Problem 13 - A History of Mathematics: An Introduction

Question

Convert Monge's form of the equations of the normal line to the surface \(z=f(x, y)\) into the modern vector equation of the line.

Step-by-Step Solution

Answer

Question: Convert the Monge's form of the equations of the normal line to the surface \(z = f(x, y)\) into the modern vector equation of the line. Answer: The modern vector equation of the normal line can be written as \(\vec{r}(t) = \begin{bmatrix} x_0+At \\ y_0+Bt \\ f(x_0, y_0)+Ct \end{bmatrix}\), where \(t\) is the parameter, \((x_0, y_0, f(x_0, y_0))\) is a point on the surface, and \((A, B, C)\) are the components of the normal vector \(\vec{N} = \frac{\partial f}{\partial x}(x_0, y_0)\hat{i} + \frac{\partial f}{\partial y}(x_0, y_0)\hat{j} - \hat{k}\).

Step 1: Find the normal vector to the surface

Given the surface equation \(z = f(x, y)\), we can find the normal vector \(\vec{N}\space (A,B,C)\) at a point \((x_0, y_0, f(x_0, y_0))\) by computing the gradient of \(f\): $$ \nabla f(x,y) = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} - \hat{k}. $$ At the point \((x_0, y_0, f(x_0, y_0))\), the normal vector is $$ \vec{N} = \frac{\partial f}{\partial x}(x_0, y_0)\hat{i} + \frac{\partial f}{\partial y}(x_0, y_0)\hat{j} - \hat{k}. $$

Step 2: Write the parametric form of the normal line

Now that we have the normal vector \(\vec{N}\) at the point \((x_0, y_0, f(x_0, y_0))\), we can write the parametric form of the normal line using a parameter \(t\): $$ \begin{cases} x = x_0 + At\\ y = y_0 + Bt\\ z = f(x_0, y_0) + Ct \end{cases}, $$ where \((A, B, C)\) are the components of the normal vector \(\vec{N}\).

Step 3: Convert the parametric form to the modern vector equation

The modern vector equation of a line can be written as \(\vec{r}(t) = \vec{r_0} + t\vec{N}\), where \(\vec{r}(t)\) is the position vector of a point on the line, \(\vec{r_0}\) is the position vector of the point \((x_0, y_0, f(x_0, y_0))\), and \(\vec{N}\) is the normal vector. Given the parametric form from Step 2, the position vector \(\vec{r}(t)\) is: $$ \vec{r}(t) = \begin{bmatrix} x_0+At \\ y_0+Bt \\ f(x_0, y_0)+Ct \end{bmatrix}. $$ The position vector of the point \((x_0, y_0, f(x_0, y_0))\) is: $$ \vec{r_0} = \begin{bmatrix} x_0 \\ y_0 \\ f(x_0, y_0) \end{bmatrix}. $$ And the normal vector \(\vec{N}\) is: $$ \vec{N} = \begin{bmatrix} A \\ B \\ C \end{bmatrix}. $$ Therefore, the modern vector equation of the normal line is: $$ \vec{r}(t) = \vec{r_0} + t\vec{N} = \begin{bmatrix} x_0 \\ y_0 \\ f(x_0, y_0) \end{bmatrix} + t \begin{bmatrix} A \\ B \\ C \end{bmatrix} = \begin{bmatrix} x_0+At \\ y_0+Bt \\ f(x_0, y_0)+Ct \end{bmatrix}. $$

Key Concepts

Normal Vector

A normal vector is a vector that is perpendicular to a surface at a given point. For a surface defined by the equation \(z = f(x, y)\), finding the normal vector involves computing the gradient of the function \(f\). The gradient, denoted by \(abla f\), gives us a direction where the function increases most steeply.
To find the normal vector using the gradient, we look at its components. It is calculated as:

• \(abla f(x,y) = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} - \hat{k}\).
• The terms \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) represent the rates of change of \(f\) with respect to \(x\) and \(y\) respectively.
• The negative sign before \(\hat{k}\) aligns the normal vector with the surface's orientation in three-dimensional space.

The gradient thus provides the normal vector \(\vec{N} = \left(\frac{\partial f}{\partial x}(x_0, y_0), \frac{\partial f}{\partial y}(x_0, y_0), -1\right)\) at any point \((x_0, y_0, f(x_0, y_0))\). This vector is essential for defining the orientation of the surface and constructing lines perpendicular to it, such as normal lines.

Parametric Equations

Parametric equations are a way to express the coordinates of the points on a line using a parameter, often denoted by \(t\). This approach provides a clearer understanding of how points change along the line.
When we have a line through a point \((x_0, y_0, z_0)\) with direction defined by a vector, the parametric equations in terms of \(t\) are:

• \(x = x_0 + At\)
• \(y = y_0 + Bt\)
• \(z = z_0 + Ct\)

Here, \((A, B, C)\) correspond to the components of the direction or normal vector \(\vec{N}\). These equations give us the position of a point on the line as \(t\) changes.
This method is particularly useful in vector calculus as it simplifies the transition from geometrical descriptions to algebraic expressions. For instance, parametric descriptions help in transforming complex surface properties into manageable algebraic forms. They make computations more accessible, especially when vector quantities are involved.

Gradient

In calculus, the gradient is a vector that shows the direction of the most significant rate of increase of a function, and its magnitude is the rate of increase. It's crucial for understanding how functions behave and change in space.
When dealing with a surface described by \(z = f(x, y)\), the gradient \(abla f\) is vital in finding the normal vector to the surface. It is given by:

• \(abla f(x, y) = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j}\)

This expression provides the direction in which the function \(f\) is increasing the fastest, moving from point \((x, y)\) across the surface.
The significance of the gradient in vector calculus cannot be overstated. It assists in determining tangent planes, optimizing functions, and is instrumental in generating differential equations that describe physical phenomena. Understanding the gradient's direction and magnitude is fundamental in visualizing and working with multi-variable systems, making it a key tool in calculus.