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Chapter 11. Vector Functions
- 11장 답안입니다. 자세한 답안이 아닌 학습 시 정답을 확인하기 위한 간략한 답안입니다. 시험에서는 자세한
§11.1. Vector Functions and Space Curves
- Find the limit lim
→ ∞
〈
sin
〉
(sol) 〈
〉
- Draw the projections of the curve 〈 sin cos 〉 on the three coordinate planes. Use
these projections to help sketch the curve.
(sol)
The projection of the curve onto the -plane is given by ^ 〈 sin 〉 whose graph is the curve
sin , . Similarly, the projection onto the -plane is ^ 〈 cos 〉 whose graph is the
curve cos , and the projection onto the -plane is ^ 〈 sin cos 〉 whose graph is
the ellipse
, .
From this we can get the following picture
3. Find a vector equation and parametric equations for the line segment that joins and
(sol) 〈
〉
≤ ≤ . parametric equations are
- Find a vector function the represents the curve of intersection of the cylinder
and the
surface .
(sol) 〈 cos sin sin 〉 ≤ ≤
- Two particles travel along the space curves
〈
〉 (^) and
(^) 〈 〉 Do
the particles collide? Do their paths intersect?
(sol) (1) For the collision of two particles^ assume
(^) for some
. Then
and
hence
Since
, we have , a contradiction. Two particles do not collide
(2) By solving a system of equation
, one can deduce that two paths of particles intersect
at a point
〈 〉 and
- Find ′, where ∙ , 〈 〉, ′ 〈 〉, and 〈
〉.
(sol) ′ ′ ∙ ∙ ′ ⇒ ′ .
- Show that if is a vector function such that ″ exists, then
× ′ × ″.
(sol)
× ′ ′ × ′ × ″ × ″
- If ≠ , show that
(sol)
- If a curve has the property that the position vector is always perpendicular to the tangent
vector ′, show that the curve lies on a sphere with center the origin where ≠
(sol)
∙ ′ for any ∈R^ because ^ is always perpendicular to the tangent
vector ′ ^ Then^ ≡ where is a positive constant^ This implies that
and the given curve lies on a sphere with the radius and center the origin
- Determine whether the following statements are true or false. If it is true, explain why.
Otherwise, explain why or give an example that disproves the statement.
(a) If ≠ , then
′.
(sol) (False) counterexample ^ 〈cos sin 〉
(b) If for all , then ′^ is constant.
(sol) (False) counterexample 〈 ^
〉
§11.3. Arc Length and Curvature
- Find the length of the curve. cos sin ln cos , ≤ ≤
(sol) ^
sin
cos
cos
sin
^
sec ln
- Reparametrize the curve
sin
cos with respect to arc length measured from
the point where in the direction of increasing .
(sol) The arc length function is computed by
. Then ln
sinln
cosln
- (a) Find the curvature of 〈
cos
sin 〉 at the point .
(sol) At the point
′
′ × ′′
〈 〉×〈 〉
^
(b) Find the curvature of the curve
at the point .
(sol)
^
- At what point does the curve
have maximum curvature? What happens to the curve as
(sol) ^
′
and ′ ^
. When ^
ln
and ^
, the curvature is maximum and (^) lim
→∞
- (a) Use the formula
′
′ × ′′
to show that the curvature of a plane parametric curve
,^ ^ is^ ^
where the dots indicate derivatives with respect to (^) .
(sol)
′
′ × ′′
〈
〉^
〈
〉×〈
〉
(b) Use the formula in (a) to find the curvature of the cycloid (^) sin , (^) cos at the top
of one of its arches.
(sol)
cos
cos
⇒ the tangent is horizontal when , so take and into the expression for :
^
cos
cos
