Formulas
Trigonometric Identities:
sin(x+y) = sin x cos y + cos x sin y
sin x sin y =
cos(x-y) - cos(x+y)
2
sin x cos y =
sin2x =
cos x cos y =
cos(x-y) + cos(x+y)
2
sin(x+y) + sin(x-y)
2
1 - cos(2x)
2
sinh x =
cos(x+y) = cos x cos y - sin x sin y
ex – e-x
2
cos2x =
1 + cos(2x)
2
cosh x =
ex + e-x
2
Geometry & Vectors:
x
 xo 
y
In the following p = (x, y, z) =   = xi + yj + zk, po = (xo, yo, zo) =  yo  = xoi + yoj + zok,
z
 zo 
x
x
1
2
 
 
p1 = (x1, y1, z1) =  y1  = x1i + y1j + z1k, p2 = (x2, y2, z2) =  y2  = x2i + y2j + z2k, and
 z1 
 z2 
a
 
n = (a, b, c) =  b  = ai + bj + ck are either points in three dimensions and their coordinates or vectors
c
and their components.
 x1 + x2 
p1 + p2 =  y1 + y2  = vector along diagonal of parallelogram whose sides are p1 and p2.
 z1 + z2 
 x1 - x2 
p1 - p2 =  y1 - y2  = vector from p2 to p1.
 z1 - z2 
 cx 
cp =  cy  = vector with length |c| times the length of p and in same direction as p if c > 0 and in the
 cz 
opposite direction if c < 0.
|p| = x2 + y2 + z2 = length of p.
p1 . p2 = x1x2 + y1y2 + z1z2 = |p1| |p2| cos = inner product of p1 and p2. ( = angle between p1 & p2)
F . (r) = Work done by constant force F on object which undergoes displacement r.
i j k 
 y1z2 - y2z1 
p1  p2 =  x1 y1 z1  =  z1x2 - z2x1  =


 x1y2 - x2y1 
 x2 y2 z2 
cross product of p1 and p2. It has length |p1| |p2| sin, and
is perpendicular to p1 and p2 with direction determined by
the right hand rule.
r  F = Torque due to force F acting on an object at a point which is a displacement r from the axis of
rotation.
Distance from p1 to p2 = |p1 - p2| =
(x1-x2)2 + (y1-y2)2 + (z1-z2)2.
p1 + p2
Midpoint of line segment joining p1 to p2 =
=
2
Scalar projection of p2 onto p1 =



x1+x2
2
y1+y2
2
z1+z2
2



p1 . p2
= (Signed) length of line segment obtained by dropping
|p1|
perpendicular from p2 onto line through p1.
Vector projection of p2 onto p1 =
p1 . p2 p
1 = Vector along p1 with length equal to the scalar
|p1|2
projection of p2 on p1.
Distance from po = (xo, yo, zo) to the plane ax + by + cz = d:
Distance =
Distance from p to the line through po having direction v = (a, b, c):
|axo + byo + czo - d|
a2 + b2 + c2
Distance =
|v  (p-po)|
|v|
Derivatives and Integrals:
d
-1
dx sin x =
1
1 - x2
d
-1
dx cos x =
ln(x) dx = x ln(x) - x

-1
1 - x2
d
1
-1
dx tan x = 1 + x2
xex dx = xex - ex

x
 cos(x) dx = x sin(x) + cos(x)
x sin(x) dx = -x cos(x) + sin(x)

2
2
x
 sin(x) dx = -x cos(x) + 2x sin(x) + 2cos(x)
2
2
x
 cos(x) dx = x sin(x) + 2x cos(x) - 2sin(x)
Curves:
x = xo + at, y = yo + bt, z = zo + ct
Line through po = (xo, yo, zo) in direction of v = (a,b,c).
x2
y2
+
a2
b2 = 1
ellipse
r = (a cos t) i + (a sin t) j +btk
helix
Projectile motion:
Maximum height =
(vo sin )2
2g
Area = ab
Flight time =
g = 32 ft/sec2 = 9.8 m/sec2
2vo sin 
g
Range =
vo2 sin 2
g
(acceleration of gravity)
dx dy dz
v = ( dt , dt , dt ) = velocity vector. It is tangent to the curve and has length equal to the speed.
d2x d2y d2z
a = ( dt2 , dt2 , dt2 ) = acceleration vector.
2
F = m a = Newton’s second law. Given F, m, vo and po it allows one to find the position p at any t.
b
Length of curve = 
|v| dt
a
 =
|x’y” - y’x”|
|dT/dt|
=
3
|v|
|v|
 = -
B = TN
N =
dT/dt
| dT/dt |
(dB/dt) . N
|v|
Surfaces:
ax + by + cz = d
plane with normal n = (a, b, c)
(x-xo)2+ (y-yo)2 +(z-zo)2 = r2
sphere with center po = (xo, yo, zo) and radius r
(x-xo)2 (y-yo)2 (z-zo)2
a2 + b2 + c2 = 1
ellipsoid
(x-xo)2 (y-yo)2 (z-zo)2
a2 + b2 - c2 = 1
hyperboloid of one sheet
(x-xo)2 (y-yo)2 (z-zo)2
a2 + b2 - c2 = -1
hyperboloid of two sheets
(x-xo)2 (y-yo)2
(z-zo)2
a2 + b2 = c2
elliptic cone
(x-xo)2 (y-yo)2
z-zo
+
=
2
2
a
b
c
elliptic paraboloid
(x-xo)2 (y-yo)2
z-zo
=
2
2
a
b
c
hyperbolic paraboloid
Partial derivatives:
z 
z
z
x +
y
x
y
w
w
w
w =  x i +  y j +  z k
(Linear approximation)
z
z x
z y
=
+
u
x u
y u
(Gradient)
Duz = z .u
3
(Chain rule)
(Directional derivative)
P Q R
div(P i + Q j + R k) =
+
+
x y z

