1 + (y y ) )2 cylindrical surface
2
(
equation of a
sphere : x + (2 2 = ra :
one var
missing
-
-
-
.
, . ,
( ) center
,, y ,,
2
,
=
Ac + By+ [22 +
(collinear) quadratic surface : Dec + Ey + Fz +
G
parallel rectors := G > O
·
ellipsoid : A , B ,
C >
dot product : B - =
a ,
b ,
+ azbe
·
one-sheet hyperboloid : A , B > 0 ; (0
- = 11/12 ·
two-sheet hyperboloid :
As0 ; B C ,
< O
G =
G
2 5 .
angle between vectors : O A B 0 (O
>
Hallibil
·
cone
=
COS :
,
,
AorBorC = 0
(corollary/ orthogonal)
perpendicular vectors : 5
:
·
elliptic paraboloid (bowl") Acc +
0 z By
: =
.
=
2
·
hyperbolic paraboloid ("saddle") : z =
Ax2-By
orthogonal decomposition : =
pric()
<
rotational surface x + :
y2 =
f(z)
work (w) =
Fo & negative
j T
( y z)
rectangular coordinates :
, ,
cross product : x5 = a ,
a
, as
cylindrical coordinates (r :
, A z) ,
b , bz by
spherical coordinates (S :
,
0 ,
9)
x = -
(ax) R =
C C- R
x = 10 ,
0 ,
0 ·
x = rcOSO ·
r = vxz yz +
·
y
= rsinG ·
tand =
orthogonal vector : x to &
- S 53C
Ilax5 f2 r2 + z3 Scost
of
parallelogram
·
·
area : =
z =
·
tand = ·
r =
Ssin
torque : x &
parallelapiped volume : (BA) A ·
rector-valued line F :
= B + at
tetrohedron : El parallelapiped volume vector-valued circle : v =
Lacost , sint) : Oct
> >
-
coplanar :
(ABAC) .*D =
O helix : =
Lacost ,
asint , t) &
:
E r
canonical equation of line Cycloid :
=
r(t-sint)
r(l- cost)
S
x + at
y
x = =
parametric equation : y =
y .
+ bt
(a ,
b ,
c)
2 =
2. + Ct
Mo(o Yo Zol , , limit of :) =
(f(t) , 9(t) ,(t)
equation of a plane : Ak -() +
B(y yo) - + ((z 2) -
=
0 derivative : '(E) =
(f'(t) g'(t) nltl) , ,
Ax +
By + Cz + D
integral Sr(t)dt (ff(t)dt Sg(t)dt ShIt)dt) Y
, , ,
distance from point to plane : d :
=
+
VA2
=
B2 C
, ,
+ +
Intell
distance from point to line : d =
IIell
, position vector : F(t)
velocity rector : Y(t) = '(t) always find
(t)
TIME
acceleration vector a(t) :
=
u(t) = "(t)
speed : 11 ELIII
projectile motion : () =
((Vosdt , (Vosindt -)
tangent unit vector : F(t) =
i
principle unit vector : (t) =
acceleration decomposition alt :
=
af an
+
⑧ calculate '(t) , "(t)
O calculate
>
-
② a+ = (t) ·
T
>
-
⑧ an =
a(t) -
a +
T
①
an
= Il step 311 : N=
arclength
: s = III at
II"Il
curvature : k =
11/113
Cor)
AN
k =
115-112
(or)
1y"l
k =
(1 +
y(z)5
2
(
equation of a
sphere : x + (2 2 = ra :
one var
missing
-
-
-
.
, . ,
( ) center
,, y ,,
2
,
=
Ac + By+ [22 +
(collinear) quadratic surface : Dec + Ey + Fz +
G
parallel rectors := G > O
·
ellipsoid : A , B ,
C >
dot product : B - =
a ,
b ,
+ azbe
·
one-sheet hyperboloid : A , B > 0 ; (0
- = 11/12 ·
two-sheet hyperboloid :
As0 ; B C ,
< O
G =
G
2 5 .
angle between vectors : O A B 0 (O
>
Hallibil
·
cone
=
COS :
,
,
AorBorC = 0
(corollary/ orthogonal)
perpendicular vectors : 5
:
·
elliptic paraboloid (bowl") Acc +
0 z By
: =
.
=
2
·
hyperbolic paraboloid ("saddle") : z =
Ax2-By
orthogonal decomposition : =
pric()
<
rotational surface x + :
y2 =
f(z)
work (w) =
Fo & negative
j T
( y z)
rectangular coordinates :
, ,
cross product : x5 = a ,
a
, as
cylindrical coordinates (r :
, A z) ,
b , bz by
spherical coordinates (S :
,
0 ,
9)
x = -
(ax) R =
C C- R
x = 10 ,
0 ,
0 ·
x = rcOSO ·
r = vxz yz +
·
y
= rsinG ·
tand =
orthogonal vector : x to &
- S 53C
Ilax5 f2 r2 + z3 Scost
of
parallelogram
·
·
area : =
z =
·
tand = ·
r =
Ssin
torque : x &
parallelapiped volume : (BA) A ·
rector-valued line F :
= B + at
tetrohedron : El parallelapiped volume vector-valued circle : v =
Lacost , sint) : Oct
> >
-
coplanar :
(ABAC) .*D =
O helix : =
Lacost ,
asint , t) &
:
E r
canonical equation of line Cycloid :
=
r(t-sint)
r(l- cost)
S
x + at
y
x = =
parametric equation : y =
y .
+ bt
(a ,
b ,
c)
2 =
2. + Ct
Mo(o Yo Zol , , limit of :) =
(f(t) , 9(t) ,(t)
equation of a plane : Ak -() +
B(y yo) - + ((z 2) -
=
0 derivative : '(E) =
(f'(t) g'(t) nltl) , ,
Ax +
By + Cz + D
integral Sr(t)dt (ff(t)dt Sg(t)dt ShIt)dt) Y
, , ,
distance from point to plane : d :
=
+
VA2
=
B2 C
, ,
+ +
Intell
distance from point to line : d =
IIell
, position vector : F(t)
velocity rector : Y(t) = '(t) always find
(t)
TIME
acceleration vector a(t) :
=
u(t) = "(t)
speed : 11 ELIII
projectile motion : () =
((Vosdt , (Vosindt -)
tangent unit vector : F(t) =
i
principle unit vector : (t) =
acceleration decomposition alt :
=
af an
+
⑧ calculate '(t) , "(t)
O calculate
>
-
② a+ = (t) ·
T
>
-
⑧ an =
a(t) -
a +
T
①
an
= Il step 311 : N=
arclength
: s = III at
II"Il
curvature : k =
11/113
Cor)
AN
k =
115-112
(or)
1y"l
k =
(1 +
y(z)5
