Question

Curvature for plane curves Use the result of Exercise 55 to find the curvature function of the following curves. $$\mathbf{r}(t)=\langle a \sin t, b \cos t\rangle(\text { ellipse })$$

Short answer

Question: Determine the curvature function for the given plane curve represented by the vector function: $$\mathbf{r}(t) = \langle a \sin t, b \cos t\rangle$$ using the result of Exercise 55. Answer: The curvature function for the given plane curve is: $$\kappa(t) = \frac{-ab}{(a^2 \cos^2 t + b^2 \sin^2 t)^{3/2}}$$

Step-by-step solution

Find the first and second derivatives of the curve

First, we need to find the first and second derivatives of the given curve. The first derivative, or velocity vector, will represent the tangent to the curve, and the second derivative, or the acceleration vector, will represent the rate of change of the tangent. Using the given curve, let's find the first and second derivatives with respect to t: $$\mathbf{r}(t) = \langle a \sin t, b \cos t\rangle$$ $$\mathbf{r}'(t) = \langle a \cos t, -b \sin t\rangle$$ $$\mathbf{r}''(t) = \langle -a \sin t, -b \cos t\rangle$$

Find the cross product of the first and second derivatives

Now, we’ll find the cross product of the first and second derivatives, which will be a scalar since we're in two dimensions. $$\mathbf{r}'(t) \times \mathbf{r}''(t) = (a \cos t)(-b \cos t) - (-b \sin t)(-a \sin t)$$ $$\mathbf{r}'(t) \times \mathbf{r}''(t) = -ab \cos^2 t - ab \sin^2 t$$

Find the magnitude of the first derivative

Next, we need to find the magnitude of the first derivative to calculate the curvature function. $$\|\mathbf{r}'(t)\| = \sqrt{(a \cos t)^2 + (-b \sin t)^2}$$ $$\|\mathbf{r}'(t)\| = \sqrt{a^2 \cos^2 t + b^2 \sin^2 t}$$

Calculate the curvature function

Finally, we will plug the values obtained in Steps 2 and 3 into the formula for the curvature function: $$\kappa(t) = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3}$$ $$\kappa(t) = \frac{-ab \cos^2 t - ab \sin^2 t}{(\sqrt{a^2 \cos^2 t + b^2 \sin^2 t})^3}$$ $$\kappa(t) = \frac{-ab(\cos^2 t + \sin^2 t)}{(a^2 \cos^2 t + b^2 \sin^2 t)^{3/2}}$$ Since \(\cos^2 t + \sin^2 t = 1\), the curvature function simplifies to: $$\kappa(t) = \frac{-ab}{(a^2 \cos^2 t + b^2 \sin^2 t)^{3/2}}$$ So, the curvature function for the given plane curve is: $$\kappa(t) = \frac{-ab}{(a^2 \cos^2 t + b^2 \sin^2 t)^{3/2}}$$

Key concepts

Ellipse

An ellipse is a geometrical shape resembling a stretched circle. It is defined based on two principal parameters: the semi-major axis (denoted as 'a') and the semi-minor axis (denoted as 'b'). These parameters determine the shape and orientation of the ellipse in the plane. The longest line that can be drawn across the ellipse passes through its center and along its major axis. Similarly, the shortest line that passes through the center runs along its minor axis.

Ellipses appear commonly in the natural world and are important in various fields such as astronomy, where the orbits of planets and celestial bodies are often elliptical. In mathematics, they provide a rich ground for exploration, particularly when analyzing their properties using calculus.

When an ellipse is expressed in a parametric form, such as \(\mathbf{r}(t) = \langle a \sin t, b \cos t\rangle\), each value of the parameter 't' gives a point on the ellipse, making it easy to analyze its properties and segment its arc.

Parametric Equations

Parametric equations provide a powerful way to represent curves, lines, and shapes by using a parameter, typically denoted as 't', to define the coordinates of points on a curve. Unlike the traditional Cartesian equations that express the relation between y and x alone, parametric equations introduce an additional dimension giving more control and flexibility in describing motion and paths efficiently.

For example, in the given exercise, the ellipse is parameterized as \(\mathbf{r}(t) = \langle a \sin t, b \cos t\rangle\). Here, 't' acts as a parameter controlling the position along the curve. With each increment in 't', we obtain successive points that together form the curve.

By enabling us to work with parameters, parametric equations simplify complex curves into more manageable forms, and allow for the exploration of curves' derivatives, curvature, and other properties in calculus making it easier to analyze motion and paths along these curves.

Calculus Derivatives

Derivatives are fundamental in calculus for measuring changes and determining the slopes of curves at specific points. They form the basis for calculating the rate at which one variable changes concerning another. In the context of parametric equations, we calculate the derivatives with respect to the parameter 't' to find slopes and curvatures of the parameterized curves.

In this exercise, the first derivative, \(\mathbf{r}'(t) = \langle a \cos t, -b \sin t\rangle\), represents the tangent or velocity vector of the curve, providing the direction and rate of change of the curve at any given point. The second derivative, \(\mathbf{r}''(t) = \langle -a \sin t, -b \cos t\rangle\), represents the acceleration, illustrating how the direction of the tangent is changing, which is directly related to the curve's curvature.

These derivatives are not only crucial for analyzing the motion along curves but also pivotal for understanding dynamic systems, applications in physics, and other engineering problems where changing rates and orientations are crucial.

Rate of Change

The rate of change is a concept in calculus describing the speed at which a quantity changes over time. This concept becomes essential when dealing with motion along curves, as it helps to understand how the position of a point on the curve evolves with respect to the parameter 't'.

In the given exercise, the rate of change is illustrated by derivatives. The first derivative represents the instantaneous rate of change of the curve's position, indicating how fast the curve is moving at each point. The second derivative provides further insight into how this rate itself is changing, offering a lens into the dynamics of acceleration along the curve.

Understanding rates of change is fundamental in fields like physics, where it applies to velocities and accelerations. In economics, it explains how quantities like cost or revenue change over time. This concept forms the foundation for making predictions and analyzing trends across various scientific disciplines.