Problem 164 - Calculus 3

Question

Find a unit normal vector to the surface at the indicated point. $$\ln \left(\frac{x}{y-z}\right)=0 \text { when } x=y=1$$

Step-by-Step Solution

Answer

The unit normal vector is \( \left( \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \).

Step 1: Understand the Surface Equation

Given the equation \( \ln \left(\frac{x}{y-z}\right) = 0 \), we can simplify this to \( \frac{x}{y-z} = 1 \). Consequently, the equation of the surface becomes \( x = y - z \).

Step 2: Rewrite as a Level Surface

Rearranging the surface equation, we have \( f(x, y, z) = x - y + z = 0 \). Here, \( f(x, y, z) \) represents a level surface in \( \mathbb{R}^3 \).

Step 3: Calculate the Gradient

To find a normal vector to the surface at a point, we compute the gradient of \( f(x, y, z) \). The gradient is given by \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = (1, -1, 1) \).

Step 4: Evaluate the Gradient at the Indicated Point

Substitute \( x = 1, y = 1, z = 0 \) into the gradient to find its value at the given point. The gradient remains \( (1, -1, 1) \) since it is a constant vector field on this surface.

Step 5: Normalize the Gradient Vector

To find the unit normal vector, we need to normalize the gradient vector. The magnitude of \( abla f = (1, -1, 1) \) is \( \|abla f\| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{3} \). Thus, the unit normal vector is \( \left( \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \).

Key Concepts

Unit Vector

In mathematics and physics, a unit vector is a vector that has a magnitude of 1. Unit vectors are particularly useful in defining directions without considering magnitude. In our exercise, we aim to find a unit normal vector to a surface, which means a vector that is perpendicular to the surface at a given point and has a magnitude of 1.

• Unit vectors are often denoted with a hat above the letter, such as \( \hat{i} \), \( \hat{j} \), or \( \hat{k} \).
• To convert any vector into a unit vector, we use the process of normalization, which involves dividing the vector by its magnitude.
• A unit normal vector is essential in many fields, including physics and computer graphics, for calculations involving force directions and surface orientations.

The unit vector calculated in the exercise is \( \left( \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \), which serves as a fundamental building block in understanding vector spaces and surface interactions.

Gradient

The gradient is a vector that contains all the partial derivatives of a multivariable function. It points in the direction of the greatest rate of increase of the function. For a function of three variables, like in our exercise, the gradient \( abla f \) is given by \( (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}) \).

• The gradient vector is extremely crucial in determining the nature of a surface at a specific point.
• In our exercise, the gradient \( (1, -1, 1) \) tells us how the surface is oriented in space.
• It also serves as a normal vector since it's perpendicular to level surfaces, thus playing a dual role in both analytical and geometrical contexts.

Finding the gradient was a critical step in arriving at the unit normal vector to the surface because it gives us the needed vector direction before normalization.

Level Surface

A level surface, in mathematical terms, is the set of points \((x, y, z)\) in three-dimensional space where a given scalar function \(f(x, y, z)\) is constant. For example, in the exercise, the level surface is \(x - y + z = 0\).

• Level surfaces are vital for visualizing and understanding complex mathematical surfaces and contours in 3D space.
• They can represent things like potential fields in physics or topographic maps in geography.
• A normal vector to the level surface is derived from its gradient, providing further analytical insight into the surface's properties.

By considering the equation as a level surface, we are equipped to find normal vectors and vectors parallel to the surface, helping in a myriad of applied science and engineering solutions.

Normalization

Normalization is the process used to convert a vector into a unit vector. This involves dividing each component of the vector by its magnitude. It ensures that the resulting vector has a length of 1, making it a unit vector.

• The formula for normalizing a vector \(\vec{v} = (a, b, c)\) is \( \frac{\vec{v}}{\|\vec{v}\|} \), where \(\|\vec{v}\|\) is the magnitude and is calculated as \( \sqrt{a^2 + b^2 + c^2} \).
• Normalization keeps the direction of the vector unchanged, which is particularly useful when direction is vital in applications.
• In our context, normalization was needed to ensure the gradient became a unit normal vector, crucial when applying constraints like equilibrium or field consistency.

Understanding normalization is key in using vectors effectively, allowing precise manipulation in both analytical and applied mathematical fields.