sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = sin α cos β − cos α sin β
sin α + sin β = 2 sin(
α−β
α+β
) cos(
)
2
2
sin α − sin β = 2 sin(
α−β
α+β
) cos(
)
2
2
cos(α + β) = cos α cos β − sin α sin β
cos α + cos β = 2 cos(
cos(α − β) = cos α cos β + sin α sin β
√
sin α + cos α = 2 cos(α − π4 )
sin α + sin β = −2 sin(
α
1 − cos α
sin2 ( ) =
2
2
sin(2α) = 2 sin α cos α
cos(2α) = cos2 α − sin2 α
α+β
α−β
) cos(
)
2
2
sin α =
α
1 + cos α
cos ( ) =
2
2
α+β
α−β
) sin(
)
2
2
2 tan(α/2)
1 + tan2 (α/2)
Formule di Gauss-Green e calcolo dell’area
Z
m(D) =
Z Z
Z
fx dx dy =
D
m(D) = −
y dx
Z
fy dx dy = −
D
+∂D
fd
+∂D
1
2
m(D) =
Z
x dy − y dx
+∂D
Teorema della divergenza
Z Z
Z
divF dx dy dz =
< F, ne > dl
D
1 − tan(α/2)
cos α =
1 + tan2 (α/2)
2
Z
+∂D
Z Z
x dy
+∂D
f dy
in R2
∂D
Z Z Z
Z
divF dx dy dz =
< F, νe > dσ
T
in R3
∂T
Coordinate sferiche
α
sin α
cos α
tan α
π/6
1/2
√
3/2
√
3/3
d 3
(a ) = ax log a
dx
π/4
√
2/2
√
2/2
1
π/2
1
0
−
2π/3
√
3/2
−1/2
√
− 3
∞
X
xn
n!
n=0
ex =
1
d
(loga x) =
dx
x log a
log(1 + x) =
d
1
(arcsin x) = √
dx
1 − x2
π/3
√
3/2
1/2
√
3
sin x =
∞
X
(−1)n
n=0
xn
n
cos x =
∞
X
(−1)n 2n+1
x
(2n + 1)!
n=0
ρ=


x = ρ sin φ cos θ
y = ρ sin φ sin θ


z = ρ cos φ
1
=
xn
1 − x n=0
p
x2 + y 2 + z 2 , J = ρ2 sin φ
φ ∈ [0, π]
θ ∈ [0, 2π]
Integrali di superficie
φu × φv = Aî + B ĵ + C k̂
Piano tangente A(x0 , y0 , z0 )(x − x0 ) + B(x0 , y0 , z0 )(y − y0 ) + C(x0 , y0 , z0 )(z − z0 ) = 0
Z
Z Z
f dσ =
f (φ(u, v))|φu × φv | du dv
∞
X
(−1)n 2n
x
(2n)!
n=0
S
D
Versore normale ν =
∞
X
(−1)n 2n+1
arctan x =
x
2n
+1
n=0
∞
X
d
1
(arccos x) = − √
dx
1 − x2
3π/4
√
2/2
√
− 2/2
−1
φu × φv
||φu × φv ||
Z Z
|φu × φv | du dv
Area superficie A(φ) =
D
Formula di Stokes
Integrali curvilinei e forme differenziali
Z Z b
Z
D
||φ′ (t)|| dt
L(φ) =
b
Z
′
a
< rotF, ν > dσ =
p
f (x(t), y(t)) · (x′ (t))2 + (y ′ (t))2 dt
Baricentro (xB , yB ) =
Γ
1
x dl,
L(φ)
γ
Z
y dl
k=
k=
z = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )
T (t) =
ab
(a2 sin2 t + b2 cos2 t)3/2
−Fy2 Fxx
+ 2Fxx Fxy −
(Fx2 + Fy2 )3/2
′′
′
φ (t)
||φ′ (t)||
k=
′
Fx2 Fyy
τ (t) = −
Ellisse
Curve definite impl.
−ρ(θ)ρ (θ) + 2(ρ (θ)) + (ρ(θ))2
(ρ(θ)2 + ρ′ (θ)2 )3/2
Torsione
1
in R3
||φ′ (t) × φ′′ (t)||
||φ′ (t)||3
k(t) =
γ
a
Versore tangente
< F, T > dS
∂+S
k(s) = ||γ ′′ (s)|| dove s è l’ascissa curvilinea
1
L(φ)
Z
ω = a(x, y, z)dx + b(x, y, z)dy + c(x, y, z)dz con φ(t) = (x(t), y(t), z(t))
Z
Z b
ω=
[a(φ(t))x′ (t) + b(φ(t))y ′ (t) + c(φ(t))z ′ (t) dt]
Piano tangente
in R2
Curvatura
a
F1 dx + F2 dy
+∂D
Z
S
b
Z
f (φ(t)) · ||φ (t)|| dt =
f dl =
γ
Z
dx dy =
Z
a
Z
∂F2
∂F1
−
∂x
∂y
2
Coord. polari
< φ′ (t) × φ′′ (t) , φ′′′ (t) >
||φ′ (t) × φ′′ (t)||2
2