If we have a function
f
→
:
R
→
R
n
{\displaystyle {\vec {f}}:\mathbb {R} \to \mathbb {R} ^{n}}
f
→
{\displaystyle {\vec {f}}}
{
f
→
(
t
)
|
t
∈
R
}
{\displaystyle {\Big \{}{\vec {f}}(t){\Big |}t\in \mathbb {R} {\Big \}}}
R
{\displaystyle \mathbb {R} }
curve in
R
n
{\displaystyle \mathbb {R} ^{n}}
f
→
{\displaystyle {\vec {f}}}
Parameterizations are not necessarily unique - for example,
f
→
(
t
)
=
(
cos
(
t
)
,
sin
(
t
)
)
{\displaystyle {\vec {f}}(t)=(\cos(t),\sin(t))}
t
∈
[
0
,
2
π
)
{\displaystyle t\in [0,2\pi )}
g
→
(
t
)
=
(
cos
(
a
t
)
,
sin
(
a
t
)
)
{\displaystyle {\vec {g}}(t)=(\cos(at),\sin(at))}
t
∈
[
0
,
2
π
a
)
{\displaystyle t\in \left[0,{\frac {2\pi }{a}}\right)}
Collision and intersection points
edit
Say we have two different curves. It may be important to consider
points the two curves share - where they intersect
intersections which occur for the same value of
t
{\displaystyle t}
collide . Firstly, we have two parameterizations
f
→
(
t
)
{\displaystyle {\vec {f}}(t)}
g
→
(
t
)
{\displaystyle {\vec {g}}(t)}
f
→
(
t
)
=
g
→
(
s
)
{\displaystyle {\vec {f}}(t)={\vec {g}}(s)}
because we're seeking the function values independent of the times they intersect.
For example, if we have
f
→
(
t
)
=
(
t
,
3
t
)
{\displaystyle {\vec {f}}(t)=(t,3t)}
g
→
(
t
)
=
(
t
,
t
2
)
{\displaystyle {\vec {g}}(t)=(t,t^{2})}
f
→
(
t
)
=
g
→
(
s
)
{\displaystyle {\vec {f}}(t)={\vec {g}}(s)}
(
t
,
3
t
)
=
(
s
,
s
2
)
{\displaystyle (t,3t)=(s,s^{2})}
t
=
s
,
3
t
=
s
2
{\displaystyle t=s\ ,\ 3t=s^{2}}
with solutions
(
t
,
s
)
=
(
0
,
0
)
,
(
3
,
9
)
{\displaystyle (t,s)=(0,0)\ ,\ (3,9)}
So, the two curves intersect at the points
(
0
,
0
)
,
(
3
,
9
)
{\displaystyle (0,0)\ ,\ (3,9)}
However, if we want to know when the points "collide", with
f
→
(
t
)
{\displaystyle {\vec {f}}(t)}
g
→
(
t
)
{\displaystyle {\vec {g}}(t)}
and the times are the same, so we need to solve instead
f
→
(
t
)
=
g
→
(
t
)
{\displaystyle {\vec {f}}(t)={\vec {g}}(t)}
For example, using the same functions as before,
f
→
(
t
)
=
(
t
,
3
t
)
{\displaystyle {\vec {f}}(t)=(t,3t)}
g
→
(
t
)
=
(
t
,
t
2
)
{\displaystyle {\vec {g}}(t)=(t,t^{2})}
f
→
(
t
)
=
g
→
(
t
)
{\displaystyle {\vec {f}}(t)={\vec {g}}(t)}
(
t
,
3
t
)
=
(
t
,
t
2
)
{\displaystyle (t,3t)=(t,t^{2})}
t
=
t
,
3
t
=
t
2
{\displaystyle t=t\ ,\ 3t=t^{2}}
which gives solutions
t
=
0
,
3
{\displaystyle t=0,3}
(
0
,
0
)
,
(
3
,
9
)
{\displaystyle (0,0)\ ,\ (3,9)}
We may want to do this to actually model physical problems, such as in ballistics.
Intersection vector functions
edit
We can also use vector functions to represent the curve of intersection of two surfaces. For example, we want to know the curve of intersection of the cylinder
x
2
+
y
2
=
1
{\displaystyle x^{2}+y^{2}=1}
y
+
z
=
2
{\displaystyle y+z=2}
Vector functions rely on parameterizations, so we can rewrite the equation of the cylinder into:
cos
2
t
+
sin
2
t
=
1
{\displaystyle \cos ^{2}t+\sin ^{2}t=1}
x
=
cos
t
,
y
=
sin
t
{\displaystyle x=\cos t,y=\sin t}
z
=
2
−
y
=
2
−
sin
t
{\displaystyle z=2-y=2-\sin t}
r
(
t
)
=
⟨
cos
t
,
sin
t
,
2
−
sin
t
⟩
{\displaystyle \mathbf {r} (t)=\langle \cos t,\sin t,2-\sin t\rangle }
Limits and Continuity
edit
The limit of a vector function
r
{\displaystyle \mathbf {r} }
The limit of a vector function
There is a vector function
r
(
t
)
=
⟨
f
(
t
)
,
g
(
t
)
,
h
(
t
)
⟩
{\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }
lim
t
→
a
f
(
t
)
,
lim
t
→
a
g
(
t
)
,
lim
t
→
a
h
(
t
)
{\displaystyle \lim _{t\rightarrow a}f(t),\lim _{t\rightarrow a}g(t),\lim _{t\rightarrow a}h(t)}
lim
t
→
a
r
(
t
)
=
⟨
lim
t
→
a
f
(
t
)
,
lim
t
→
a
g
(
t
)
,
lim
t
→
a
h
(
t
)
⟩
{\displaystyle \lim _{t\rightarrow a}\mathbf {r} (t)=\langle \lim _{t\rightarrow a}f(t),\lim _{t\rightarrow a}g(t),\lim _{t\rightarrow a}h(t)\rangle }
And the requirement for continuity is also simple:
A vector function
r
{\displaystyle \mathbf {r} }
a
{\displaystyle a}
lim
t
→
a
r
(
t
)
=
r
(
a
)
{\displaystyle \lim _{t\rightarrow a}\mathbf {r} (t)=\mathbf {r} (a)}
Recall that the first derivative of a scalar function
f
(
x
)
{\displaystyle f(x)}
f
′
(
x
)
=
lim
Δ
x
→
0
f
(
x
+
Δ
x
)
−
f
(
x
)
Δ
x
{\displaystyle f'(x)=\lim _{\Delta x\rightarrow 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}
The first derivative of a vector function
r
(
t
)
{\displaystyle \mathbf {r} (t)}
r
′
(
t
)
=
lim
Δ
t
→
0
r
(
t
+
Δ
t
)
−
r
(
t
)
Δ
t
{\displaystyle \mathbf {r} '(t)=\lim _{\Delta t\rightarrow 0}{\frac {\mathbf {r} (t+\Delta t)-\mathbf {r} (t)}{\Delta t}}}
We can use this definition to prove that the derivative of a vector function can be presented as the derivative of its component functions.
r
′
(
t
)
=
lim
Δ
t
→
0
1
Δ
t
[
r
(
t
+
Δ
t
)
−
r
(
t
)
]
definition
=
lim
Δ
t
→
0
1
Δ
t
[
⟨
f
(
t
+
Δ
t
)
,
g
(
t
+
Δ
t
)
,
h
(
t
+
Δ
t
)
⟩
−
⟨
f
(
t
)
,
g
(
t
)
,
h
(
t
)
⟩
]
component form
=
lim
Δ
t
→
0
⟨
f
(
t
+
Δ
t
)
−
f
(
t
)
Δ
t
,
g
(
t
+
Δ
t
)
−
g
(
t
)
Δ
t
,
h
(
t
+
Δ
t
)
−
h
(
t
)
Δ
t
⟩
vector addition and multiplication
=
⟨
lim
Δ
t
→
0
f
(
t
+
Δ
t
)
−
f
(
t
)
Δ
t
,
lim
Δ
t
→
0
g
(
t
+
Δ
t
)
−
g
(
t
)
Δ
t
,
lim
Δ
t
→
0
h
(
t
+
Δ
t
)
−
h
(
t
)
Δ
t
⟩
definition
=
⟨
f
′
(
t
)
,
g
′
(
t
)
,
h
′
(
t
)
⟩
definition
{\displaystyle {\begin{aligned}\mathbf {r} '(t)&=\lim _{\Delta t\rightarrow 0}{\frac {1}{\Delta t}}[\mathbf {r} (t+\Delta t)-\mathbf {r} (t)]\quad {\text{definition}}\\&=\lim _{\Delta t\rightarrow 0}{\frac {1}{\Delta t}}[\langle f(t+\Delta t),g(t+\Delta t),h(t+\Delta t)\rangle -\langle f(t),g(t),h(t)\rangle ]\quad {\text{component form}}\\&=\lim _{\Delta t\rightarrow 0}\langle {\frac {f(t+\Delta t)-f(t)}{\Delta t}},{\frac {g(t+\Delta t)-g(t)}{\Delta t}},{\frac {h(t+\Delta t)-h(t)}{\Delta t}}\rangle \quad {\text{vector addition and multiplication}}\\&=\langle \lim _{\Delta t\rightarrow 0}{\frac {f(t+\Delta t)-f(t)}{\Delta t}},\lim _{\Delta t\rightarrow 0}{\frac {g(t+\Delta t)-g(t)}{\Delta t}},\lim _{\Delta t\rightarrow 0}{\frac {h(t+\Delta t)-h(t)}{\Delta t}}\rangle \quad {\text{definition}}\\&=\langle f'(t),g'(t),h'(t)\rangle \quad {\text{definition}}\\\end{aligned}}}
Thus, using the same method, we can derive the second derivative and so on.
