A.6. VECTOR ARITHMETIC 171
A.6 VECTOR ARITHMETIC
VECTOR ARITHMETIC AND ALGEBRA
• c Ev D .cv
1
; cv
2
; : : : ; cv
n
/
• Ev C Ew D .v
1
C w
1
; v
2
C w
2
; : : : ; v
n
C w
n
/
• Ev Ew D .v
1
w
1
; v
2
w
2
; : : : ; v
n
w
n
/
• Ev Ew D v
1
w
1
C v
2
w
2
C : : : C v
n
w
n
• c .Ev C Ew/ D c Ev Cc Ew
• c .d Ev/ D .cd/ Ev
• Ev C Ew D Ew C Ev
• E
u
.
E
v
C E
w/
D E
u
E
v
C E
u
E
w
CROSS PRODUCT OF VECTORS
• Ev Ew D . v
2
w
3
v
3
v
2
; v
3
w
1
v
1
w
3
; v
1
w
2
v
2
w
1
/
Formula for the angle between vectors
cos./ D
Ev Ew
jvjjwj
A.7 POLAR AND RECTANGULAR CONVERSION
• x D r cos./
• y D r sin. /
• r D
p
x
2
C y
2
• D tan
1
.y=x/
A.8 INTEGRAL RULES
BASIC INTEGRATION RULES
•
Z
x
n
dx D
1
n C1
x
nC1
C C
•
Z
a f .x/ dx D a
Z
f .x/ dx
•
Z
.
f .x/ C g.x/
/
dx D
Z
f .x/ dx C
Z
g.x/ dx
LOG AND EXPONENT
•
Z
1
x
dx D ln.x/ CC
•
Z
e
x
dx D e
x
C C
172 A. USEFUL FORMULAS
TRIG AND INVERSE TRIG
•
Z
sin.x/ dx D cos.x/ C C
•
Z
cos.x/ dx D sin.x/ CC
•
Z
sec
2
.x/ dx D tan.x/ CC
•
Z
csc
2
.x/ dx D cot.x/ C C
•
Z
sec.x/ tan.x/ dx D sec.x/ C C
•
Z
csc.x/ cot.x/ dx D csc.x/ CC
•
Z
1
p
1 x
2
dx D sin
1
.x/ C C
•
Z
1
1 Cx
2
dx D tan
1
.x/ C C
•
Z
1
x
p
x
2
1
dx D sec
1
.jxj/ C C
•
Z
tan.x/ dx D ln jsec.x/jC C
•
Z
sec.x/ dx D ln jsec.x/ C tan.x/j C C
INTEGRATION BY PARTS
Z
U dV D U V
Z
V d U
EXPONENTIAL/POLYNOMIAL SHORTCUT
Z
p.x/e
x
dx D
p.x/ p
0
.x/ C p
00
.x/ p
000
.x/ C
e
x
C C
VOLUME, SURFACE, ARC LENGTH
Volume of Rotation, Disks
V D
Z
b
a
f .x/
2
dx
Volume of Rotation, Cylindrical Shells
V D 2
Z
xDb
xDa
x f .x/ dx
Volume of Rotation with Washers
V D
Z
b
a
f
1
.x/
2
f
2
.x/
2
dx
Differential of Arc Length
ds D
q
.
y
0
/
2
dx D
q
.
f
0
.x/
/
2
dx
Arc Length
S D
Z
ds
Surface Area of Rotation
A D 2
Z
b
a
f .x/ ds
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