Calculus Volume 3 · free preview
Arc Length and Curvature
Space curves have arc length integrating the speed, and curvature measuring the bending per unit distance with the unit tangent and the principal normal framing the curve's turning geometry.
1. Arc length of a space curve
Arc length of a space curve
2. Curvature
Curvature
3. The Frenet frame
The Frenet frame
4. Reparametrization by arc length
Reparametrization by arc length
5. Space arc length
The arc length of r(t) from a to b equals which integral?
6. Curvature of a circle
The circle of radius a has curvature equal to what?
7. Unit tangent
The unit tangent T equals which expression?
8. Normal direction
The principal normal N points in which direction?
9. Straight line curvature
A straight line parametrized as r(t) = r_0 + t v has curvature equal to what?
10. Helix speed value
The helix (cos t, sin t, t) has constant speed root 2. As a number, what is under the root? Answer with a number.
11. Curvature formula denominator
The curvature kappa equals |r prime cross r double prime| over |r prime| cubed. As a power, what exponent? Answer with a number.
12. Binormal definition
The binormal B equals T cross N. As a single letter, what is it? Answer with an uppercase letter.
13. Osculating circle meaning
The osculating circle at a point of a curve is best described as what?
14. Torsion meaning
Torsion measures the curve's twist out of which plane? Answer with two words.
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