Parameterization, Folium of Descartes, etc.

• dr721
In summary, the problem is to show that lines with direction vectors u = [1, 0, b] and v = [0, 1, a] through point P = [a, b, ab] are contained in the surface z = xy. The solution is not provided, but the attempt at a solution discusses the limit as t approaches -1 for the curve x^3 + y^3 = 3axy and the symmetry of the curve about y = x. The parametrized equations for the curve are x = 3at / (1 + t^3) and y = 3at^2 / (1 + t^3). The limit of x(t) + y(t) as t

Homework Statement

Consider the surface z = f$\left(x y\right)$ = xy. Given a point P = $\left[a, b, ab\right]$ on this surface, show that the lines with direction vectors u = $\left[1, 0, b\right]$ and v = $\left[0, 1, a\right]$ through P are entirely contained in the surface.

2. The attempt at a solution

To be honest, I'm not really sure what I'm supposed to do here or how to go about doing it, if someone could explain it and/or give me a starting point, that'd be very helpful.

Homework Statement

Consider the curve given by x3 + y3 = 3axy.

What happens as t$\rightarrow$-1? Consider the limit as t$\rightarrow$-1 of x(t) + y(t). What do you conclude?

The curve has obvious symmetry. Verify this using your parametrization.

Homework Equations

The parametrized equations are:

x = 3at / 1 + t3

y = 3at2 / 1 + t3

2. The attempt at a solution

For the limit, after adding and simplifying the equations, it equals -a. I believe the equation is asymptotic at t = -1, but I guess I'm not quite sure what I'm supposed to conclude or what -a means.

For the symmetry part, it is symmetric about y = x, so that for every point (x, y) there is a point (y, x). How do I show this with the parametric equations?

Sorry for the poor formatting. I don't know how to format column vectors or fractions.

1. What is parameterization?

Parameterization is a mathematical technique used to represent a curve or surface in terms of one or more independent variables called parameters. It allows for the description of complex shapes and structures in a more simplified manner.

2. How is parameterization used in scientific research?

Parameterization is used in various fields of science, such as physics, engineering, and computer graphics, to model and analyze complex systems. It is particularly useful in numerical simulations and computer modeling, where it allows for the efficient representation of complex shapes and structures.

3. What is the Folium of Descartes?

The Folium of Descartes is a famous curve named after French mathematician René Descartes. It is a geometric shape that resembles a cloverleaf with three leaves and is defined by the equation x³ + y³ = 3axy, where a is a constant. It has interesting properties such as being self-intersecting and having a cusp at the origin.

4. How is the Folium of Descartes related to parameterization?

The Folium of Descartes can be parameterized by using the polar coordinates r = a(1 + cosθ) and θ = 3t, where t is the parameter. This allows for the representation of the curve in terms of a single variable, making it easier to analyze and manipulate mathematically.

5. What are some real-world applications of the Folium of Descartes?

The Folium of Descartes has been used in various applications, such as designing mechanical linkages and creating smooth curves in computer graphics. It also has applications in physics, such as modeling the motion of a particle under the influence of a central force. Additionally, the Folium of Descartes has been studied for its aesthetic and mathematical beauty, making it a popular topic in recreational mathematics.

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