Question

Using polar coordinates, describe the level curves of the function defined byf ( x , y ) = 2 x y ( x 2 + y 2 ) if ( x , y ) ≠ ( 0 , 0 ) and f ( 0 , 0 ) = 0.

Answer 1

The level curves of the function f(x, y) = 2xy(x^2 + y^2) can be described in polar coordinates by replacing x and y with r cos(θ) and r sin(θ). This results in the equation 2r^3 cos(θ) sin(θ) = c, where c is a constant representing a specific level curve. Different values of r and θ that satisfy this equation for a particular c will determine the shape of each level curve in the polar coordinate system.

Step-by-step explanation:

To describe the level curves of the function f(x, y) = 2xy(x2 + y2) using polar coordinates, we first need to express x and y in terms of the polar coordinates r and θ. Recall that x = r cos(θ) and y = r sin(θ). Substituting these into the function gives us:

f(r cos(θ), r sin(θ)) = 2(r cos(θ))(r sin(θ))(r2 cos2(θ) + r2 sin2(θ))

This simplifies to f(r, θ) = 2r3 cos(θ) sin(θ) since cos2(θ) + sin2(θ) = 1. The level curves of the function are the sets of points for which f(r, θ) = c, where c is a constant. Therefore:

2r3 cos(θ) sin(θ) = c

We can see that the level curves are dependent on the values of r and θ that satisfy this equation for different constants c. The level curves in the polar coordinate system represent the same values of the function f(x, y) at different points (r, θ) on the plane.

Answer 2

The level curves of the given function in polar coordinates are found by setting the transformed function f(r, \(\theta\)) equal to a constant. The transformed function f(r, \(\theta\)) = r^4sin(2\(\theta\)) results from converting x and y to their polar equivalents. Consequently, different level curves correspond to different constants and reveal the function's behavior in various quadrants.

Step-by-step explanation:

The question involves describing level curves of a given function in polar coordinates. To do this, we first express the function f(x, y) = 2xy(x2 + y2) in terms of polar coordinates. Using the relationships x = rcos(\(\theta\)) and y = rsin(\(\theta\)), the function becomes f(r, \(\theta\)) = 2(rcos(\(\theta\)))(rsin(\(\theta\)))(r2). Simplifying, we get f(r, \(\theta\)) = 2r4sin(\(\theta\))cos(\(\theta\)), which can be further simplified to f(r, \(\theta\)) = r4sin(2\(\theta\)) since 2sin(\(\theta\))cos(\(\theta\)) = sin(2\(\theta\)).

To find the level curves, we set f(r, \(\theta\)) equal to a constant c. This gives us r4sin(2\(\theta\)) = c. For different values of c, we get different level curves. These correspond to various shapes depending on the value of c and \(\theta\). When c is positive, the level curves will be situated in quadrants where sin(2\(\theta\)) is positive (first and third quadrants), and when c is negative, they are in the second and fourth quadrants.