Vectors
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Curvature
Basis Vectors
dT = 1 κ= ds |v|
i = h1, 0, 0i j = h0, 1, 0i
y = f (x)
k = h0, 0, 1i u = hu1 , u2 , u3 i = u1 i + u2 j + u3 k
N=
q
u21 + u22 + u23
1 dT dT/dt = κ ds |dT/dt|
radius: ρ =
u · w = u1 w1 + u2 w2 + u3 w3
Projection
u·w w w·w
B=T×N Torsion
Cross Product i u × w = u1 w1
j u2 w2
k u3 w3
τ =−
|u × w| = |u||w| sin θ
dB ·N= ds
|v × a|2
a = aT T + aN N
r(t) = x(t)i + y(t)j + z(t)k
aT =
v(t) = r0 (t) = x0 (t)i + y 0 (t)j + z 0 (t)k a(t) = r00 (t) = x00 (t)i + y 00 (t)j + z 00 (t)k Arc Length
v·a d |v| = dt |v|
aN = κ|v|2 =
q
|a|2 − a2T =
|v × a| |v|
Projectile Motion
s Z b a
dx dt
2
+
dy dt
2
+
dz dt
2
dt
r(t) = ((v0 cos θ)t + x0 ) i 1 + − gt2 + (v0 sin θ)t + y0 j 2
b
|v(t)| dt
a
Gradient Vector Z
t
|v(τ )| dτ,
t0
Unit Tangent Vector T=
x0 (t) y 0 (t) z 0 (t) 00 x (t) y 00 (t) z 00 (t) 000 x (t) y 000 (t) z 000 (t)
Acceleration
Position, Velocity, Acceleration
s(t) =
1 N(t0 ) κ(t0 )
Unit Binormal Vector
projw u =
Z
1 κ(t0 )
center: C = r(t0 ) +
u · w = |u||w| cos θ
L=
|f 00 (x)| [1 + (f 0 (x))2 ]3/2
Osculating Circle
Dot Product
L=
κ=
Principal Unit Normal Vector
Magnitude |u| =
⇒
dT |v × a| dt = |v|3
v dr = ds |v|
ds = |v(t)| dt
∇f =
∂f ∂f ∂f i+ j+ k ∂x ∂y ∂z
Directional Derivative Du f =
1 (∇f · u) |u|
Basis Vectors
dT = 1 κ= ds |v|
i = h1, 0, 0i j = h0, 1, 0i
y = f (x)
k = h0, 0, 1i u = hu1 , u2 , u3 i = u1 i + u2 j + u3 k
N=
q
u21 + u22 + u23
1 dT dT/dt = κ ds |dT/dt|
radius: ρ =
u · w = u1 w1 + u2 w2 + u3 w3
Projection
u·w w w·w
B=T×N Torsion
Cross Product i u × w = u1 w1
j u2 w2
k u3 w3
τ =−
|u × w| = |u||w| sin θ
dB ·N= ds
|v × a|2
a = aT T + aN N
r(t) = x(t)i + y(t)j + z(t)k
aT =
v(t) = r0 (t) = x0 (t)i + y 0 (t)j + z 0 (t)k a(t) = r00 (t) = x00 (t)i + y 00 (t)j + z 00 (t)k Arc Length
v·a d |v| = dt |v|
aN = κ|v|2 =
q
|a|2 − a2T =
|v × a| |v|
Projectile Motion
s Z b a
dx dt
2
+
dy dt
2
+
dz dt
2
dt
r(t) = ((v0 cos θ)t + x0 ) i 1 + − gt2 + (v0 sin θ)t + y0 j 2
b
|v(t)| dt
a
Gradient Vector Z
t
|v(τ )| dτ,
t0
Unit Tangent Vector T=
x0 (t) y 0 (t) z 0 (t) 00 x (t) y 00 (t) z 00 (t) 000 x (t) y 000 (t) z 000 (t)
Acceleration
Position, Velocity, Acceleration
s(t) =
1 N(t0 ) κ(t0 )
Unit Binormal Vector
projw u =
Z
1 κ(t0 )
center: C = r(t0 ) +
u · w = |u||w| cos θ
L=
|f 00 (x)| [1 + (f 0 (x))2 ]3/2
Osculating Circle
Dot Product
L=
κ=
Principal Unit Normal Vector
Magnitude |u| =
⇒
dT |v × a| dt = |v|3
v dr = ds |v|
ds = |v(t)| dt
∇f =
∂f ∂f ∂f i+ j+ k ∂x ∂y ∂z
Directional Derivative Du f =
1 (∇f · u) |u|
