Calculus Volume 3 · 免费预览
向量值函数与空间曲线
向量值函数把参数映射为向量，从而描绘出空间曲线；其极限、连续性以及逐分量的结构，把函数概念扩展到三维轨迹。
1. Vector-valued functions
Vector-valued functions
2. Limits and continuity
Limits and continuity
3. Graphing space curves
Graphing space curves
4. Domains and standard parametrizations
Domains and standard parametrizations
5. Componentwise limit
The limit of r(t) = (f(t), g(t), h(t)) as t approaches a equals what?
6. Helix components
The helix (cos t, sin t, t) has which projection onto the xy plane?
7. Continuity definition
The vector function r is continuous at a when which condition holds?
8. Domain intersection
The domain of r(t) = (1/t, root t, ln t) is which interval?
9. Magnitude continuity insufficiency
Why is continuity of |r(t)| alone not enough for r's continuity?
10. Twisted cubic middle component
The twisted cubic (t, t^2, t^3) has which middle component? Answer like t^2.
11. Helix rise per turn
The helix (a cos t, a sin t, ct) rises by 2 pi c per turn. As a multiple of pi c, what is the rise? Answer with a number.
12. Domain intersection count
The domain of (1/t, root t, ln t) excludes zero and negatives. As a number, how many components restrict it? Answer with a number.
13. Curve versus path distinction
The helix's projection being a circle illustrates which distinction?
14. Position vector viewpoint
A space curve's points are the tips of which vectors as the parameter varies? Answer with two words.