Problem 35 - Advanced Engineering Mathematics

Question

Find symmetric equations for the normal line. $$ z=4 x^{2}+9 y^{2}+1 ;\left(\frac{1}{2}, \frac{1}{3}, 3\right) $$

Step-by-Step Solution

Answer

The symmetric equations are \( \frac{x - \frac{1}{2}}{4} = \frac{y - \frac{1}{3}}{6} = \frac{z - 3}{-1} \).

Step 1: Understanding the Surface and Point

The given surface equation is that of an elliptic paraboloid, described by \( z = 4x^2 + 9y^2 + 1 \). You need to find the normal line at the point \( \left(\frac{1}{2}, \frac{1}{3}, 3\right) \) on this surface.

Step 2: Finding the Gradient Vector

The gradient vector of the function \( z = f(x, y) = 4x^2 + 9y^2 + 1 \) gives the direction of the normal. First, compute the partial derivatives: \( f_x = \frac{\partial f}{\partial x} = 8x \) and \( f_y = \frac{\partial f}{\partial y} = 18y \). At the point given, \( x = \frac{1}{2} \) and \( y = \frac{1}{3} \), calculate the gradient vector: \( abla f = (8 \cdot \frac{1}{2}, 18 \cdot \frac{1}{3}) = (4, 6) \). The direction vector for the normal line is \( \langle 4, 6, -1 \rangle \) since the function \( z \) is missing from \( f \), we add -1 for the derivative in the direction of \( z \).

Step 3: Writing the Symmetric Equations

The normal line can be parameterized using the point \( \left(\frac{1}{2}, \frac{1}{3}, 3\right) \) and the direction vector \( \langle 4, 6, -1 \rangle \). The parameterized equations of the line are: \( x = \frac{1}{2} + 4t \), \( y = \frac{1}{3} + 6t \), \( z = 3 - t \). The symmetric equations are obtained by eliminating the parameter \( t \): \( \frac{x - \frac{1}{2}}{4} = \frac{y - \frac{1}{3}}{6} = \frac{z - 3}{-1} \).

Key Concepts

Normal Line

The normal line to a surface at a given point is a line that is perpendicular to the tangent plane at that point. In our exercise, finding the normal line for the elliptic paraboloid involves determining the direction in which the surface changes most rapidly at the given point, \( \left(\frac{1}{2}, \frac{1}{3}, 3\right) \). This is achieved through the gradient vector, which points in this direction.

• The normal line is crucial in understanding the geometric relationship between a surface and the space around it.
• It spans the height of the surface, adding a third dimension to a two-dimensional tangent plane.

These properties allow us to express the normal line through symmetric equations, offering a means to capture its spatial orientation and positioning. This line is useful in various applications, including optimization problems and surface analysis. The symmetric equations succinctly describe the normal line's pathway utilizing its direction vector and the initial point.

Elliptic Paraboloid

An elliptic paraboloid is a three-dimensional surface emerging from the equation \( z = ax^2 + by^2 + c \). In our particular exercise, \( z = 4x^2 + 9y^2 + 1 \) defines the surface. This kind of surface resembles an upward-facing bowl.

• The constants in the equation define the curvature and orientation of the paraboloid.
• It is shaped like a parabola in two vertical cross-sections that intersect at the origin.

Understanding its properties helps in identifying the nature of curves and planes that interact with it, such as finding tangent planes or normal lines. The ***elliptic paraboloid*** plays an important role in disciplines like physics and engineering, where surface characteristics such as maximum height impact system responses or processes.

Gradient Vector

The gradient vector is a fundamental concept in calculus and vector analysis, representing the direction of the steepest ascent of a function. For a function \( f(x, y) \), the gradient vector \( abla f \) is composed of its partial derivatives \( f_x \) and \( f_y \). In our exercise: - \( f_x = 8x \) - \( f_y = 18y \)At the point \( \left(\frac{1}{2}, \frac{1}{3}\right) \), the gradient vector is calculated as \( (4, 6) \), indicating the direction and magnitude of the maximal rate of increase.

• In three-dimensional contexts, the gradient vector helps establish the normal vector for a surface.
• This information is fundamental in creating tangent planes and assisting in optimization problems.

The role of the gradient vector goes beyond mathematics; it's applied to fields such as artificial intelligence and machine learning, where optimization is key.

Parameterization

Parameterization is the process of defining a curve, surface, or line using parameters typically denoted as \( t \) (for lines or curves) or other variables for surfaces. In this exercise, parameterization enables the description of the normal line passing through a given point on the elliptic paraboloid. For the point \( \left(\frac{1}{2}, \frac{1}{3}, 3\right) \), and direction vector \( \langle 4, 6, -1 \rangle \), the parameterized equations are:- \( x = \frac{1}{2} + 4t \)- \( y = \frac{1}{3} + 6t \)- \( z = 3 - t \)

• This approach provides a straightforward way to describe every point on a line or path.
• It simplifies complex three-dimensional paths into one-dimensional functions.

Parameterization plays a critical role in computational models and simulations, enabling the resolution of dynamic systems over time or along specific trajectories.