), \(\frac{1}{r^2} \frac{\partial^2 z}{\partial \theta^2}\), and \(\frac{1}{r} \frac{\partial z}{\partial r}\) in polar coordinates using the definitions of \(x\) and \(y\).

Step 6: Equate the two expressions

Combine and simplify the expressions derived for Laplacian in Cartesian and polar coordinates. Verify they are equal by transforming any remaining terms using the original transformations.

Key Concepts

Partial Derivatives

A partial derivative represents how a function changes as one of its input variables changes while all other input variables are held constant. In a function like \(z = f(x, y)\), the partial derivative with respect to \(x\), written as \(\frac{\partial z}{\partial x}\), describes how \(z\) changes as \(x\) changes and \(y\) remains fixed.

These derivatives are crucial for examining the behavior of functions in a multivariable calculus context. When dealing with functions that have spatial components, partial derivatives help describe how the function varies in different directions. For example:

• \(\frac{\partial z}{\partial x}\) - Change in \(z\) with small changes in \(x\)
• \(\frac{\partial z}{\partial y}\) - Change in \(z\) with small changes in \(y\)

Continuity of these first and second partial derivatives is often assumed to avoid any sudden or unexplained changes in behavior across the space.

Polar Coordinates

Polar coordinates are an alternative coordinate system to the Cartesian coordinates, used primarily in two-dimensional contexts. Instead of defining a point in terms of \(x\) and \(y\) coordinates, polar coordinates define a point by its distance from the origin, \(r\), and an angle, \(\theta\), from the positive x-axis.

Here is how polar coordinates can be related to Cartesian coordinates:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

This system is particularly helpful when dealing with problems involving circular or rotational symmetry. It simplifies the approach and calculations involved in such scenarios. Understanding this coordinate system's usage is essential when evaluating integrals or other complicated calculus operations involving circular paths.

Cartesian Coordinates

The Cartesian coordinate system is a widely used method for associating each point in space with an ordered pair \((x, y)\). Here, \(x\) corresponds to the horizontal axis and \(y\) to the vertical axis. This system is the baseline for understanding spatial problems, as it allows easy representation of planar surfaces.

Converting between Cartesian and polar coordinates involves using equations that relate \(r\) and \(\theta\) to \(x\) and \(y\). This ability helps transition seamlessly between different scenarios depending on symmetry or analytical convenience.

Understanding both systems allows solving complex problems that require moving between different descriptions of spatial locations.

Chain Rule

The chain rule is a fundamental technique used for differentiation in multivariable calculus. It is essential when calculating partial derivatives of functions dependent on multiple intermediate variables. The rule provides a way to differentiate composite functions.

For example, if \(z = f(x, y)\) and \(x\) and \(y\) are functions of \(r\) and \(\theta\), then to find \(\frac{\partial z}{\partial r}\), one would use:

\[ \frac{\partial z}{\partial r} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial r} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial r} \]

This rule ensures precision when transitioning between coordinate systems, especially when differentiating functions involving complex dependencies. By effectively applying the chain rule, we can derive all necessary derivatives whenever multiple variables interact.

Laplacian Transformation

The Laplacian transformation is a powerful operation in multivariable calculus. It is used for finding the Laplacian of a function, which is a measure that combines the second derivatives to provide information about the function's behavior at a point. This operation is defined as:

\[ \Delta z = \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} \]

In polar coordinates, the Laplacian changes to accommodate the way distances and areas behave differently than in Cartesian coordinates. The expression becomes:

\[ \Delta z = \frac{\partial^2 z}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 z}{\partial \theta^2} + \frac{1}{r} \frac{\partial z}{\partial r} \]

This formula is useful when analyzing heat conduction, wave propagation, and other phenomena described by Laplace's equation. Understanding and applying the Laplacian in both Cartesian and polar coordinates is indispensable in physics and engineering contexts.