Find Curvature, find the curvature k of the curve, where s is the arc length parameter.
r(s)=(3+s)i+j
we want d T/ds
where T = r'(s)/|r'(s)| the unit tangent vector
r'(s) = 1 i + 0 j
|r'(s)| = 1
so
T = i + 0 j
dT/ds = 0
so
k = 0
Well, I'm a bot, so I don't have eyes to see the curve. But I can definitely bend over backward to try and help you find its curvature!
Now, let's get serious for a moment. To find the curvature of a curve defined by an arc length parameter, we need to calculate the second derivative of the position vector with respect to s.
Given r(s) = (3 + s)i + j, the first derivative dr/ds is simply 1i. Differentiate it again, and you'll get 0. Therefore, the second derivative d²r/ds² is equal to 0.
Curvature, denoted by k, is defined as the magnitude of the second derivative divided by the magnitude of the first derivative raised to the power of 3/2.
In this case, since the second derivative is zero, we have k = 0 / |1|^3/2 = 0.
So, the curvature of this curve is zero. It's as straight as a ruler!
To find the curvature of the curve given by r(s) = (3+s)i + j, where s is the arc length parameter, we need to compute the second derivative of r(s).
Step 1: Compute the first derivative of r(s)
To find the first derivative, we differentiate each component of r(s) with respect to s:
r'(s) = d/ds[(3+s)i + j]
= 1i + 0j
= i
Step 2: Compute the second derivative of r(s)
To find the second derivative, we differentiate each component of r'(s) with respect to s:
r''(s) = d/ds[1i]
= 0i
= 0
Step 3: Calculate the magnitude of the second derivative
The curvature k of a curve is given by the magnitude of its second derivative. Since the second derivative r''(s) is 0, the curvature k is also 0.
Therefore, the curvature k of the curve r(s) = (3+s)i + j, where s is the arc length parameter, is 0.
To find the curvature of the curve given by the position vector r(s) = (3 + s)i + j, where s is the arc length parameter, we can follow these steps:
Step 1: Find the velocity vector v(s) and the magnitude of the velocity vector, ||v(s)||.
The velocity vector v(s) is obtained by taking the derivative of the position vector r(s) with respect to s.
v(s) = dr(s)/ds = d(3 + s)i/ds + dj/ds = i + 0j = i
The magnitude of the velocity vector ||v(s)|| is equal to 1.
Step 2: Find the acceleration vector a(s) and the magnitude of the acceleration vector, ||a(s)||.
The acceleration vector a(s) is obtained by taking the derivative of the velocity vector v(s) with respect to s.
a(s) = dv(s)/ds = d(i)/ds = 0
The magnitude of the acceleration vector ||a(s)|| is equal to 0.
Step 3: Calculate the curvature k using the formula k = ||a(s)||/||v(s)||^3.
In this case, since ||a(s)|| = 0 and ||v(s)|| = 1, k = 0.
Therefore, the curvature of the curve r(s) = (3 + s)i + j is 0.