Question

Parametrize the surface defined by the function \(z=f(x, y)\) over each of the given regions \(R\) of the \(x\) -y plane. \(z=3 x^{2} y\) (a) \(R\) is the rectangle bounded by \(-1 \leq x \leq 1\) and \(0 \leq y \leq 2\) (b) \(R\) is the circle of radius \(3,\) centered at (1,2) . (c) \(R\) is the triangle with vertices (0,0),(1,0) and (0,2) . (d) \(R\) is the region bounded by the \(x\) -axis and the graph of \(y=1-x^{2}\)

Short answer

Parametrize as follows: (a) \((-1 \leq x \leq 1, 0 \leq y \leq 2)\) (b) Use \((1 + r\cos(\theta), 2 + r\sin(\theta))\) (c) With \(u, v, u + \frac{v}{2} \leq 1\) (d) Restricted by \(y=1-x^2\).

Step-by-step solution

Understand the function and regions

We have the surface defined by the function \( z = 3x^2y \) and we need to parametrize it over different regions \( R \) in the \( x-y \) plane.

Parametrize the surface over rectangle

For region \( R \) defined by the rectangle \(-1 \leq x \leq 1\) and \(0 \leq y \leq 2\), the surface can be parametrized by setting \( x = u \) and \( y = v \), where \( u \) and \( v \) range over the given intervals. Thus, the parametrization is \( (u, v, 3u^2v), \; -1 \leq u \leq 1, \; 0 \leq v \leq 2 \).

Parametrize the surface over circle

For region \( R \) defined as a circle of radius 3 centered at (1,2), use polar coordinates: let \( x = 1 + r \cos(\theta) \) and \( y = 2 + r \sin(\theta) \) where \( 0 \leq r \leq 3 \) and \( 0 \leq \theta < 2\pi \). The parametrization becomes \( (1 + r\cos(\theta), 2 + r\sin(\theta), 3(1 + r\cos(\theta))^2(2 + r\sin(\theta))) \).

Parametrize the surface over triangle

For the triangle with vertices (0,0), (1,0), and (0,2), use barycentric coordinates: let \( x = u \) and \( y = v \) where \( u \geq 0, \; v \geq 0 \) and \( u + \frac{v}{2} \leq 1 \). The parametrization is \( (u, v, 3u^2v) \).

Parametrize the surface over bounded region

For the region bounded by the \( x \)-axis and the graph of \( y = 1-x^2 \), set \( x = u \) and \( y = v \) where \( -1 \leq u \leq 1 \) and \( 0 \leq v \leq 1 - u^2 \). The parametrization is \( (u, v, 3u^2v) \).

Key concepts

Surface parametrization

Surface parametrization is a technique used to describe surfaces in a mathematical space in terms of parameters or variables. It is especially useful for analyzing complex surfaces by breaking them down into more manageable pieces. To parametrize a surface, you assign variables to represent each coordinate of a point on the surface. For the given surface function \( z = 3x^2y \), parametrization involves determining the values of \( x \), \( y \), and \( z \) for each condition set by different regions in the \( xy \)-plane.
The rectangle and triangle mentioned in the exercise offer straightforward intervals and inequalities, while circles and curves like \( y = 1 - x^2 \) require more creative approaches, such as converting to polar or other coordinate systems for effective parametrization. Understanding the boundaries of each region helps tailor the parametrization to adequately describe the surface over said region.

Polar coordinates

Polar coordinates provide an alternative to the standard Cartesian coordinates, making it easier to work with circular regions or symmetries. They describe a point in terms of its distance from the origin (radius \( r \)) and the angle \( \theta \) from a reference direction (usually positive \( x \)-axis).
In this exercise, to parametrize the region defined by a circle of radius 3 centered at (1, 2), we translate the center to the origin by adding to the polar formulas. Thus, \( x = 1 + r \cos(\theta) \) and \( y = 2 + r \sin(\theta) \). For \( z \), substitute \( x \) and \( y \) into the original equation and adjust \( r \) and \( \theta \) constraints \( 0 \leq r \leq 3 \) and \( 0 \leq \theta < 2\pi \).
This method is particularly advantageous when dealing with questions involving circles as it simplifies both visualization and computation of points on a circular path.

Barycentric coordinates

Barycentric coordinates are often used when dealing with triangles and relate a point inside a triangle to the vertices of that triangle. They provide a means to express any point's position as a combination of the triangle's vertices.
In the bounding triangle with vertices (0,0), (1,0), and (0,2), a parametrization employs barycentric coordinates, ensuring that \( u \geq 0 \), \( v \geq 0 \), and \( u + \frac{v}{2} \leq 1 \). By doing this, every point inside or on the triangle can be described as a linear combination of these vertices, which ensures that the surface definition \( z = 3u^2v \) applies consistently.
Barycentric coordinates simplify calculations by maintaining linear constraints relative to triangle sides, lending themselves to clear geometric interpretation and easy computation.

Regions in the xy-plane

Understanding regions in the \( xy \)-plane is a vital skill in parametrization as it defines the domain over which the surface is analyzed. Regions can come in various simple or complex shapes, each with unique methods for defining boundaries.
For parametrization, one might deal with:

• Rectangles, with straight interval boundaries, are directly expressible in parameter ranges.
• Circles, where polar coordinates provide a natural choice to account for symmetry.
• Triangles, often employing barycentric coordinates to leverage geometric properties.
• Complex bounded regions, such as beneath curves like \( y = 1-x^2 \), which require careful analysis of inequalities to define bounds.

Choosing appropriate parameter formats helps align closely with region shapes, facilitating more efficient and accurate computation of the parametrized surfaces. Addressing these initial region definitions thoroughly eases downstream complexity in surface equations.