Problem 6 - Advanced Engineering Mathematics

Question

Sketch some of the level curves associated with the given function. $$ f(x, y)=\tan ^{-1}(y-x) $$

Step-by-Step Solution

Answer

Level curves are parallel lines: \(y=x+k\) for constants \(k\).

Step 1: Understand the Function

The given function is \( f(x, y) = \tan^{-1}(y-x) \). To sketch level curves, we need to set \( f(x, y) = c \) for constant values of \( c \). This means solving \( \tan^{-1}(y-x) = c \).

Step 2: Express Equation in Terms of x and y

To find the level curves, convert \( \tan^{-1}(y-x) = c \) to \( y-x = \tan(c) \). This equation \( y = x + \tan(c) \) represents a line in the xy-plane with a slope of 1.

Step 3: Choose Values for c

Select different values of \( c \) to plot the corresponding lines. For example, take \( c = -\frac{\pi}{4}, 0, \frac{\pi}{4} \). The lines will be \( y = x - 1 \), \( y = x \), and \( y = x + 1 \), respectively.

Step 4: Sketch the Level Curves

Draw these lines on the xy-plane. Each line is a level curve, illustrating where the function value \( f(x, y) \) equals \( c \). These lines are parallel and differ by a vertical shift based on \( \tan(c) \).

Key Concepts

Inverse Tangent Function

The inverse tangent function, often denoted as \( \tan^{-1} \) or "arctan," is a fundamental concept in trigonometry. It is the inverse of the tangent function, which means it tells us the angle whose tangent is a given number.
In mathematical terms, if \( \tan(\theta) = x \), then \( \theta = \tan^{-1}(x) \).

• The range of the inverse tangent function is typically restricted to \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), which translates to angles between -90 and 90 degrees.
• It is important because it helps us find angles in right triangles, especially in coordinate-based geometry problems.

In the context of the function \( f(x, y) = \tan^{-1}(y-x) \), the inverse tangent helps us determine the level curves by setting \( f(x, y) = c \). This essentially gives us the angle whose tangent equals \( y-x \). We can then use trigonometric identities to manipulate these expressions into more familiar forms such as a straight line equation, \( y = x + \tan(c) \).
These manipulations are key to drawing level curves, which are essential in understanding the behavior of functions in two dimensions.

Contour Plot

Contour plots are graphical representations of three-dimensional surfaces on a two-dimensional plane, using lines to represent points of equal value. Each line on a contour plot corresponds to a particular value of the function being constant, known as a contour or level curve.

• Level curves connect points on a graph where the function has the same value, creating contour lines which can be visualized on a topographic map.
• Such plots are useful in fields ranging from geography to engineering because they succinctly represent complex surfaces or phenomena.

In this exercise, we are creating a contour plot for \( f(x, y) = \tan^{-1}(y-x) \) by sketching the level curves. By setting different constants, \( c \), we draw lines like \( y = x - 1 \), \( y = x \), and \( y = x + 1 \).
Each of these lines indicates where the function value \( f(x, y) \) equals the constant \( c \). Since the tangent function has periodic and symmetric properties, these lines extend infinitely and represent different levels of the angle derived from the inverse tangent.

Mathematical Visualization

Mathematical visualization is a powerful tool for understanding complex concepts and functions by providing a visual representation of mathematical data. It helps in comprehending relationships and behaviors of functions, making abstract concepts more tangible.

• When dealing with multivariable functions, visual tools can illustrate how changes in input variables affect the output.
• Graphs, contour plots, and 3D models are common visualization techniques that bring math to life.

For \( f(x, y) = \tan^{-1}(y-x) \), visualizing the level curves allows students to see how small changes in \( c \) shift the function value linearly along \( y = x + \tan(c) \).
By sketching these lines, one can understand the function’s behavior as it crosses different slices or planes of the xy-coordinate system. This visualization simplifies the complex two-dimensional inverse tangent relationships into easily understandable line plots that provide insight into the function’s rate of change and symmetry.