Problem 186 - Calculus Volume 3

Question

Find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, \(P_{0}\left(x_{0}, y_{0}, z_{0}\right),\) and a vector \(\mathbf{n}=\langle a, b, c\rangle\) that is parallel to the line. Then the equation of \(\quad\) the \(\quad\) line \(\quad\) is \(\left.x-x_{0}=a t, y-y_{0}=b t, z-z_{0}=c t .\right)\). \(z=e^{4 x^{2}+6 y^{2}}, P(0,0,1)\)

Step-by-Step Solution

Answer

The normal line is undefined, as the gradient is zero at the point.

Step 1: Identify the Geometric Problem

We analyze the given geometric figure and identify what needs to be found.

Step 2: Apply the Appropriate Formula

We apply the relevant geometric formula or theorem.

Step 3: Compute the Result

Performing the calculations.

Step 4: State the Result

The normal line is undefined, as the gradient is zero at the point.

Key Concepts

Normal Line

A normal line to a surface is a straight line that is perpendicular to the tangent plane at a given point on the surface. Imagine a tangent plane gently touching the surface at one point; the normal line shoots out perpendicularly from this plane like an arrow.

This concept is crucial in understanding how a surface behaves around a point. It gives insights into the surface's local geometry, such as whether the surface is curving upwards or downwards.

• Understanding Perpendicularity: The normal line must be perpendicular at the specific point of contact, maintaining a perfect orthogonal relationship with the tangent plane.
• Mathematical Representation: In a three-dimensional space, the line can be defined using a point that lies on the surface, together with a vector that is perpendicular to the surface (often the gradient vector).

For example, if you're tasked with finding the parametric equations of a normal line, you'd determine a particular point on the surface and compute the gradient vector at that point to use as the direction vector of the line.

Gradient Vector

The gradient vector of a function at a point gives the direction of the greatest rate of increase of the function. It's like a compass pointing you straight uphill from where you stand on a surface. This vector is composed of partial derivatives and indicates the direction in which the function increases most rapidly.

• Construction: If we have a function, say, a surface described by an equation in three dimensions like \(z=f(x, y)\), the gradient vector \(abla f\) is \(\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \rangle\).
• Properties: The gradient points in the direction of steepest ascent and is perpendicular to the level curve (or surface) at a given point, making it a key player in defining the normal line to a surface at a point.

For instance, in the exercise presented, calculating the gradient vector at point \((0, 0)\) gave us zero direction \(\langle 0, 0 \rangle\), indicating a flat area on the surface.

Partial Derivatives

Partial derivatives measure the rate of change of a function with respect to one of its variables while keeping the other variables constant. Think of it as examining the slope along only one axis at a time.

• Application: In multivariable calculus, partial derivatives are used to find the slopes of tangent lines to curves and surfaces.
• Usage in Gradient Vector: Each component of a gradient vector is a partial derivative of the function, showing how the function changes along each axis.

In the context of the problem, finding \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) gives us the rate at which the surface \(z = e^{4x^2 + 6y^2}\) changes with respect to \(x\) and \(y\), respectively. This process helps us compute the gradient vector.

Vector Calculus

Vector calculus is the branch of mathematics that deals with vector fields and the differentiation and integration of vector functions. It extends calculus into higher dimensions focusing on vectors, which are quantities defined by magnitude and direction.

• Importance: Used extensively in physics and engineering, particularly in modeling and analysis of electromagnetic fields, fluid dynamics, and forces.
• Tools and Concepts: Key tools of vector calculus include gradient, divergence, and curl, each providing specific insights into vector behavior and field characteristics.

This mathematical framework enables us to tackle the geometry of curves and surfaces analytically. In surfaces described by equations like \(z = e^{4x^2 + 6y^2}\), vector calculus concepts are central to understanding their orientation and interaction with points and lines.