Problem 6 - Advanced Calculus

Question

Investigate the behavior of \(F(x, y)\) at \((0,0)\) if (a) \(F(x, y)=\frac{x^{2} y}{2 x^{2}+y^{2}}\) (b) \(F(x, y)=\frac{x^{2} y}{3 x^{4}+2 y^{2}}\)

Step-by-Step Solution

Answer

The behavior of the function \(F(x, y)\) at the point (0,0) for (a) is defined and equals 0, because the value of the function approaches 0 as (x,y) approaches (0,0) from any direction. For (b) the limit is not defined at (0,0), meaning the behaviour of \(F(x, y)\) at (0,0) is dependent on the direction from which (x,y) approaches (0,0).

Step 1: Convert to Polar Coordinates (a)

First convert \(F(x, y)\) into polar coordinates \(F(r, θ)\) using the coordinate transformation \(x = rcosθ, y=rsinθ\): \[F(r, θ)=\frac{r^{2} cos^{2} θ \cdot r sinθ}{2 r^{2} cos^{2} θ + r^{2} sin^{2} θ} = \frac{rcos^2θsinθ}{2cos^2θ+sin^2θ}\]

Step 2: Calculate Limit as r approaches 0 (a)

Next, as r goes to zero, solve the limit: \[\lim_{r \to 0}F(r, θ)=\lim_{r \to 0} \frac{rcos^2θsinθ}{2cos^2θ+sin^2θ} = 0\] Since r has been factored out in the numerator, it goes to zero as r approaches zero, no matter the value of θ.

Step 3: Convert to Polar Coordinates (b)

Similarly for (b), convert \(F(x, y)\) into polar coordinates \(F(r, θ)\): \[F(r, θ)=\frac{r^{2} cos^{2} θ \cdot r sinθ}{3 r^{4} cos^{4} θ + 2r^{2} sin^{2} θ} = \frac{rcos^2θsinθ}{3rcos^4θ+2sin^2θ}\]

Step 4: Calculate Limit as r approaches 0 (b)

Finally, calculate the limit as r approaches zero: \[\lim_{r \to 0}F(r, θ)=\lim_{r \to 0} \frac{rcos^2θsinθ}{3 r cos^4θ+2sin^2θ}\] This limit is not defined, because when r approaches zero, the limit depends on the direction, or the value of θ. For example, for θ=0 and θ=π the values are different.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to express points in the plane using a system based on radius and angle. Instead of using the traditional Cartesian coordinates, denoted as \(x, y\), polar coordinates use \(r\) and \(\theta\). Here, \(r\) represents the distance from the origin to the point, while \(\theta\) represents the angle from the positive x-axis.

A simple conversion formula allows us to switch between Cartesian and polar: \(x = r\cos\theta\) and \(y = r\sin\theta\). This transformation simplifies the analysis of functions with radial symmetry and is particularly useful in multivariable calculus for understanding behavior as we approach the origin.

Using polar coordinates simplifies the expression of \(F(x, y)\) considerably, as it emphasizes the dependency on distance (\(r\)) and the angular component \(\theta\). Also, solving limits in polar coordinates can sometimes simplify complex expressions.

Limits in Multivariable Calculus

In multivariable calculus, limits help us understand the behavior of a function as inputs approach a certain point, often where two or more variables change together. This process is more complicated than single-variable limits due to the many possible paths to a point.

For a limit \(\lim_{(x, y) \to (a, b)} f(x, y)\) to exist, the function must approach the same value, no matter the path taken towards \(a, b\). If different paths yield different limits, the overall limit does not exist.

In the step-by-step solution, converting to polar coordinates allows us to employ a uniform approach to explore these limits. By observing the behavior as \(r\) approaches zero, we can simplify the complexity of these multivariable functions, shedding light on their true behavior at given points.

Directional Limits

Directional limits are an essential concept when studying limits involving multiple variables. These limits consider different paths towards a particular point, allowing us to see how the limit changes depending on the direction of approach.

In the example provided, the functions were explored using various directions given by \(\theta\) in polar coordinates.

A directional limit does not guarantee the overall limit's existence. To determine if a limit exists at a point, it must be consistent across all possible paths. Therefore, understanding directional limits helps pinpoint where discrepancies might occur. They reveal whether a change in direction influences the limit, as seen in part (b) of the exercise, where different angles lead to differing results.

Behavior at a Point

Examining behavior at a specific point is crucial for understanding the nuances of multivariable functions. The behavior at a point can help identify discontinuities or singularities that might not be apparent from a function's general form.

In the exercises, the behavior at \( (0,0) \) was investigated.

For the first function, converting to polar coordinates allowed us to see a uniform approach towards zero, confirming the consistent behavior at the origin. The limit calculation revealed that the function approaches zero irrespective of the path taken.

Conversely, the second function demonstrated inconsistency as the direction changed, reflecting unstable behavior at the origin. This difference underscores the importance of thoroughly investigating every potential path to determine the true nature of a function’s behavior.