Problem 11 - A History of Mathematics: An Introduction

Question

Find the curvature of the ellipse \(x^{2}+4 y^{2}=1\) by using Newton's procedure.

Step-by-Step Solution

Answer

Question: Find the curvature of the ellipse \(x^2 + 4y^2 = 1\) in terms of the parameter \(t\). Answer: The curvature of the ellipse is given by the expression \(k = \frac{1/2}{(\sin^2{t} + \frac{1}{4}\cos^2{t})^{\frac{3}{2}}}\).

Step 1: Find the parametric equation of the ellipse

First, we need to express the ellipse in parametric form. Let $$ x = \cos{t}, $$ and $$ y = \frac{1}{2} \sin{t}. $$ Now, we have an equation for the position on the ellipse as a function of a parameter \(t\): \((x(t), y(t)) = (\cos{t}, \frac{1}{2} \sin{t})\).

Step 2: Find the first derivatives

Now we find the first derivatives \(x'(t)\) and \(y'(t)\). Differentiating \(x(t)\) with respect to \(t\), we get $$ x'(t) = -\sin{t}. $$ Differentiating \(y(t)\) with respect to \(t\), we get $$ y'(t) = \frac{1}{2} \cos{t}. $$

Step 3: Find the second derivatives

Next, we find the second derivatives \(x''(t)\) and \(y''(t)\). Differentiating \(x'(t)\) with respect to \(t\), we get $$ x''(t) = -\cos{t}. $$ Differentiating \(y'(t)\) with respect to \(t\), we get $$ y''(t) = -\frac{1}{2} \sin{t}. $$

Step 4: Apply Newton's procedure to find the curvature

According to Newton's procedure, the curvature of a curve is given by the formula $$ k = \frac{|x'y'' - x''y'|}{(x'^{2} + y'^{2})^{\frac{3}{2}}}. $$ Substituting the expressions for the first and second derivatives, we have $$ k = \frac{|(-\sin{t})(- \frac{1}{2} \sin{t}) - (-\cos{t}) (\frac{1}{2}\cos{t})|}{((- \sin{t})^{2} + (\frac{1}{2}\cos{t})^{2})^{\frac{3}{2}}}. $$ Simplifying the expression, we get $$ k = \frac{\frac{1}{2} \sin^2{t} + \frac{1}{2} \cos^2{t}}{(\sin^2{t} + \frac{1}{4}\cos^2{t})^{\frac{3}{2}}}. $$ As \(\sin^2{t} + \cos^2{t} = 1\), the result can be simplified further: $$ k = \frac{1/2}{(\sin^2{t} + \frac{1}{4}\cos^2{t})^{\frac{3}{2}}}. $$ This is the curvature of the ellipse \(x^2 + 4y^2 = 1\) as a function of the parameter \(t\).

Key Concepts

Parametric Equations

Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. For example, in the case of an ellipse, the parametric equations are often used to represent its shape because they can neatly describe the x and y coordinates separately for any point along the curve.

In the exercise at hand, the ellipse defined by the equation \(x^{2}+4y^{2}=1\) is expressed in parametric form with the parameter \(t\) representing the angle at which a point on the ellipse is located from the positive x-axis. The parametric equations given are \(x(t) = \cos{t}\) and \(y(t) = \frac{1}{2}\sin{t}\). This simplifies the process of working with the curve, especially when calculating derivatives or integrals.

Derivatives in Calculus

Derivatives represent the rate at which a function is changing at any given point and play a crucial role in calculus. They are fundamental in finding slopes of tangent lines, velocities, and acceleration among other rates of change. In the exercise solution, derivatives are used to find the rate of change of the x and y coordinates with respect to the parameter \(t\).

First derivatives \(x'(t)\) and \(y'(t)\) tell us how quickly the coordinates change along the ellipse. The second derivatives \(x''(t)\) and \(y''(t)\), which are the derivatives of the first derivatives, provide information on the curvature of the path at any given point. When these derivatives are plugged into Newton's formula for curvature, they yield the curvature \(k\) of the ellipse at a specific point.

Newton's Procedure in Mathematics

Newton's procedure, also known as Newton's method, is a technique used in mathematics to find the roots of a function or, in this context, to find the curvature of a curve. For curves defined by parametric equations, such as our ellipse, the curvature \(k\) at a particular point is given by the formula \[k = \frac{|x'y'' - x''y'|}{(x'^{2} + y'^{2})^{\frac{3}{2}}}.\]
This formula is derived from the principles of differential calculus and represents the inverse of the radius of the osculating circle at a point on the curve. The osculating circle is the circle that best approximates the curve at that point. The curvature tells us how quickly the curve is changing direction at that point. Applying this procedure to our ellipse, we obtain an expression for \(k\), which indicates how the curvature of the ellipse changes as we move along its path.