Calculus Volume 3 · 免费预览
弧长与曲率
空间曲线的弧长由速率积分得到，曲率则度量单位长度上的弯曲程度，而单位切向量与主法向量共同刻画曲线的转向几何。
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.