Problem 14 - Advanced Engineering Mathematics

Question

For \(u\left(e^{i \theta}\right)=\cos 2 \theta,\) the Fourier series solution (6) reduces to \\[ u(r, \theta)=r^{2} \cos 2 \theta=\operatorname{Re}\left(z^{2}\right) \quad \text { or } \quad u(x, y)=x^{2}-y^{2} \\] The corresponding system of level curves is shown in the figure.

Step-by-Step Solution

Answer

The Fourier series solution is \( u(x, y) = x^2 - y^2 \).

Step 1: Understanding the Fourier Series

The Fourier series representation of a function decomposes it into a sum of sines and cosines. In this case, we have the function \( u(e^{i \theta}) = \cos 2\theta \). The crucial step here is understanding that this function can be expressed in terms of complex exponentials.

Step 2: Express in Complex Form

The expression \( \cos 2\theta \) can be written in terms of the complex exponential using Euler's formula: \( e^{i\theta} = \cos \theta + i\sin \theta \). Thus, \( \cos 2\theta = \operatorname{Re}(e^{2i\theta}) \).

Step 3: Substitute and Simplify

In the exercise, we are substituting \( r = e^{i\theta} \), leading to \( u(r, \theta) = r^2 \cos 2\theta \). Since \( r^2 = (e^{i\theta})^2 = e^{2i\theta} \), we have \( u(r, \theta) = \operatorname{Re}(z^2) \).

Step 4: Translate to Cartesian Coordinates

To express \( u(x, y) \) in terms of Cartesian coordinates, recall that for a complex number \( z = x + iy \), \( z^2 = (x+iy)^2 = x^2 - y^2 + 2ixy \). The real part, which is what \( u(x, y) \) represents, is \( x^2 - y^2 \).

Step 5: Conclusion about Level Curves

The level curves for the function \( u(x, y) = x^2 - y^2 \) are hyperbolas, due to the form of the equation suggesting a difference of squares. These curves represent the values of \( u \) that remain constant for different \( (x, y) \).

Key Concepts

Complex Exponentials

Complex exponentials are a powerful way to represent periodic functions and signals. They arise when dealing with expressions involving the imaginary unit, commonly denoted as \( i \), which satisfies \( i^2 = -1 \). Complex numbers are typically expressed in the form \( z = x + iy \), where \( x \) and \( y \) are real numbers.

• The beauty of complex exponentials is their ability to model oscillations and waves.
• They simplify many mathematical operations, particularly those involving trigonometric functions.

In the context of Fourier series, expressing functions in terms of complex exponentials can yield elegant and compact solutions. For example, periodic functions like sine and cosine can be represented using Euler's formula in terms of complex exponentials. This provides a straightforward method for analyzing and understanding the behavior of these functions over different intervals.

Euler's Formula

Euler's Formula is a key mathematical tool that bridges the gap between trigonometry and exponential functions. It is defined by the equation \( e^{i\theta} = \cos \theta + i\sin \theta \). This formula allows us to convert complex trigonometric expressions into their exponential equivalents.

• Euler's Formula makes it simple to transform cosine and sine functions into exponential form.
• This transformation is particularly useful in solving problems related to Fourier series, oscillations, and wave functions.

For the exercise in question, Euler's formula helps us rewrite \( \cos 2\theta \) as \( \operatorname{Re}(e^{2i\theta}) \). It highlights the real part of the complex exponential, linking it directly with the trigonometric expression. This simplification is fundamental in manipulating such functions smoothly within various domains.

Cartesian Coordinates

Cartesian coordinates provide a system for describing the location of points on a plane using two numerical coordinates. These coordinates are usually represented as \( (x, y) \), where \( x \) is the horizontal axis and \( y \) is the vertical axis.

• In mathematics, converting polar coordinates (like \( e^{i\theta} \)) to Cartesian form is essential for simplifying and solving real-world problems.
• For complex numbers, this involves expressing \( z = r(e^{i\theta}) \) in the form \( z = x + iy \).

When transforming functions from polar expressions to Cartesian coordinates, you often need to use transformations such as \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). For instance, in the step-by-step solution, the function expressed in complex form \( z^2 \) translates into Cartesian coordinates as \( x^2 - y^2 \). This real part is critical for expressing the solution in the simpler, more familiar Cartesian coordinate system.

Level Curves

Level curves, also known as contour lines, are an essential concept in multivariable calculus. A level curve represents a set of points in a plane where a function holds the same value. In simpler terms, they are the maps of constant heights in a function landscape.

• Level curves are particularly useful in visualizing the behavior and distribution of functions across a plane.
• They help identify regions of the plane with particular values, effectively giving a 'contour map' of a function.

In our exercise, the level curves of the function \( u(x, y) = x^2 - y^2 \) are hyperbolas. This form indicates a difference of squares that is characteristic of hyperbolic geometry. Understanding the shape of these level curves helps in interpreting and predicting the function's behavior over a set region, making problem-solving more intuitive and visual.