Use Theorem 10 to find the curvature. 𝐫(t)=3 t 𝐢+4 sint 𝐣+4 cost 𝐤

Name: Problem 19, Chapter 10: Essential Calculus Early Transcendentals
Uploaded: 2020-07-07T23:14:31-08:00
Duration: 7 min 42 s
Channel: Anna Marie Morra
Description: Use Theorem 10 to find the curvature. 𝐫(t)=3 t 𝐢+4 sint 𝐣+4 cost 𝐤

Use Theorem 10 to find the curvature. 𝐫(t)=3 t 𝐢+4 sint 𝐣+4...

Key Concepts

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in a vector perpendicular to both. Its magnitude is related to the area of the parallelogram spanned by the vectors, and in the curvature formula it helps measure the rate at which the direction of the tangent vector changes along the curve.

Curvature

Curvature is a measure of how sharply a curve bends at a given point. In the context of vector calculus, it is often computed using the derivatives of a vector-valued function that represents the curve, commonly using the formula involving the cross product of the first and second derivatives. This measurement is essential in understanding the local geometrical properties of curves.

Differentiation of Vector-Valued Functions

Differentiation of vector-valued functions involves taking the derivative of each of the component functions separately. This process yields the velocity vector (first derivative) and acceleration vector (second derivative), which are critical for analyzing the dynamics and geometry of curves in space.

Transcript

00:01 In this problem, we want to use theorem 10 to find the curvature of the curve represented by the vector r of t, which can be written as 3t times vector i plus 4 sine of t times vector j, plus 4 cosine of t times vector k.

00:25 I'm going to go ahead and write that in my angle bracket notation as 3t, 4, sine of of t and four cosine of t.

00:40 So let's start off by taking the first derivative of r prime, which we'll need for the numerator of our, the numerator and denominator actually of our curvature function by theorem 10.

00:52 So what we'll have is 3, 4 times cosine of t and negative 4 times sine of t.

01:05 And that's just taking the first derivative with respect to t of each of the component functions in r.

01:11 And then we're going to take the derivative again to get the second derivative of r, so r double prime, and we'll have 0, negative 4 times sine of t, and negative 4 times cosine of t.

01:28 And we want to take the cross product of r prime and our double prime.

01:35 So r prime of t crossed with our double prime of t.

01:41 We can write that as the determinant of a 3 by 3 matrix, so we're going to have i 3 and 0 in our first column j 4 times cosine of t and negative 4 times sign of t in our second column and vector k negative 4 times sine of t and negative 4 times cosine of t in that third column and then we'll go through and what we'll have for our first term is we're going to look in collins j and k to get the 2 by 2 determinant 4 times cosine of t, negative 4 times sine of t, negative 4 times sine of t, and negative 4 times cosine of t is the appropriate elements, and that 2 by 2 determinant is going to be multiplied by vector i.

02:45 Then we're going to subtract a 2 by 2 determinant times vector j.

02:51 So we're looking in the i and k column, so we'll have 3, 0, negative 4, sine of t, and negative 4 cosine of t times vector j, and then plus the 2 by 2 determinant of the items in vector, in columns i and j, so we're going to have 3, 0, 4 cosine of t, and negative 4 sine of t, and that's going to be times vector k.

03:24 And then let's go through and take those two by two determinants.

03:29 So for our first one, we're going to have negative 16 cosine squared of t minus 16 sine squared of t.

03:44 That's times vector p.

03:47 And we're going to subtract negative 12 times cosine of t minus zero.

03:55 We don't have to necessarily write that.

04:00 Plus negative 12 times sine of t minus zero.

04:05 And again, we don't have to write that to save some space.

04:09 So let's simplify that a little bit.

04:10 And i'm going to go ahead and write it in the angle bracket notation.

04:15 Notice that we can factor a negative 16 out of this term up here...