Exam 1 Review Session
Week 05, Day 2
Class 13
1
1
PMR

8.02 Math (P)Review:
Outline
Hour 1:
Vector Review (Dot, Cross Products)
Review of 1D Calculus
Scalar Functions in higher dimensions
Vector Functions
Differentials
Purpose
: Provide conceptual
framework NOT teach mechanics
2
PMR

Vectors
• Magnitude
and
Direction
• Typically written using unit vectors:
• Unit vector just direction vector:
ˆˆ
ˆ
ˆˆˆ
x
yz x y z
=
++ = + +
ri
j
kx
y
z
G
r
r
=
⇒=
r
rr
r
G
G
Length = 1
3
PMR

Dot (Scalar) Product
• How Parallel?
How much is
r
along
s
?
• Ex: Work from force.
How much does
force push along direction of motion?
r
G
s
G
cos
r
θ
( )
cos
sr
⋅=
rs
G G
Note: If
r
,
s
perpendicular
0
⋅
=
G G
dW
=
⋅
Fd
s
JJG
G
4
PMR

Cross (Vector) Product
• How Perpendicular?
• Direction Perpendicular to both
r
,
s
r
G
s
G
sin
s
()
sin
×=
GG
Note: If
r
,
s
parallel
0
Which perpendicular?
Into or out of page?
Use a right hand rule.
There are many versions.
5
PMR

Review:
1D Calculus
• Think about scalar functions in 1D:
Think of this as height of mountain vs position
f x
x
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PMR

Derivatives
How does function change with position?
dx
df
'( )
slope
xa
df
fa
dx
=
==
f x
x
x
a
=
Rate of change of
at
?
f
=
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Week 05, Day 2
Class 13
2
7
PMR

By the way… Taylor Series
• Approximate function? Copy derivatives!
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 Fall '08
 RABE
 Physics, Derivative, Work, way… Taylor Series

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