Derivatives of a vector function
There is a vector function
r
(
t
)
=
⟨
f
(
t
)
,
g
(
t
)
,
h
(
t
)
⟩
{\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }
r
′
(
t
)
=
lim
Δ
t
→
0
r
(
t
+
Δ
t
)
−
r
(
t
)
Δ
t
=
⟨
f
′
(
t
)
,
g
′
(
t
)
,
h
′
(
t
)
⟩
{\displaystyle \mathbf {r} '(t)=\lim _{\Delta t\rightarrow 0}{\frac {\mathbf {r} (t+\Delta t)-\mathbf {r} (t)}{\Delta t}}=\langle f'(t),g'(t),h'(t)\rangle }
So the
n
{\displaystyle n}
r
(
n
)
(
t
)
=
⟨
f
(
n
)
(
t
)
,
g
(
n
)
(
t
)
,
h
(
n
)
(
t
)
⟩
{\displaystyle \mathbf {r} ^{(n)}(t)=\langle f^{(n)}(t),g^{(n)}(t),h^{(n)}(t)\rangle }
Differentiation rules
edit
Just like real-valued functions, there are some differentiation rules in the world of vector functions. The factor that makes vector differentiation rules slight more complicated is the product rule because there are two kinds of multiplication in vectors: dot product and cross product.
Naturally, we will prove that those rules are correct. Let us assume that
u
(
t
)
=
⟨
f
1
(
t
)
,
f
2
(
t
)
,
f
3
(
t
)
⟩
{\displaystyle \mathbf {u} (t)=\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle }
v
(
t
)
=
⟨
g
1
(
t
)
,
g
2
(
t
)
,
g
3
(
t
)
⟩
{\displaystyle \mathbf {v} (t)=\langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle }
Rule 1: the addition rule
d
d
t
[
u
(
t
)
+
v
(
t
)
]
=
d
d
t
[
⟨
f
1
(
t
)
,
f
2
(
t
)
,
f
3
(
t
)
⟩
+
⟨
g
1
(
t
)
,
g
2
(
t
)
,
g
3
(
t
)
⟩
]
component form
=
d
d
t
⟨
f
1
(
t
)
+
g
1
(
t
)
,
f
2
(
t
)
+
g
2
(
t
)
,
f
3
(
t
)
+
g
3
(
t
)
⟩
vector addition
=
⟨
d
d
t
(
f
1
(
t
)
+
g
1
(
t
)
)
,
d
d
t
(
f
2
(
t
)
+
g
2
(
t
)
)
,
d
d
t
(
f
3
(
t
)
+
g
3
(
t
)
)
⟩
distribution
=
⟨
f
1
′
(
t
)
+
g
1
′
(
t
)
,
f
2
′
(
t
)
+
g
2
′
(
t
)
,
f
3
′
(
t
)
+
g
3
′
(
t
)
⟩
addition rule for real-valued functions
=
⟨
f
1
′
(
t
)
,
f
2
′
(
t
)
,
f
3
′
(
t
)
⟩
+
⟨
g
1
′
(
t
)
,
g
2
′
(
t
)
,
g
3
′
(
t
)
⟩
vector addition
=
u
′
(
t
)
+
v
′
(
t
)
vector form
{\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)+\mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle +\langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}\langle f_{1}(t)+g_{1}(t),f_{2}(t)+g_{2}(t),f_{3}(t)+g_{3}(t)\rangle \quad {\text{vector addition}}\\&=\langle {\frac {d}{dt}}(f_{1}(t)+g_{1}(t)),{\frac {d}{dt}}(f_{2}(t)+g_{2}(t)),{\frac {d}{dt}}(f_{3}(t)+g_{3}(t))\rangle \quad {\text{distribution}}\\&=\langle f_{1}'(t)+g_{1}'(t),f_{2}'(t)+g_{2}'(t),f_{3}'(t)+g_{3}'(t)\rangle \quad {\text{addition rule for real-valued functions}}\\&=\langle f_{1}'(t),f_{2}'(t),f_{3}'(t)\rangle +\langle g_{1}'(t),g_{2}'(t),g_{3}'(t)\rangle \quad {\text{vector addition}}\\&=\mathbf {u} '(t)+\mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}
Rule 2: the scalar multiplication rule
d
d
t
[
f
(
t
)
u
(
t
)
]
=
d
d
t
[
f
(
t
)
(
f
1
(
t
)
f
2
(
t
)
f
3
(
t
)
)
]
component form
=
d
d
t
(
f
(
t
)
f
1
(
t
)
f
(
t
)
f
2
(
t
)
f
(
t
)
f
3
(
t
)
)
scalar multiplication
=
(
d
d
t
(
f
(
t
)
f
1
(
t
)
)
d
d
t
(
f
(
t
)
f
2
(
t
)
)
d
d
t
(
f
(
t
)
f
3
(
t
)
)
)
distribution
=
(
f
′
(
t
)
f
1
(
t
)
+
f
(
t
)
f
1
′
(
t
)
f
′
(
t
)
f
2
(
t
)
+
f
(
t
)
f
2
′
(
t
)
f
′
(
t
)
f
3
(
t
)
+
f
(
t
)
f
3
′
(
t
)
)
multiplication rule for real-valued functions
=
(
f
′
(
t
)
f
1
(
t
)
f
′
(
t
)
f
2
(
t
)
f
′
(
t
)
f
3
(
t
)
)
+
(
f
(
t
)
f
1
′
(
t
)
f
(
t
)
f
2
′
(
t
)
f
(
t
)
f
3
′
(
t
)
)
vector addition
=
f
′
(
t
)
(
f
1
(
t
)
f
2
(
t
)
f
3
(
t
)
)
+
f
(
t
)
(
f
1
′
(
t
)
f
2
′
(
t
)
f
3
′
(
t
)
)
scalar multiplication
=
f
′
(
t
)
u
(
t
)
+
f
(
t
)
u
′
(
t
)
vector form
{\displaystyle {\begin{aligned}{\frac {d}{dt}}[f(t)\mathbf {u} (t)]&={\frac {d}{dt}}[f(t){\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\end{pmatrix}}]\quad {\text{component form}}\\&={\frac {d}{dt}}{\begin{pmatrix}f(t)f_{1}(t)\\f(t)f_{2}(t)\\f(t)f_{3}(t)\end{pmatrix}}\quad {\text{scalar multiplication}}\\&={\begin{pmatrix}{\frac {d}{dt}}(f(t)f_{1}(t))\\{\frac {d}{dt}}(f(t)f_{2}(t))\\{\frac {d}{dt}}(f(t)f_{3}(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f'(t)f_{1}(t)+f(t)f_{1}'(t)\\f'(t)f_{2}(t)+f(t)f_{2}'(t)\\f'(t)f_{3}(t)+f(t)f_{3}'(t)\\\end{pmatrix}}\quad {\text{multiplication rule for real-valued functions}}\\&={\begin{pmatrix}f'(t)f_{1}(t)\\f'(t)f_{2}(t)\\f'(t)f_{3}(t)\end{pmatrix}}+{\begin{pmatrix}f(t)f_{1}'(t)\\f(t)f_{2}'(t)\\f(t)f_{3}'(t)\end{pmatrix}}\quad {\text{vector addition}}\\&=f'(t){\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\end{pmatrix}}+f(t){\begin{pmatrix}f_{1}'(t)\\f_{2}'(t)\\f_{3}'(t)\end{pmatrix}}\quad {\text{scalar multiplication}}\\&=f'(t)\mathbf {u} (t)+f(t)\mathbf {u} '(t)\quad {\text{vector form}}\\\end{aligned}}}
Rule 3: the dot product rule
d
d
t
[
u
(
t
)
⋅
v
(
t
)
]
=
d
d
t
[
⟨
f
1
(
t
)
,
f
2
(
t
)
,
f
3
(
t
)
⟩
⋅
⟨
g
1
(
t
)
,
g
2
(
t
)
,
g
3
(
t
)
⟩
]
component form
=
d
d
t
(
f
1
(
t
)
g
1
(
t
)
+
f
2
(
t
)
g
2
(
t
)
+
f
3
(
t
)
g
3
(
t
)
)
dot product
=
d
d
t
∑
i
=
1
3
f
i
(
t
)
g
i
(
t
)
simplification
=
∑
i
=
1
3
d
d
t
f
i
(
t
)
g
i
(
t
)
distribution
=
∑
i
=
1
3
[
f
i
′
(
t
)
g
i
(
t
)
+
f
i
(
t
)
g
i
′
(
t
)
]
multiplication rule for real-valued functions
=
∑
i
=
1
3
f
i
(
t
)
g
i
(
t
)
+
∑
i
=
1
3
f
i
(
t
)
g
i
′
(
t
)
vector addition
=
u
′
(
t
)
⋅
v
(
t
)
+
u
(
t
)
⋅
v
′
(
t
)
vector form
{\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)\cdot \mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle \cdot \langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}(f_{1}(t)g_{1}(t)+f_{2}(t)g_{2}(t)+f_{3}(t)g_{3}(t))\quad {\text{dot product}}\\&={\frac {d}{dt}}\sum _{i=1}^{3}f_{i}(t)g_{i}(t)\quad {\text{simplification}}\\&=\sum _{i=1}^{3}{\frac {d}{dt}}f_{i}(t)g_{i}(t)\quad {\text{distribution}}\\&=\sum _{i=1}^{3}[f_{i}'(t)g_{i}(t)+f_{i}(t)g_{i}'(t)]\quad {\text{multiplication rule for real-valued functions}}\\&=\sum _{i=1}^{3}f_{i}(t)g_{i}(t)+\sum _{i=1}^{3}f_{i}(t)g_{i}'(t)\quad {\text{vector addition}}\\&=\mathbf {u} '(t)\cdot \mathbf {v} (t)+\mathbf {u} (t)\cdot \mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}
