1
Curves is ℝ2 and ℝ3
We are going to study functions where the domain is a subset of the real
numbers and the range is points in ℝ3 , i.e. a vectors. As we saw earlier these are
called vector valued functions. You’ve already seen this with parametric
equations.
Ex. 𝑥 = cos 𝑡 This is really 𝑟⃑: ℝ → ℝ2 , 𝑡 → (cos 𝑡 , sin 𝑡), so we
𝑦 = sin 𝑡 could think of this as a vector valued function in ℝ2 ,
𝑟⃑(𝑡) = < cos 𝑡 , sin 𝑡 >.
(𝑐𝑜𝑠𝑡, 𝑠𝑖𝑛𝑡)
𝑡
In ℝ3 we will represent a vector valued function by:
𝑟⃑(𝑡) = < 𝑓 (𝑡), 𝑔(𝑡), ℎ(𝑡) > = 𝑓(𝑡)𝑖⃑ + 𝑔(𝑡)𝑗⃑ + ℎ(𝑡)𝑘⃑⃑ .
Ex. Let 𝑟⃑(𝑡) = < 𝑡 2 , √𝑡 2 − 9, ln 𝑡 >, find the domain of 𝑟⃑.
The domain is all values 𝑡 where 𝑟⃑(𝑡) is defined:
The domain of 𝑡 2 is all real numbers,
The domain of √𝑡 2 − 9 is |𝑡| ≥ 3 ,
The domain of ln 𝑡 is 𝑡 > 0.
So the domain of 𝑟⃑ is the intersection of these 3 sets, 𝑡 ≥ 3 (i.e. [3, ∞)).
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Def. If 𝑟⃑(𝑡) =< 𝑓 (𝑡), 𝑔(𝑡), ℎ(𝑡) > , then:
⃑⃑ (𝒕) = < 𝐥𝐢𝐦 𝒇 (𝒕), 𝐥𝐢𝐦 𝒈 (𝒕) , 𝐥𝐢𝐦 𝒉 (𝒕) >
𝐥𝐢𝐦 𝒓
𝒕→𝒂 𝒕→𝒂 𝒕→𝒂 𝒕→𝒂
(provided the limits of the components exist).
Def. 𝑟⃑(𝑡) is continuous at 𝒕 = 𝒂 if lim 𝑟⃑ (𝑡) = 𝑟⃑(𝑎) (i.e. it’s continuous if all of
𝑡→𝑎
the components are continuous.).
We can think of this vector valued function on a subset of ℝ as a curve in ℝ3
defined by:
𝑟⃑(𝑡) = < 𝑓(𝑡), 𝑔(𝑡), ℎ(𝑡) >
𝑥 = 𝑓 (𝑡)
𝑦 = 𝑔(𝑡)
𝑧 = ℎ(𝑡).
Ex. Describe the curve defined by: 𝑟⃑(𝑡) = < 3 − 𝑡, 2𝑡 + 1, −3𝑡 >.
𝑥 =3−𝑡 The parametric form of a line through (3, 1, 0)
𝑦 = 1 + 2𝑡 with direction vector < −1, 2, −3 >.
𝑧 = −3𝑡
Curves is ℝ2 and ℝ3
We are going to study functions where the domain is a subset of the real
numbers and the range is points in ℝ3 , i.e. a vectors. As we saw earlier these are
called vector valued functions. You’ve already seen this with parametric
equations.
Ex. 𝑥 = cos 𝑡 This is really 𝑟⃑: ℝ → ℝ2 , 𝑡 → (cos 𝑡 , sin 𝑡), so we
𝑦 = sin 𝑡 could think of this as a vector valued function in ℝ2 ,
𝑟⃑(𝑡) = < cos 𝑡 , sin 𝑡 >.
(𝑐𝑜𝑠𝑡, 𝑠𝑖𝑛𝑡)
𝑡
In ℝ3 we will represent a vector valued function by:
𝑟⃑(𝑡) = < 𝑓 (𝑡), 𝑔(𝑡), ℎ(𝑡) > = 𝑓(𝑡)𝑖⃑ + 𝑔(𝑡)𝑗⃑ + ℎ(𝑡)𝑘⃑⃑ .
Ex. Let 𝑟⃑(𝑡) = < 𝑡 2 , √𝑡 2 − 9, ln 𝑡 >, find the domain of 𝑟⃑.
The domain is all values 𝑡 where 𝑟⃑(𝑡) is defined:
The domain of 𝑡 2 is all real numbers,
The domain of √𝑡 2 − 9 is |𝑡| ≥ 3 ,
The domain of ln 𝑡 is 𝑡 > 0.
So the domain of 𝑟⃑ is the intersection of these 3 sets, 𝑡 ≥ 3 (i.e. [3, ∞)).
, 2
Def. If 𝑟⃑(𝑡) =< 𝑓 (𝑡), 𝑔(𝑡), ℎ(𝑡) > , then:
⃑⃑ (𝒕) = < 𝐥𝐢𝐦 𝒇 (𝒕), 𝐥𝐢𝐦 𝒈 (𝒕) , 𝐥𝐢𝐦 𝒉 (𝒕) >
𝐥𝐢𝐦 𝒓
𝒕→𝒂 𝒕→𝒂 𝒕→𝒂 𝒕→𝒂
(provided the limits of the components exist).
Def. 𝑟⃑(𝑡) is continuous at 𝒕 = 𝒂 if lim 𝑟⃑ (𝑡) = 𝑟⃑(𝑎) (i.e. it’s continuous if all of
𝑡→𝑎
the components are continuous.).
We can think of this vector valued function on a subset of ℝ as a curve in ℝ3
defined by:
𝑟⃑(𝑡) = < 𝑓(𝑡), 𝑔(𝑡), ℎ(𝑡) >
𝑥 = 𝑓 (𝑡)
𝑦 = 𝑔(𝑡)
𝑧 = ℎ(𝑡).
Ex. Describe the curve defined by: 𝑟⃑(𝑡) = < 3 − 𝑡, 2𝑡 + 1, −3𝑡 >.
𝑥 =3−𝑡 The parametric form of a line through (3, 1, 0)
𝑦 = 1 + 2𝑡 with direction vector < −1, 2, −3 >.
𝑧 = −3𝑡