curl(P i + Q j + R k) = 

(Divergence)

P i
x

Q j
y

R k
z



(Curl)
z
z
= 0 and
= 0.
x
y
Interior maximum or minimum 
z
z
= 0 and
= 0 and AC > B2 and A > 0 and C > 0  local minimum
x
y
z
z
= 0 and
= 0 and AC > B2 and A < 0 and C < 0  local maximum
x
y
z
z
= 0 and
= 0 and AC < B2  saddle point
x
y
2z
A =
 x2
2z
B =
 x y
2z
C =
 y2
Double Integrals:
(x,y) = density at x, y (mass per unit area) of thin plate occupying region R in xy plane
Mass = M =
R (x,y) dx dy
_ _
Center of mass = (x, y)
_
1
x = M
R
x (x,y) dx dy
_
1
y = M
Moments of inertia:
About x axis:
Ix =
R y
(x,y) dx dy
About y axis:
Iy =
R x
(x,y) dx dy
About origin (or z axis):
2
2
Io = Iz =
4
R (x
2
+ y2) (x,y) dx dy
R y (x,y) dx dy
x = r cos 
Polar coordinates:
=
R f (x,y) dx dy
=
y = r sin 
r = 2( )
 f (r cos , r sin ) r dr d



 =  r = 1( )
Triple Integrals:
(x,y,z) = density at x, y, z (mass per unit volume) of solid occupying region R in space
Mass = M =
R  (x,y,z) dx dy dz
_ _ _
Center of mass = (x, y, z )
_
1
x = M
R 
_
_
and similarly for y and z
x (x,y,z) dx dy dz
Moments of inertia:
About x axis:
About line L:
Ix =
IL =
R  (y
2
R  D
2
(x,y,z) dx dy dz
=
r = 2( )




f (x,y,z) dx dy dz =
 =  r = 1( )
R  f (x,y,z) dx dy dz


z=z


f (r cos , r sin , z) dz r dr d
z = h1(,r)
y =  sin  sin 
z =  cos 
=
 =   = 2( )  = h2(,)


y = r sin 
z = h2(,r)
x =  sin  cos 
Spherical coordinates:
and similarly for Iy and Iz.
D = D(x,y,z) = distance of (x,y,z) to L.
x = r cos 
Cylindrical coordinates:
R 
+ z2) (x,y,z) dx dy dz


f ( sin cos,  sin sin,  cos) 2 sin d d d
 =   = 1( )  = h1(,)
5
Line Integrals:
C is the curve given by x = x(t), y = y(t), z = z(t) as t goes from a to b
F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k
f(x, y, z) is a real valued function
b

=  f(x(t), y(t), z(t))


 f(x,y,z) ds
C
dy 2
dz 2
2
( dx
dt ) + ( dt ) + ( dt )
dt
a
b

 (F . T) ds
= 
 F . dr
C
=
C
 [P dx + Q dy + R dz] dt

dt
dt
dt

= Work
a
b

 F . n ds
=
C
 [P dy - Q dx] dt

dt
dt

= Flux (in two dimensions)
a
 (f . T) ds = f(x(b), y(b), z(b)) - f(x(a), y(a), z(a))

C

 (F . T) ds =
C
Q P
R  x - y  dxdy

 F . n ds =
C
P
Q
R  x + y  dxdy
Surface Integrals:
x(s,t)
S is the surface given by r = r(s,t) = y(s,t) as (s,t) varies over R
z(s,t)
r r
S f(x,y,z) dS =
f(x(s,t), y(s,t), z(s,t))    dsdt
s t 
R
6
(Green’s Theorem)