Rule 4: the cross product rule
d
d
t
[
u
(
t
)
×
v
(
t
)
]
=
d
d
t
[
⟨
f
1
(
t
)
,
f
2
(
t
)
,
f
3
(
t
)
⟩
×
⟨
g
1
(
t
)
,
g
2
(
t
)
,
g
3
(
t
)
⟩
]
component form
=
d
d
t
|
i
^
j
^
k
^
f
1
(
t
)
f
2
(
t
)
f
3
(
t
)
g
1
(
t
)
g
2
(
t
)
g
3
(
t
)
|
cross product
=
d
d
t
(
f
2
(
t
)
g
3
(
t
)
−
f
3
(
t
)
g
2
(
t
)
f
3
(
t
)
g
1
(
t
)
−
f
1
(
t
)
g
3
(
t
)
f
1
(
t
)
g
2
(
t
)
−
f
2
(
t
)
g
1
(
t
)
)
=
(
d
d
t
(
f
2
(
t
)
g
3
(
t
)
−
f
3
(
t
)
g
2
(
t
)
)
d
d
t
(
f
3
(
t
)
g
1
(
t
)
−
f
1
(
t
)
g
3
(
t
)
)
d
d
t
(
f
1
(
t
)
g
2
(
t
)
−
f
2
(
t
)
g
1
(
t
)
)
)
distribution
=
(
f
2
′
(
t
)
g
3
(
t
)
+
f
2
(
t
)
g
3
′
(
t
)
−
f
3
′
(
t
)
g
2
(
t
)
−
f
3
(
t
)
g
2
′
(
t
)
f
3
′
(
t
)
g
1
(
t
)
+
f
3
(
t
)
g
1
′
(
t
)
−
f
1
′
(
t
)
g
3
(
t
)
−
f
1
(
t
)
g
3
′
(
t
)
f
2
′
(
t
)
g
3
(
t
)
+
f
2
(
t
)
g
3
′
(
t
)
−
f
3
′
(
t
)
g
2
(
t
)
−
f
3
(
t
)
g
2
′
(
t
)
)
multiplication rule for real-valued functions
=
(
f
2
′
(
t
)
g
3
(
t
)
−
f
3
′
(
t
)
g
2
(
t
)
+
f
2
(
t
)
g
3
′
(
t
)
−
f
3
(
t
)
g
2
′
(
t
)
f
3
′
(
t
)
g
1
(
t
)
−
f
1
′
(
t
)
g
3
(
t
)
+
f
3
(
t
)
g
1
′
(
t
)
−
f
1
(
t
)
g
3
′
(
t
)
f
2
′
(
t
)
g
3
(
t
)
−
f
3
′
(
t
)
g
2
(
t
)
+
f
2
(
t
)
g
3
′
(
t
)
−
f
3
(
t
)
g
2
′
(
t
)
)
rearrangement
=
(
f
2
′
(
t
)
g
3
(
t
)
−
f
3
′
(
t
)
g
2
(
t
)
f
3
′
(
t
)
g
1
(
t
)
−
f
1
′
(
t
)
g
3
(
t
)
f
2
′
(
t
)
g
3
(
t
)
−
f
3
′
(
t
)
g
2
(
t
)
)
+
(
f
2
(
t
)
g
3
′
(
t
)
−
f
3
(
t
)
g
2
′
(
t
)
f
3
(
t
)
g
1
′
(
t
)
−
f
1
(
t
)
g
3
′
(
t
)
f
2
(
t
)
g
3
′
(
t
)
−
f
3
(
t
)
g
2
′
(
t
)
)
vector addtion
=
|
i
^
j
^
k
^
f
1
′
(
t
)
f
2
′
(
t
)
f
3
′
(
t
)
g
1
(
t
)
g
2
(
t
)
g
3
(
t
)
|
+
|
i
^
j
^
k
^
f
1
(
t
)
f
2
(
t
)
f
3
(
t
)
g
1
′
(
t
)
g
2
′
(
t
)
g
3
′
(
t
)
|
cross product
=
(
f
1
′
(
t
)
f
2
′
(
t
)
f
3
′
(
t
)
)
×
(
g
1
(
t
)
g
2
(
t
)
g
3
(
t
)
)
+
(
f
1
(
t
)
f
2
(
t
)
f
3
(
t
)
)
×
(
g
1
′
(
t
)
g
2
′
(
t
)
g
3
′
(
t
)
)
=
u
′
(
t
)
×
v
(
t
)
+
u
(
t
)
×
v
′
(
t
)
vector form
{\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)\times \mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle \times \langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}{\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}(t)&f_{2}(t)&f_{3}(t)\\g_{1}(t)&g_{2}(t)&g_{3}(t)\\\end{vmatrix}}\quad {\text{cross product}}\\&={\frac {d}{dt}}{\begin{pmatrix}f_{2}(t)g_{3}(t)-f_{3}(t)g_{2}(t)\\f_{3}(t)g_{1}(t)-f_{1}(t)g_{3}(t)\\f_{1}(t)g_{2}(t)-f_{2}(t)g_{1}(t)\\\end{pmatrix}}\quad \\&={\begin{pmatrix}{\frac {d}{dt}}(f_{2}(t)g_{3}(t)-f_{3}(t)g_{2}(t))\\{\frac {d}{dt}}(f_{3}(t)g_{1}(t)-f_{1}(t)g_{3}(t))\\{\frac {d}{dt}}(f_{1}(t)g_{2}(t)-f_{2}(t)g_{1}(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)+f_{2}(t)g_{3}'(t)-f_{3}'(t)g_{2}(t)-f_{3}(t)g_{2}'(t)\\f_{3}'(t)g_{1}(t)+f_{3}(t)g_{1}'(t)-f_{1}'(t)g_{3}(t)-f_{1}(t)g_{3}'(t)\\f_{2}'(t)g_{3}(t)+f_{2}(t)g_{3}'(t)-f_{3}'(t)g_{2}(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{multiplication rule for real-valued functions}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)+f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\f_{3}'(t)g_{1}(t)-f_{1}'(t)g_{3}(t)+f_{3}(t)g_{1}'(t)-f_{1}(t)g_{3}'(t)\\f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)+f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{rearrangement}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)\\f_{3}'(t)g_{1}(t)-f_{1}'(t)g_{3}(t)\\f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)\\\end{pmatrix}}+{\begin{pmatrix}f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\f_{3}(t)g_{1}'(t)-f_{1}(t)g_{3}'(t)\\f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{vector addtion}}\\&={\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}'(t)&f_{2}'(t)&f_{3}'(t)\\g_{1}(t)&g_{2}(t)&g_{3}(t)\\\end{vmatrix}}+{\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}(t)&f_{2}(t)&f_{3}(t)\\g_{1}'(t)&g_{2}'(t)&g_{3}'(t)\\\end{vmatrix}}\quad {\text{cross product}}\\&={\begin{pmatrix}f_{1}'(t)\\f_{2}'(t)\\f_{3}'(t)\\\end{pmatrix}}\times {\begin{pmatrix}g_{1}(t)\\g_{2}(t)\\g_{3}(t)\\\end{pmatrix}}+{\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\\\end{pmatrix}}\times {\begin{pmatrix}g_{1}'(t)\\g_{2}'(t)\\g_{3}'(t)\\\end{pmatrix}}\quad \\&=\mathbf {u} '(t)\times \mathbf {v} (t)+\mathbf {u} (t)\times \mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}
Rule 5: the chain rule
d
d
t
[
u
(
f
(
t
)
)
]
=
d
d
t
(
f
1
(
f
(
t
)
)
f
2
(
f
(
t
)
)
f
3
(
f
(
t
)
)
)
component form
=
(
d
d
t
f
1
(
f
(
t
)
)
d
d
t
f
2
(
f
(
t
)
)
d
d
t
f
3
(
f
(
t
)
)
)
distribution
=
(
f
′
(
t
)
f
1
′
(
f
(
t
)
)
f
′
(
t
)
f
2
′
(
f
(
t
)
)
f
′
(
t
)
f
3
′
(
f
(
t
)
)
)
chain rule for real-valued functions
=
f
′
(
t
)
(
f
1
′
(
f
(
t
)
)
f
2
′
(
f
(
t
)
)
f
3
′
(
f
(
t
)
)
)
scalar multiplication
=
f
′
(
t
)
u
′
(
f
(
t
)
)
vector form
{\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (f(t))]&={\frac {d}{dt}}{\begin{pmatrix}f_{1}(f(t))\\f_{2}(f(t))\\f_{3}(f(t))\\\end{pmatrix}}\quad {\text{component form}}\\&={\begin{pmatrix}{\frac {d}{dt}}f_{1}(f(t))\\{\frac {d}{dt}}f_{2}(f(t))\\{\frac {d}{dt}}f_{3}(f(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f'(t)f_{1}'(f(t))\\f'(t)f_{2}'(f(t))\\f'(t)f_{3}'(f(t))\\\end{pmatrix}}\quad {\text{chain rule for real-valued functions}}\\&=f'(t){\begin{pmatrix}f_{1}'(f(t))\\f_{2}'(f(t))\\f_{3}'(f(t))\\\end{pmatrix}}\quad {\text{scalar multiplication}}\\&=f'(t)\mathbf {u} '(f(t))\quad {\text{vector form}}\\\end{aligned}}}
