Use Theorem 10 to find the curvature. 𝐫(t)=t^3 𝐣+t^2 𝐤 |...

Name: Problem 17, Chapter 10: Essential Calculus Early Transcendentals
Uploaded: 2020-07-07T22:41:11-08:00
Duration: 5 min 56 s
Channel: Anna Marie Morra
Description: Use Theorem 10 to find the curvature. 𝐫(t)=t^3 𝐣+t^2 𝐤

Key Concepts

Curvature

Curvature is a measure of how sharply a curve bends at a given point. In the context of differential geometry, it quantifies the rate of change of the tangent vector with respect to arc length. One commonly used formula for curvature, especially when dealing with parametrized curves, is the magnitude of the cross product of the first and second derivatives divided by the cube of the magnitude of the first derivative.

Cross Product Method for Curvature

The cross product method for computing curvature involves using the first and second derivatives of the curve to form a vector whose magnitude directly relates to the change in direction of the curve. Specifically, the curvature is given by the formula kappa = |r'(t) x r''(t)| / |r'(t)|^3. This method is particularly useful in three-dimensional space, but it can also be adapted to planar curves by considering the appropriate interpretation of the cross product.

Parametrized Curve

A parametrized curve is a function that maps a real parameter (often time) to points in space, providing a way to represent curves by tracking how position changes with respect to the parameter. This concept is fundamental in differential geometry as it lays the groundwork for analyzing the properties of curves, such as tangents, lengths, and curvature.

Vector Differentiation

Vector differentiation involves finding the derivative of a vector-valued function with respect to its parameter. The first derivative represents the velocity (or tangent vector) of the curve, and the second derivative provides information about the acceleration, which are both crucial for determining the curvature at a point along the curve.

Transcript

00:01 Alright, in this problem we want to use theorem 10, which is in the top right corner of the screen, to find the curvature of the curve represented by the vector r of t.

00:13 T cubed times vector j plus t squared times vector k.

00:22 I'm going to go ahead and write this in the angle bracket notation.

00:26 So we would have zero, since we have no vector i component that's written, t -quist.

00:34 And t squared is our third component function.

00:38 So to use theorem 10 we need to have the first and second derivatives of the vector r.

00:43 So i'm going to take the first derivative.

00:45 So we'll have r prime of t, which is going to be zero as our first component function.

00:51 Second component function is going to be three t squared and third component function is going to be two t.

00:59 And then let's go ahead and take the second derivative as well.

01:02 So we're going to have our double prime.

01:04 Of t, which is going to be 0, 6t, and 2.

01:13 So to get the numerator, we need to take the cross product of those two vectors of the first and second derivatives.

01:19 So i'm gonna take our prime crossed with our double prime.

01:28 And what we're gonna do is we're gonna write it in this notation, so that way it's easier to take the determinant...

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