Name
Function
Vectors
Magnitude
‖
a
→
‖
=
a
x
2
+
a
y
2
{\displaystyle \left\Vert {\vec {a}}\right\|={\sqrt {a_{x}^{2}+a_{y}^{2}}}}
Dot Product
⟨
a
,
b
⟩
⋅
⟨
c
,
d
⟩
=
a
c
+
b
d
{\displaystyle \left\langle a,b\right\rangle \,\cdot \,\left\langle c,d\right\rangle =a\,c+b\,d}
Angle between 2 vectors
cos
(
θ
)
=
a
→
⋅
b
→
‖
a
→
‖
‖
b
→
‖
{\displaystyle \cos {(\theta )}={\frac {{\vec {a}}\cdot {\vec {b}}}{\left\Vert {\vec {a}}\right\|\,\left\Vert {\vec {b}}\right\|}}}
Cross Product
‖
a
→
×
b
→
‖
=
‖
a
→
‖
‖
b
→
‖
sin
(
θ
)
{\displaystyle \left\Vert {\vec {a}}\times {\vec {b}}\right\|=\left\Vert {\vec {a}}\right\|\,\left\Vert {\vec {b}}\right\|\,\sin {(\theta )}}
Vector Functions
Velocity
v
→
=
d
d
t
r
→
{\displaystyle {\vec {v}}={\frac {d}{dt}}\,{\vec {r}}}
Tangent Vector
T
→
=
v
→
‖
v
→
‖
{\displaystyle {\vec {T}}={\frac {\vec {v}}{\left\Vert {\vec {v}}\right\|}}}
Normal Vector
N
→
=
d
T
→
d
t
‖
d
T
→
d
t
‖
{\displaystyle {\vec {N}}={\frac {\frac {d{\vec {T}}}{dt}}{\left\Vert {\frac {d{\vec {T}}}{dt}}\right\|}}}
Acceleration
a
→
=
d
d
t
v
→
{\displaystyle {\vec {a}}={\frac {d}{dt}}\,{\vec {v}}}
Partial Derivatives
A
B
Multiple Integrals
Average Value
∬
R
f
(
x
)
d
A
∬
R
d
A
{\displaystyle {\frac {\iint \limits _{R}\,f(x)\,dA}{\iint \limits _{R}\,dA}}}
Area
∬
R
d
A
{\displaystyle \iint \limits _{R}\,dA}
Volume
∭
R
d
V
{\displaystyle \iiint \limits _{R}\,dV}
Mass
∭
R
σ
d
V
{\displaystyle \iiint \limits _{R}\,\sigma \,dV}
First Moment
∭
R
r
σ
d
V
{\displaystyle \iiint \limits _{R}\,r\,\sigma \,dV}
Second Moment
∭
R
r
2
σ
d
V
{\displaystyle \iiint \limits _{R}\,r^{2}\,\sigma \,dV}