◼
{\displaystyle \blacksquare }
Angle between curves
edit
We can then formulate the concept of the angle between two curves by considering the angle between the two tangent vectors. If two curves, parametrized by
f
1
→
{\displaystyle {\vec {f_{1}}}}
f
2
→
{\displaystyle {\vec {f_{2}}}}
f
1
→
(
s
)
=
f
2
→
(
t
)
=
c
{\displaystyle {\vec {f_{1}}}(s)={\vec {f_{2}}}(t)=c}
the angle between these two curves at
c
{\displaystyle c}
f
1
′
→
(
s
)
{\displaystyle {\vec {f_{1}'}}(s)}
f
2
′
→
(
t
)
{\displaystyle {\vec {f_{2}'}}(t)}
arccos
(
f
1
′
→
(
s
)
⋅
f
2
′
→
(
t
)
‖
f
1
′
→
(
s
)
‖
‖
f
2
′
→
(
t
)
‖
)
{\displaystyle \arccos \left({\frac {{\vec {f_{1}'}}(s)\cdot {\vec {f_{2}'}}(t)}{{\Big \|}{\vec {f_{1}'}}(s){\Big \|}{\Big \|}{\vec {f_{2}'}}(t){\Big \|}}}\right)}
Similar to real-valued functions, the definite integral of a vector function
r
(
t
)
{\displaystyle \mathbf {r} (t)}
∫
a
b
r
(
t
)
d
t
=
lim
n
→
∞
∑
i
=
1
n
r
(
t
i
∗
)
Δ
t
=
lim
n
→
∞
[
(
∑
i
=
1
n
f
(
t
i
∗
)
Δ
t
)
i
+
(
∑
i
=
1
n
g
(
t
i
∗
)
Δ
t
)
j
+
(
∑
i
=
1
n
h
(
t
i
∗
)
Δ
t
)
k
]
=
(
∫
a
b
f
(
t
)
d
t
)
i
+
(
∫
a
b
g
(
t
)
d
t
)
j
+
(
∫
a
b
h
(
t
)
d
t
)
k
{\displaystyle {\begin{aligned}\int _{a}^{b}\mathbf {r} (t)dt&=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}\mathbf {r} (t_{i}^{*})\Delta t\\&=\lim _{n\rightarrow \infty }{\biggl [}{\biggl (}\sum _{i=1}^{n}f(t_{i}^{*})\Delta t{\biggr )}\mathbf {i} +{\biggl (}\sum _{i=1}^{n}g(t_{i}^{*})\Delta t{\biggr )}\mathbf {j} +{\biggl (}\sum _{i=1}^{n}h(t_{i}^{*})\Delta t{\biggr )}\mathbf {k} {\biggr ]}\\&={\biggl (}\int _{a}^{b}f(t)dt{\biggr )}\mathbf {i} +{\biggl (}\int _{a}^{b}g(t)dt{\biggr )}\mathbf {j} +{\biggl (}\int _{a}^{b}h(t)dt{\biggr )}\mathbf {k} \end{aligned}}}
We can extend the Fundamental Theorem of Calculus to continuous vector functions as follows:
The Fundamental Theorem of Calculus (including vector functions)
∫
a
b
r
(
t
)
d
t
=
R
(
t
)
]
a
b
=
R
(
b
)
−
R
(
a
)
{\displaystyle \int _{a}^{b}\mathbf {r} (t)dt=\mathbf {R} (t){\biggr ]}_{a}^{b}=\mathbf {R} (b)-\mathbf {R} (a)}
For indefinite integrals, the definition is:
∫
r
(
t
)
d
t
=
(
∫
f
(
t
)
d
t
)
i
+
(
∫
g
(
t
)
d
t
)
j
+
(
∫
h
(
t
)
d
t
)
k
+
C
{\displaystyle \int \mathbf {r} (t)dt={\biggl (}\int f(t)dt{\biggr )}\mathbf {i} +{\biggl (}\int g(t)dt{\biggr )}\mathbf {j} +{\biggl (}\int h(t)dt{\biggr )}\mathbf {k} +\mathbf {C} }
Recall that in Chapter 5.3 , we deduced the length of a curve with parametric equations
{
x
=
f
(
t
)
y
=
g
(
t
)
{\displaystyle {\begin{cases}x=f(t)\\y=g(t)\end{cases}}}
a
≤
t
≤
b
{\displaystyle a\leq t\leq b}
L
=
∫
a
b
(
d
x
d
t
)
2
+
(
d
y
d
t
)
2
d
t
{\displaystyle L=\int _{a}^{b}{\sqrt {{\biggl (}{\frac {dx}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dy}{dt}}{\biggr )}^{2}}}dt}
Since vector functions are fundamentally parametric equations with directions, we can utilize the formula above into the length of a space curve.
Arc length of a space curve
If the curve has the vector equation
r
(
t
)
=
⟨
f
(
t
)
,
g
(
t
)
,
h
(
t
)
⟩
,
a
≤
t
≤
b
{\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle ,a\leq t\leq b}
x
=
f
(
t
)
,
y
=
g
(
t
)
,
z
=
h
(
t
)
{\displaystyle x=f(t),y=g(t),z=h(t)}
f
′
,
g
′
,
h
′
{\displaystyle f',g',h'}
t
=
a
{\displaystyle t=a}
t
=
b
{\displaystyle t=b}
L
=
∫
a
b
[
f
′
(
t
)
]
2
+
[
g
′
(
t
)
]
2
+
[
h
′
(
t
)
]
2
d
t
=
∫
a
b
(
d
x
d
t
)
2
+
(
d
y
d
t
)
2
+
(
d
x
d
z
)
2
d
t
{\displaystyle L=\int _{a}^{b}{\sqrt {[f'(t)]^{2}+[g'(t)]^{2}+[h'(t)]^{2}}}dt=\int _{a}^{b}{\sqrt {{\biggl (}{\frac {dx}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dy}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dx}{dz}}{\biggr )}^{2}}}dt}
For those who prefer simplicity, the formula can be rewritten into:
L
=
∫
a
b
|
r
′
(
t
)
|
d
t
{\displaystyle L=\int _{a}^{b}|\mathbf {r} '(t)|dt\quad }
d
L
d
t
=
|
r
′
(
t
)
|
{\displaystyle \quad {\frac {dL}{dt}}=|\mathbf {r} '(t)|}
Suppose that there is a vector function which describes the displacement of a particle with respect to time and has the equation:
r
(
t
)
=
cos
t
i
+
sin
t
j
+
t
k
{\displaystyle \mathbf {r} (t)=\cos t\ \mathbf {i} +\sin t\ \mathbf {j} +t\ \mathbf {k} }
However, for some reason, we do not want to know the displacement of this particle with respect to time. Instead, we want to know its displacement with respect to its traveled distance (
s
{\displaystyle s}
(
1
,
0
,
0
)
{\displaystyle (1,0,0)}
t
{\displaystyle t}
t
=
t
(
s
)
{\displaystyle t=t(s)}
According to the definition, the arc length function for our curve
r
(
t
)
{\displaystyle \mathbf {r} (t)}
s
(
t
)
=
∫
0
t
|
r
′
(
u
)
|
d
u
=
∫
0
t
(
−
sin
u
)
2
+
(
cos
u
)
2
+
1
2
d
u
=
2
u
]
0
t
=
2
t
{\displaystyle s(t)=\int _{0}^{t}|\mathbf {r} '(u)|du=\int _{0}^{t}{\sqrt {(-\sin u)^{2}+(\cos u)^{2}+1^{2}}}du={\sqrt {2}}u{\bigg ]}_{0}^{t}={\sqrt {2}}t}
Note that
a
=
0
{\displaystyle a=0}
(
1
,
0
,
0
)
{\displaystyle (1,0,0)}
t
=
0
{\displaystyle t=0}
t
{\displaystyle t}
0
{\displaystyle 0}
t
{\displaystyle t}
t
(
s
)
=
s
2
{\displaystyle t(s)={\frac {s}{\sqrt {2}}}}
Then we substitute the original function and get the answer:
r
(
s
)
=
cos
s
2
i
+
sin
s
2
j
+
s
2
k
{\displaystyle \mathbf {r} (s)=\cos {\frac {s}{\sqrt {2}}}\ \mathbf {i} +\sin {\frac {s}{\sqrt {2}}}\ \mathbf {j} +{\frac {s}{\sqrt {2}}}\ \mathbf {k} }
Reparametrization has important applications in real life because sometimes we want to know a value with respect to different variables. In this case, instead of describing the path of a particle with respect to time, we described its path with respect to its distance, which will be very useful in certain situations.
