Problem 10 - Calculus: Early Transcendental Functions

Question

Sketch a graph of the parametric surface. \(x=u, y=v, z=4-u^{2}-v^{2}\)

Step-by-Step Solution

Answer

The resultant graph is a downward-opening 3D paraboloid, having its peak at the point (0,0,4) in 3D space.

Step 1: Interpret the Parameters

First, it is important to understand the role of parameters 'u' and 'v' in these equations. In these expressions, 'u' is equal to 'x' and 'v' is equal to 'y'. 'z' value depends on 'u' and 'v' according to equation \(z=4-u^{2}-v^{2}\). This gives a clear relationship between 'x', 'y', 'z', 'u' and 'v'.

Step 2: Understand the Form of the Surface

The equation for 'z' appears to be a negative quadratic in 'u' and 'v'. This suggests that the surface is part of a downward-opening paraboloid, with '4' as its maximum point.

Step 3: Sketch the Surface

Sketch the surface by first creating a circular outline with origin at (0,0) for both 'u' and 'v' ranging from -2 to 2. Then make the paraboloid shape, where the topmost point is '4' on the 'z' axis. Make sure the sketch shows the paraboloid opening downwards, as determined by the equation \(z=4-u^{2}-v^{2}\).

Key Concepts

Paraboloid Sketching

Paraboloids are a type of surface that have a distinct shape defined by their mathematical equations. To sketch a paraboloid, start by analyzing the equation given. In our exercise, the parametric surface is defined by \(z = 4 - u^2 - v^2\). This equation indicates a paraboloid because it involves quadratic terms with \(u\) and \(v\). The surface opens downward, as the coefficients of \(u^2\) and \(v^2\) are negative.
To sketch this surface effectively:

• Recognize it as a 3D object shaped like a bowl, inverted to open downward.
• Determine that the vertex, or peak of the paraboloid in this case, is at \(z = 4\) when \(u = 0\) and \(v = 0\).
• Set a range for \(u\) and \(v\) from -2 to 2 to encompass the main features of the paraboloid within a manageable graph area.

Start your sketch by drawing a circle on the \(xy\)-plane. The circle represents the boundary of the paraboloid's surface as it intersects the plane at \(z=0\) when \(u^2 + v^2 = 4\). Next, draw curves circling downwards to simulate the surface opening, with decreasing values for \(z\) as \(u\) and \(v\) move away from zero. This visualization helps to understand how the paraboloid's downward facing shape is formed.

Coordinate Systems

Coordinate systems are crucial as they enable us to define and locate specific points in space. For a parametric surface like a paraboloid, understanding the coordinate system used is essential for accurate graph depiction. The problem we're handling involves the Cartesian coordinate system, where points are defined by coordinates \((x, y, z)\).

In our exercise:

• 'u' directly corresponds to the 'x' coordinate.
• 'v' directly corresponds to the 'y' coordinate.
• 'z' is expressed in terms of 'u' and 'v', giving us \(z = 4 - u^2 - v^2\).

This setup helps us realize the transformation between parameters \(u, v\) and coordinates \(x, y, z\). Each different pair \((u, v)\) provides a distinct point \((x, y, z)\) on the surface of the paraboloid. By mapping these parameters across a specified range (such as -2 to 2 in this example), you capture the full stretch and shape of the surface. Mastery of coordinate systems allows you to visualize 3D structures accurately on a 2D plane.

Quadratic Surfaces

Quadratic surfaces are a broad category encompassing various forms, including spheres, cylinders, and hyperboloids. A paraboloid, like the one we're exploring in this problem, is a specific type of quadratic surface where the defining equation exhibits a quadratic relationship among variables.

A closer look at the given paraboloid equation, \(z = 4 - u^2 - v^2\), reveals key traits:

• It's a circular paraboloid due to having symmetrical \(u^2\) and \(v^2\) terms.
• The surface reaches a peak at point \(z = 4\), above the origin in the \(uv\)-plane.
• The negative leading coefficients indicate a downward opening.

Quadratic surfaces like paraboloids appear frequently in physics and engineering. They describe optics, potential fields, and even surface reflectors. Understanding their geometric properties aids in comprehending concepts such as concavity, foci, and how they interact with spatial constraints. By reducing complex shapes into simpler quadratic components, you can tackle intricate spatial analysis and modeling more effectively.