Before we start discussing curvature, there are some important vectors and concepts we need to be at least aware of.
The unit tangent vector
edit
In the differentiation section of this chapter, we discussed the derivatives of a vector function. We know that
v
(
t
)
=
r
′
(
t
)
{\displaystyle \mathbf {v} (t)=\mathbf {r} '(t)}
t
=
t
0
{\displaystyle t=t_{0}}
r
(
t
)
{\displaystyle \mathbf {r} (t)}
t
=
t
0
{\displaystyle t=t_{0}}
r
′
(
t
)
{\displaystyle \mathbf {r} '(t)}
T
(
t
)
=
r
′
(
t
)
|
r
′
(
t
)
|
{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{|\mathbf {r} '(t)|}}}
As we can see, the magnitude of the unit tangent vector is always
1
{\displaystyle 1}
r
(
t
)
{\displaystyle \mathbf {r} (t)}
The unit normal vector
edit
The unit normal vector is defined as
N
(
t
)
=
T
′
(
t
)
|
T
′
(
t
)
|
{\displaystyle \mathbf {N} (t)={\frac {\mathbf {T} '(t)}{|\mathbf {T} '(t)|}}}
The unit normal is orthogonal to the unit tangent because since
|
T
(
t
)
|
=
1
{\displaystyle |\mathbf {T} (t)|=1}
d
d
t
|
T
(
t
)
|
2
=
0
=
d
d
t
[
T
(
t
)
⋅
T
(
t
)
]
=
2
T
′
(
t
)
⋅
T
(
t
)
{\displaystyle {\frac {d}{dt}}|\mathbf {T} (t)|^{2}=0={\frac {d}{dt}}[\mathbf {T} (t)\cdot \mathbf {T} (t)]=2\mathbf {T} '(t)\cdot \mathbf {T} (t)}
⇔
T
′
(
t
)
⋅
T
(
t
)
=
0
{\displaystyle \Leftrightarrow \mathbf {T} '(t)\cdot \mathbf {T} (t)=0}
This means that
T
′
(
t
)
{\displaystyle \mathbf {T} '(t)}
T
(
t
)
{\displaystyle \mathbf {T} (t)}
N
(
t
)
{\displaystyle \mathbf {N} (t)}
T
(
t
)
{\displaystyle \mathbf {T} (t)}
The binormal vector is defined as
B
(
t
)
=
T
(
t
)
×
N
(
t
)
{\displaystyle \mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)}
The binormal vector is perpendicular to both the unit tangent and the unit normal because of the properties of the cross product. The magnitude of the binormal is always 1 because
|
B
(
t
)
|
=
|
T
(
t
)
|
|
N
(
t
)
|
sin
θ
=
1
{\displaystyle |\mathbf {B} (t)|=|\mathbf {T} (t)||\mathbf {N} (t)|\sin \theta =1}
edit
The normal plane is the plane determined by the normal and binormal vectors
N
and
B
{\displaystyle \mathbf {N} {\text{ and }}\mathbf {B} }
T
{\displaystyle \mathbf {T} }
The osculating plane is the plane determined by the unit tangent and unit normal
T
and
N
{\displaystyle \mathbf {T} {\text{ and }}\mathbf {N} }
t
=
t
0
{\displaystyle t=t_{0}}
The osculating circle is the circle that lies in the osculating plane towards the direction of
N
{\displaystyle \mathbf {N} }
r
=
1
κ
{\displaystyle r={\frac {1}{\kappa }}}
t
=
t
0
{\displaystyle t=t_{0}}
These concepts are very important in the branch of differential geometry and in its applications to the motion of spacecraft.
The curvature of a curve at a given point is a measure of how quickly the curve changes direction at that point. We define it to be the magnitude of the rate of change of the unit tangent with respect to arc length. We use arc length so that the curvature will be independent of parametrization.
Suppose that a space curve has vector function
r
{\displaystyle \mathbf {r} }
T
{\displaystyle \mathbf {T} }
s
{\displaystyle s}
κ
=
|
d
T
(
s
)
d
s
|
{\displaystyle \kappa ={\biggl |}{\frac {d\mathbf {T} (s)}{ds}}{\biggr |}}
There are two other ways to express the curvature. We can express curvature in terms of
t
{\displaystyle t}
s
{\displaystyle s}
d
L
d
t
=
|
r
′
(
t
)
|
{\displaystyle {\frac {dL}{dt}}=|\mathbf {r} '(t)|}
κ
=
|
d
T
d
s
|
=
|
d
T
d
s
d
s
d
t
d
s
d
t
|
=
|
d
T
(
t
)
d
t
d
s
d
t
|
=
|
T
′
(
t
)
|
|
r
′
(
t
)
|
{\displaystyle \kappa ={\biggl |}{\frac {d\mathbf {T} }{ds}}{\biggr |}={\Biggl |}{\frac {{\frac {d\mathbf {T} }{ds}}{\frac {ds}{dt}}}{\frac {ds}{dt}}}{\Biggr |}={\Biggl |}{\frac {\frac {d\mathbf {T} (t)}{dt}}{\frac {ds}{dt}}}{\Biggr |}={\frac {|\mathbf {T} '(t)|}{|\mathbf {r} '(t)|}}}
The third way is more complicated to deduce, but it is often more convenient to apply because it only requires
r
(
t
)
{\displaystyle \mathbf {r} (t)}
κ
(
t
)
=
|
r
′
(
t
)
×
r
″
(
t
)
|
|
r
′
(
t
)
|
3
{\displaystyle \kappa (t)={\frac {|\mathbf {r} '(t)\times \mathbf {r} ''(t)|}{|\mathbf {r} '(t)|^{3}}}}
And now for the proof for this theorem:
According to the definition of the unit tangent vector, we know that
r
′
=
|
r
′
|
T
{\displaystyle \mathbf {r} '=|\mathbf {r} '|\mathbf {T} }
r
{\displaystyle \mathbf {r} }
r
″
=
(
|
r
′
|
T
)
′
=
|
r
″
|
T
+
|
r
′
|
T
′
the product rule
{\displaystyle {\begin{aligned}\mathbf {r} ''&=(|\mathbf {r} '|\ \mathbf {T} )'\\&=|\mathbf {r} ''|\ \mathbf {T} +|\mathbf {r} '|\ \mathbf {T} '\quad {\text{the product rule}}\\\end{aligned}}}
Now we calculate
r
′
×
r
″
{\displaystyle \mathbf {r} '\times \mathbf {r} ''}
r
′
×
r
″
=
(
|
r
′
|
T
)
×
(
|
r
″
|
T
+
|
r
′
|
T
′
)
substitution
=
(
|
r
′
|
T
×
|
r
″
|
T
)
+
(
|
r
′
|
T
×
|
r
′
|
T
′
)
distribution
=
|
r
′
|
|
r
″
|
(
T
×
T
)
+
|
r
′
|
2
(
T
×
T
′
)
rearrangement
=
|
r
′
|
2
(
T
×
T
′
)
realizing that
T
×
T
=
0
{\displaystyle {\begin{aligned}\mathbf {r} '\times \mathbf {r} ''&={\bigl (}|\mathbf {r} '|\mathbf {T} {\bigr )}\times {\bigl (}|\mathbf {r} ''|\ \mathbf {T} +|\mathbf {r} '|\ \mathbf {T} '{\bigr )}\quad {\text{substitution}}\\&={\bigl (}|\mathbf {r} '|\mathbf {T} \times |\mathbf {r} ''|\mathbf {T} {\bigr )}+{\bigl (}|\mathbf {r} '|\mathbf {T} \times |\mathbf {r} '|\mathbf {T} '{\bigr )}\quad {\text{distribution}}\\&=|\mathbf {r} '||\mathbf {r} ''|(\mathbf {T} \times \mathbf {T} )+|\mathbf {r} '|^{2}(\mathbf {T} \times \mathbf {T} ')\quad {\text{rearrangement}}\\&=|\mathbf {r} '|^{2}(\mathbf {T} \times \mathbf {T} ')\quad {\text{realizing that }}\mathbf {T} \times \mathbf {T} =0\\\end{aligned}}}
And then we calculate
|
r
′
×
r
″
|
{\displaystyle |\mathbf {r} '\times \mathbf {r} ''|}
|
r
′
×
r
″
|
=
|
r
′
|
2
|
T
×
T
′
|
substitution
=
|
r
′
|
2
|
T
|
|
T
′
|
sin
θ
the magnitude for the cross product
=
|
r
′
|
2
|
T
′
|
when you realize
|
T
|
=
1
and
T
⊥
T
′
{\displaystyle {\begin{aligned}|\mathbf {r} '\times \mathbf {r} ''|&=|\mathbf {r} '|^{2}|\mathbf {T} \times \mathbf {T} '|\quad {\text{substitution}}\\&=|\mathbf {r} '|^{2}|\mathbf {T} ||\mathbf {T} '|\sin \theta \quad {\text{the magnitude for the cross product}}\\&=|\mathbf {r} '|^{2}|\mathbf {T} '|\quad {\text{when you realize }}|\mathbf {T} |=1{\text{ and }}\mathbf {T} \perp \mathbf {T} '\\\end{aligned}}}
We rearrange the equation into:
|
T
′
|
=
|
r
′
×
r
″
|
|
r
′
|
2
{\displaystyle |\mathbf {T} '|={\frac {|\mathbf {r} '\times \mathbf {r} ''|}{|\mathbf {r} '|^{2}}}}
Since
κ
=
|
T
′
|
|
r
′
|
{\displaystyle \kappa ={\frac {|\mathbf {T} '|}{|\mathbf {r} '|}}}
|
T
′
|
{\displaystyle |\mathbf {T} '|}
|
r
′
×
r
″
|
|
r
′
|
2
{\displaystyle {\frac {|\mathbf {r} '\times \mathbf {r} ''|}{|\mathbf {r} '|^{2}}}}
κ
=
|
r
′
×
r
″
|
|
r
′
|
3
{\displaystyle \kappa ={\frac {|\mathbf {r} '\times \mathbf {r} ''|}{|\mathbf {r} '|^{3}}}}
Here is a little summary on ways to calculate the curvature.
Definition
With parametrization with respect to
t
{\displaystyle \mathbf {t} }
In terms of
r
(
t
)
{\displaystyle \mathbf {r} (t)}
κ
=
|
d
T
d
s
|
{\displaystyle \kappa ={\biggl |}{\frac {d\mathbf {T} }{ds}}{\biggr |}}
κ
=
|
T
′
|
|
r
′
|
{\displaystyle \kappa ={\frac {|\mathbf {T} '|}{|\mathbf {r} '|}}}
κ
=
|
r
′
×
r
″
|
|
r
′
|
3
{\displaystyle \kappa ={\frac {|\mathbf {r} '\times \mathbf {r} ''|}{|\mathbf {r} '|^{3}}}}
Velocity and acceleration
edit
Remember in 2-dimensional calculus, we mentioned that a particle with displacement function
f
(
x
)
{\displaystyle f(x)}
f
′
(
x
)
{\displaystyle f'(x)}
f
″
(
x
)
{\displaystyle f''(x)}
t
{\displaystyle t}
r
(
t
)
{\displaystyle \mathbf {r} (t)}
v
(
t
)
=
lim
h
→
0
r
(
t
+
h
)
−
r
(
t
)
h
=
r
′
(
t
)
{\displaystyle \mathbf {v} (t)=\lim _{h\rightarrow 0}{\frac {\mathbf {r} (t+h)-\mathbf {r} (t)}{h}}=\mathbf {r} '(t)\quad }
a
(
t
)
=
lim
h
→
0
v
(
t
+
h
)
−
v
(
t
)
h
=
v
′
(
t
)
=
r
″
(
t
)
{\displaystyle \quad \mathbf {a} (t)=\lim _{h\rightarrow 0}{\frac {\mathbf {v} (t+h)-\mathbf {v} (t)}{h}}=\mathbf {v} '(t)=\mathbf {r} ''(t)}
Simply put:
r
″
(
t
)
=
v
′
(
t
)
=
a
(
t
)
{\displaystyle \mathbf {r} ''(t)=\mathbf {v} '(t)=\mathbf {a} (t)}
|
v
(
t
)
|
{\displaystyle |\mathbf {v} (t)|}
t
=
a
to
b
{\displaystyle t=a{\text{ to }}b}
∫
a
b
|
v
(
t
)
|
d
t
{\displaystyle \int _{a}^{b}|\mathbf {v} (t)|dt}
With the help of the Fundamental Theorem of Calculus, we can deduce the velocity function and position function when we know the particle's acceleration.
v
(
t
)
=
v
(
t
0
)
+
∫
t
0
t
a
(
u
)
d
u
{\displaystyle \mathbf {v} (t)=\mathbf {v} (t_{0})+\int _{t_{0}}^{t}\mathbf {a} (u)du\quad }
r
(
t
)
=
r
(
t
0
)
+
∫
t
0
t
v
(
u
)
d
u
{\displaystyle \quad \mathbf {r} (t)=\mathbf {r} (t_{0})+\int _{t_{0}}^{t}\mathbf {v} (u)du}
Tangential and normal acceleration
edit
We can split the acceleration vector into two components: the tangential acceleration
a
T
{\displaystyle a_{T}}
a
N
{\displaystyle a_{N}}
T
{\displaystyle \mathbf {T} }
N
{\displaystyle \mathbf {N} }
T
{\displaystyle \mathbf {T} }
N
{\displaystyle \mathbf {N} }
a
=
a
T
T
+
a
N
N
{\displaystyle \mathbf {a} =a_{T}\mathbf {T} +a_{N}\mathbf {N} }
Our goal is to figure out how to describe the two components.
Recall that
T
=
v
|
v
|
{\displaystyle \mathbf {T} ={\frac {\mathbf {v} }{|\mathbf {v} |}}}
v
=
|
v
|
T
{\displaystyle \mathbf {v} =|\mathbf {v} |\mathbf {T} }
Now we differentiate both sides of the equation,
v
′
=
(
|
v
|
T
)
′
=
|
v
|
′
T
+
|
v
|
T
′
{\displaystyle {\begin{aligned}\mathbf {v} '&={\bigl (}|\mathbf {v} |\mathbf {T} {\bigr )}'\\&=|\mathbf {v} |'\mathbf {T} +|\mathbf {v} |\mathbf {T} '\\\end{aligned}}}
So we get
a
=
|
v
|
′
T
+
|
v
|
T
′
{\displaystyle \mathbf {a} =|\mathbf {v} |'\mathbf {T} +|\mathbf {v} |\mathbf {T} '}
Recall that
κ
=
|
T
′
|
|
r
′
|
=
|
T
′
|
|
v
|
{\displaystyle \kappa ={\frac {{\big |}\mathbf {T} '{\big |}}{{\big |}\mathbf {r} '{\big |}}}={\frac {{\big |}\mathbf {T} '{\big |}}{{\big |}\mathbf {v} {\big |}}}}
|
T
′
|
=
κ
|
v
|
{\displaystyle {\big |}\mathbf {T} '{\big |}=\kappa |\mathbf {v} |}
Recall that
N
=
T
′
|
T
′
|
{\displaystyle \mathbf {N} ={\frac {\mathbf {T} '}{{\big |}\mathbf {T} '{\big |}}}}
T
′
=
|
T
′
|
N
=
κ
|
v
|
N
{\displaystyle \mathbf {T} '={\big |}\mathbf {T} '{\big |}\mathbf {N} =\kappa |\mathbf {v} |\mathbf {N} }
We substitute
T
′
{\displaystyle \mathbf {T} '}
κ
|
v
|
N
{\displaystyle \kappa |\mathbf {v} |\mathbf {N} }
a
=
|
v
|
′
T
+
κ
|
v
|
2
N
{\displaystyle \mathbf {a} =|\mathbf {v} |'\mathbf {T} +\kappa |\mathbf {v} |^{2}\mathbf {N} }
That leaves us with:
a
T
=
|
v
|
′
{\displaystyle a_{T}=|\mathbf {v} |'\quad }
a
N
=
κ
|
v
|
2
{\displaystyle \quad a_{N}=\kappa |\mathbf {v} |^{2}}
◼
{\displaystyle \blacksquare }
Of course, it will be more convenient if those components can be written in terms if
r
(
t
)
{\displaystyle \mathbf {r} (t)}
θ
{\displaystyle \theta }
T
{\displaystyle \mathbf {T} }
a
{\displaystyle \mathbf {a} }
a
T
,
a
N
{\displaystyle a_{T},a_{N}}
a
T
=
|
a
(
t
)
|
cos
θ
=
|
v
(
t
)
|
|
a
(
t
)
|
cos
θ
|
v
(
t
)
|
algebraic manipulation
=
v
(
t
)
⋅
a
(
t
)
|
v
(
t
)
|
dot product
=
r
′
(
t
)
⋅
r
″
(
t
)
|
r
′
(
t
)
|
in terms of
r
(
t
)
and its derivatives
{\displaystyle {\begin{aligned}a_{T}&=|\mathbf {a} (t)|\cos \theta \\&={\frac {|\mathbf {v} (t)||\mathbf {a} (t)|\cos \theta }{|\mathbf {v} (t)|}}\quad {\text{algebraic manipulation}}\\&={\frac {\mathbf {v} (t)\cdot \mathbf {a} (t)}{|\mathbf {v} (t)|}}\quad {\text{dot product}}\\&={\frac {\mathbf {r} '(t)\cdot \mathbf {r} ''(t)}{|\mathbf {r} '(t)|}}\quad {\text{in terms of }}\mathbf {r} (t){\text{ and its derivatives}}\\\end{aligned}}}
a
N
=
|
a
(
t
)
|
sin
θ
=
|
v
(
t
)
|
|
a
(
t
)
|
sin
θ
|
v
(
t
)
|
algebraic manipulation
=
|
v
(
t
)
×
a
(
t
)
|
|
v
(
t
)
|
cross product
=
|
r
′
(
t
)
×
r
″
(
t
)
|
|
r
′
(
t
)
|
in terms of
r
(
t
)
and its derivatives
{\displaystyle {\begin{aligned}a_{N}&=|\mathbf {a} (t)|\sin \theta \\&={\frac {|\mathbf {v} (t)||\mathbf {a} (t)|\sin \theta }{|\mathbf {v} (t)|}}\quad {\text{algebraic manipulation}}\\&={\frac {|\mathbf {v} (t)\times \mathbf {a} (t)|}{|\mathbf {v} (t)|}}\quad {\text{cross product}}\\&={\frac {|\mathbf {r} '(t)\times \mathbf {r} ''(t)|}{|\mathbf {r} '(t)|}}\quad {\text{in terms of }}\mathbf {r} (t){\text{ and its derivatives}}\\\end{aligned}}}
◼
{\displaystyle \blacksquare }
To sum it up:
a
T
=
|
v
|
′
=
r
′
(
t
)
⋅
r
″
(
t
)
|
r
′
(
t
)
|
{\displaystyle a_{T}=|\mathbf {v} |'={\frac {\mathbf {r} '(t)\cdot \mathbf {r} ''(t)}{|\mathbf {r} '(t)|}}\quad }
a
N
=
κ
|
v
|
2
=
|
r
′
(
t
)
×
r
″
(
t
)
|
|
r
′
(
t
)
|
{\displaystyle \quad a_{N}=\kappa |\mathbf {v} |^{2}={\frac {|\mathbf {r} '(t)\times \mathbf {r} ''(t)|}{|\mathbf {r} '(t)|}}}
![{\displaystyle {\begin{aligned}\mathbf {r} '(t)&=\lim _{\Delta t\rightarrow 0}{\frac {1}{\Delta t}}[\mathbf {r} (t+\Delta t)-\mathbf {r} (t)]\quad {\text{definition}}\\&=\lim _{\Delta t\rightarrow 0}{\frac {1}{\Delta t}}[\langle f(t+\Delta t),g(t+\Delta t),h(t+\Delta t)\rangle -\langle f(t),g(t),h(t)\rangle ]\quad {\text{component form}}\\&=\lim _{\Delta t\rightarrow 0}\langle {\frac {f(t+\Delta t)-f(t)}{\Delta t}},{\frac {g(t+\Delta t)-g(t)}{\Delta t}},{\frac {h(t+\Delta t)-h(t)}{\Delta t}}\rangle \quad {\text{vector addition and multiplication}}\\&=\langle \lim _{\Delta t\rightarrow 0}{\frac {f(t+\Delta t)-f(t)}{\Delta t}},\lim _{\Delta t\rightarrow 0}{\frac {g(t+\Delta t)-g(t)}{\Delta t}},\lim _{\Delta t\rightarrow 0}{\frac {h(t+\Delta t)-h(t)}{\Delta t}}\rangle \quad {\text{definition}}\\&=\langle f'(t),g'(t),h'(t)\rangle \quad {\text{definition}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10a21eb6248156159fc752280510f5f3f010aa32)
![{\displaystyle {\frac {d}{dt}}[\mathbf {u} (t)+\mathbf {v} (t)]=\mathbf {u} '(t)+\mathbf {v} '(t)\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0641a88b16fb528f995ebbffb0980f7a1a2530)
![{\displaystyle {\frac {d}{dt}}[f(t)\mathbf {u} (t)]=f'(t)\mathbf {u} (t)+f(t)\mathbf {u} '(t)\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/66ee060fe66607674c1f961c23c1faff5bca5ef7)
![{\displaystyle {\frac {d}{dt}}[\mathbf {u} (t)\cdot \mathbf {v} (t)]=\mathbf {u} '(t)\cdot \mathbf {v} (t)+\mathbf {u} (t)\cdot \mathbf {v} '(t)\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/03f6b70044d45671d7572a1745bf280d1dd7512e)
![{\displaystyle {\frac {d}{dt}}[\mathbf {u} (t)\times \mathbf {v} (t)]=\mathbf {u} '(t)\times \mathbf {v} (t)+\mathbf {u} (t)\times \mathbf {v} '(t)\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e5cd0b27dd8912b1b19550646a1e79000026f6)
![{\displaystyle {\frac {d}{dt}}[\mathbf {u} (f(t))]=f'(t)\mathbf {u} '(f(t))\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cdd72d02b7364bf6a1ca9b4dda7f685486fd916)
![{\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)+\mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle +\langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}\langle f_{1}(t)+g_{1}(t),f_{2}(t)+g_{2}(t),f_{3}(t)+g_{3}(t)\rangle \quad {\text{vector addition}}\\&=\langle {\frac {d}{dt}}(f_{1}(t)+g_{1}(t)),{\frac {d}{dt}}(f_{2}(t)+g_{2}(t)),{\frac {d}{dt}}(f_{3}(t)+g_{3}(t))\rangle \quad {\text{distribution}}\\&=\langle f_{1}'(t)+g_{1}'(t),f_{2}'(t)+g_{2}'(t),f_{3}'(t)+g_{3}'(t)\rangle \quad {\text{addition rule for real-valued functions}}\\&=\langle f_{1}'(t),f_{2}'(t),f_{3}'(t)\rangle +\langle g_{1}'(t),g_{2}'(t),g_{3}'(t)\rangle \quad {\text{vector addition}}\\&=\mathbf {u} '(t)+\mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5406b79f26115ef7e349244ce57fb8bceb737d39)
![{\displaystyle {\begin{aligned}{\frac {d}{dt}}[f(t)\mathbf {u} (t)]&={\frac {d}{dt}}[f(t){\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\end{pmatrix}}]\quad {\text{component form}}\\&={\frac {d}{dt}}{\begin{pmatrix}f(t)f_{1}(t)\\f(t)f_{2}(t)\\f(t)f_{3}(t)\end{pmatrix}}\quad {\text{scalar multiplication}}\\&={\begin{pmatrix}{\frac {d}{dt}}(f(t)f_{1}(t))\\{\frac {d}{dt}}(f(t)f_{2}(t))\\{\frac {d}{dt}}(f(t)f_{3}(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f'(t)f_{1}(t)+f(t)f_{1}'(t)\\f'(t)f_{2}(t)+f(t)f_{2}'(t)\\f'(t)f_{3}(t)+f(t)f_{3}'(t)\\\end{pmatrix}}\quad {\text{multiplication rule for real-valued functions}}\\&={\begin{pmatrix}f'(t)f_{1}(t)\\f'(t)f_{2}(t)\\f'(t)f_{3}(t)\end{pmatrix}}+{\begin{pmatrix}f(t)f_{1}'(t)\\f(t)f_{2}'(t)\\f(t)f_{3}'(t)\end{pmatrix}}\quad {\text{vector addition}}\\&=f'(t){\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\end{pmatrix}}+f(t){\begin{pmatrix}f_{1}'(t)\\f_{2}'(t)\\f_{3}'(t)\end{pmatrix}}\quad {\text{scalar multiplication}}\\&=f'(t)\mathbf {u} (t)+f(t)\mathbf {u} '(t)\quad {\text{vector form}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7533c89950e1572f4d2b64dc3e7fc83e7c2844)
![{\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)\cdot \mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle \cdot \langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}(f_{1}(t)g_{1}(t)+f_{2}(t)g_{2}(t)+f_{3}(t)g_{3}(t))\quad {\text{dot product}}\\&={\frac {d}{dt}}\sum _{i=1}^{3}f_{i}(t)g_{i}(t)\quad {\text{simplification}}\\&=\sum _{i=1}^{3}{\frac {d}{dt}}f_{i}(t)g_{i}(t)\quad {\text{distribution}}\\&=\sum _{i=1}^{3}[f_{i}'(t)g_{i}(t)+f_{i}(t)g_{i}'(t)]\quad {\text{multiplication rule for real-valued functions}}\\&=\sum _{i=1}^{3}f_{i}(t)g_{i}(t)+\sum _{i=1}^{3}f_{i}(t)g_{i}'(t)\quad {\text{vector addition}}\\&=\mathbf {u} '(t)\cdot \mathbf {v} (t)+\mathbf {u} (t)\cdot \mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48d9654f1e96837de47bf00a4d2144cc8df87ea6)
![{\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (t)\times \mathbf {v} (t)]&={\frac {d}{dt}}[\langle f_{1}(t),f_{2}(t),f_{3}(t)\rangle \times \langle g_{1}(t),g_{2}(t),g_{3}(t)\rangle ]\quad {\text{component form}}\\&={\frac {d}{dt}}{\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}(t)&f_{2}(t)&f_{3}(t)\\g_{1}(t)&g_{2}(t)&g_{3}(t)\\\end{vmatrix}}\quad {\text{cross product}}\\&={\frac {d}{dt}}{\begin{pmatrix}f_{2}(t)g_{3}(t)-f_{3}(t)g_{2}(t)\\f_{3}(t)g_{1}(t)-f_{1}(t)g_{3}(t)\\f_{1}(t)g_{2}(t)-f_{2}(t)g_{1}(t)\\\end{pmatrix}}\quad \\&={\begin{pmatrix}{\frac {d}{dt}}(f_{2}(t)g_{3}(t)-f_{3}(t)g_{2}(t))\\{\frac {d}{dt}}(f_{3}(t)g_{1}(t)-f_{1}(t)g_{3}(t))\\{\frac {d}{dt}}(f_{1}(t)g_{2}(t)-f_{2}(t)g_{1}(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)+f_{2}(t)g_{3}'(t)-f_{3}'(t)g_{2}(t)-f_{3}(t)g_{2}'(t)\\f_{3}'(t)g_{1}(t)+f_{3}(t)g_{1}'(t)-f_{1}'(t)g_{3}(t)-f_{1}(t)g_{3}'(t)\\f_{2}'(t)g_{3}(t)+f_{2}(t)g_{3}'(t)-f_{3}'(t)g_{2}(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{multiplication rule for real-valued functions}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)+f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\f_{3}'(t)g_{1}(t)-f_{1}'(t)g_{3}(t)+f_{3}(t)g_{1}'(t)-f_{1}(t)g_{3}'(t)\\f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)+f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{rearrangement}}\\&={\begin{pmatrix}f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)\\f_{3}'(t)g_{1}(t)-f_{1}'(t)g_{3}(t)\\f_{2}'(t)g_{3}(t)-f_{3}'(t)g_{2}(t)\\\end{pmatrix}}+{\begin{pmatrix}f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\f_{3}(t)g_{1}'(t)-f_{1}(t)g_{3}'(t)\\f_{2}(t)g_{3}'(t)-f_{3}(t)g_{2}'(t)\\\end{pmatrix}}\quad {\text{vector addtion}}\\&={\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}'(t)&f_{2}'(t)&f_{3}'(t)\\g_{1}(t)&g_{2}(t)&g_{3}(t)\\\end{vmatrix}}+{\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\f_{1}(t)&f_{2}(t)&f_{3}(t)\\g_{1}'(t)&g_{2}'(t)&g_{3}'(t)\\\end{vmatrix}}\quad {\text{cross product}}\\&={\begin{pmatrix}f_{1}'(t)\\f_{2}'(t)\\f_{3}'(t)\\\end{pmatrix}}\times {\begin{pmatrix}g_{1}(t)\\g_{2}(t)\\g_{3}(t)\\\end{pmatrix}}+{\begin{pmatrix}f_{1}(t)\\f_{2}(t)\\f_{3}(t)\\\end{pmatrix}}\times {\begin{pmatrix}g_{1}'(t)\\g_{2}'(t)\\g_{3}'(t)\\\end{pmatrix}}\quad \\&=\mathbf {u} '(t)\times \mathbf {v} (t)+\mathbf {u} (t)\times \mathbf {v} '(t)\quad {\text{vector form}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee1dd679375b3ede5be3a4de1ab1649ea440ef9)
![{\displaystyle {\begin{aligned}{\frac {d}{dt}}[\mathbf {u} (f(t))]&={\frac {d}{dt}}{\begin{pmatrix}f_{1}(f(t))\\f_{2}(f(t))\\f_{3}(f(t))\\\end{pmatrix}}\quad {\text{component form}}\\&={\begin{pmatrix}{\frac {d}{dt}}f_{1}(f(t))\\{\frac {d}{dt}}f_{2}(f(t))\\{\frac {d}{dt}}f_{3}(f(t))\\\end{pmatrix}}\quad {\text{distribution}}\\&={\begin{pmatrix}f'(t)f_{1}'(f(t))\\f'(t)f_{2}'(f(t))\\f'(t)f_{3}'(f(t))\\\end{pmatrix}}\quad {\text{chain rule for real-valued functions}}\\&=f'(t){\begin{pmatrix}f_{1}'(f(t))\\f_{2}'(f(t))\\f_{3}'(f(t))\\\end{pmatrix}}\quad {\text{scalar multiplication}}\\&=f'(t)\mathbf {u} '(f(t))\quad {\text{vector form}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c28de0e944ffb9f8c98325b11a051f2a6175d6)
![{\displaystyle {\begin{aligned}\int _{a}^{b}\mathbf {r} (t)dt&=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}\mathbf {r} (t_{i}^{*})\Delta t\\&=\lim _{n\rightarrow \infty }{\biggl [}{\biggl (}\sum _{i=1}^{n}f(t_{i}^{*})\Delta t{\biggr )}\mathbf {i} +{\biggl (}\sum _{i=1}^{n}g(t_{i}^{*})\Delta t{\biggr )}\mathbf {j} +{\biggl (}\sum _{i=1}^{n}h(t_{i}^{*})\Delta t{\biggr )}\mathbf {k} {\biggr ]}\\&={\biggl (}\int _{a}^{b}f(t)dt{\biggr )}\mathbf {i} +{\biggl (}\int _{a}^{b}g(t)dt{\biggr )}\mathbf {j} +{\biggl (}\int _{a}^{b}h(t)dt{\biggr )}\mathbf {k} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8315be6f06a58e3eb7d96ef297cc22ea27d843d8)
![{\displaystyle \int _{a}^{b}\mathbf {r} (t)dt=\mathbf {R} (t){\biggr ]}_{a}^{b}=\mathbf {R} (b)-\mathbf {R} (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de55e1dbf44acffb7113f43cca625090a008c36f)
![{\displaystyle L=\int _{a}^{b}{\sqrt {[f'(t)]^{2}+[g'(t)]^{2}+[h'(t)]^{2}}}dt=\int _{a}^{b}{\sqrt {{\biggl (}{\frac {dx}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dy}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dx}{dz}}{\biggr )}^{2}}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80fd4bf2e740ece118f4071bddb2b13995fc56d5)
![{\displaystyle s(t)=\int _{0}^{t}|\mathbf {r} '(u)|du=\int _{0}^{t}{\sqrt {(-\sin u)^{2}+(\cos u)^{2}+1^{2}}}du={\sqrt {2}}u{\bigg ]}_{0}^{t}={\sqrt {2}}t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1b01fb9b1e81100164710782d6a4a515276949)
![{\displaystyle {\frac {d}{dt}}|\mathbf {T} (t)|^{2}=0={\frac {d}{dt}}[\mathbf {T} (t)\cdot \mathbf {T} (t)]=2\mathbf {T} '(t)\cdot \mathbf {T} (t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33472ffaf40506669b8dbe56a7e80bcbfe38aef9)
![{\displaystyle \quad L=\int _{a}^{b}{\sqrt {[f_{1}'(t)]^{2}+[f_{2}'(t)]^{2}+\cdots +[f_{n}'(t)]^{2}}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9527053c0d61a77a2c68cf4df4ec12922a4691)