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: ˆ ˆ ˆ ˆ ˆ ˆ x y z x y z = + + = + + r i j k x y z G ˆ ˆ r r = ⇒ = r r r 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 θ ( ) cos sr θ ⋅ = rs G G Note: If r, s perpendicular 0 ⋅ = rs G G dW = ⋅ F ds J JG G 4 PMR- Cross (Vector) Product • How Perpendicular? • Direction Perpendicular to both r, s r G s G sin s θ θ ( ) sin rs θ × = r s G G Note: If r, s parallel 0 × = r s G G 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 6 PMR- Derivatives How does function change with position? dx df '( ) slope x a df f a dx = = = () f x x x a = Rate of change of at ? f x a =
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Exam 1 Review Session Week 05, Day 2
Class 13 1
1PMR-
8.02 Math (P)Review: Outline
Hour 1:Vector Review (Dot, Cross Products)Review of 1D CalculusScalar Functions in higher dimensionsVector FunctionsDifferentials
Purpose: Provide conceptual framework NOT teach mechanics
2PMR-
Vectors• Magnitude and Direction
• Typically written using unit vectors:
• Unit vector just direction vector:
ˆ ˆ ˆ ˆ ˆ ˆx y z x y z= + + = + +r i j k x y z
ˆ ˆrr
= ⇒ =rr r r Length = 1
3PMR-
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
scosr θ
θ( )coss r θ⋅ =r s
Note: If r, s perpendicular 0⋅ =r s
dW = ⋅F ds
4PMR-
Cross (Vector) Product• How Perpendicular?
• Direction Perpendicular to both r, s
r
s
sins θθ
( )sinr s θ× =r s
Note: If r, s parallel 0× =r s
Which perpendicular? Into or out of page?Use a right hand rule. There are many versions.
5PMR-
Review: 1D Calculus• Think about scalar functions in 1D:
Think of this as height of mountain vs position
( )f x
x
6PMR-
DerivativesHow does function change with position?
dx
df
'( ) slopex a
dff adx =
= =
( )f x
xx a=Rate of change of at ?f x a=
Exam 1 Review Session Week 05, Day 2
Class 13 2
7PMR-
By the way… Taylor Series• Approximate function? Copy derivatives!
0.00 0.25 0.50 0.75 1.00
-1.0
-0.5
0.0
0.5
1.0
X
f(x)=
sin(
2πx) What is f(x) near x=0.35?
8PMR-
0.00 0.25 0.50 0.75 1.00
-1.0
-0.5
0.0
0.5
1.0
X
f(x)=
sin(
2πx)
By the way… Taylor Series• Approximate function? Copy derivatives!
What is f(x) near x=0.35?
0 ( ) (0.35)T x f=
9PMR-
0.00 0.25 0.50 0.75 1.00
-1.0
-0.5
0.0
0.5
1.0
X
f(x)=
sin(
2πx)
By the way… Taylor Series• Approximate function? Copy derivatives!
( )1( ) (0.35)
'(0.35) 0.35T x f
f x=
+ −
What is f(x) near x=0.35?
10PMR-
0.00 0.25 0.50 0.75 1.00
-1.0
-0.5
0.0
0.5
1.0
X
f(x)=
sin(
2πx)
By the way… Taylor Series• Approximate function? Copy derivatives!
( )( )
2
212
( ) (0.35)'(0.35) 0.35
''(0.35) 0.35
T x ff x
f x
=
+ −
+ −
What is f(x) near x=0.35?
11PMR-
0.00 0.25 0.50 0.75 1.00
-1.0
-0.5
0.0
0.5
1.0
X
f(x)=
sin(
2πx)
By the way… Taylor Series• Approximate function? Copy derivatives!
( ) ( )( )
0( )
!
iiN
Ni
f a x aT x
i=
−=∑
10 ( )T x
( )( )
2
212
( ) (0.35)'(0.35) 0.35
''(0.35) 0.35
T x ff x
f x
=
+ −
+ − …
What is f(x) near x=0.35?
12PMR-
0.00 0.25 0.50 0.75 1.00
-1.0
-0.5
0.0
0.5
1.0
X
f(x)=
sin(
2πx)
By the way… Taylor Series• Approximate function? Copy derivatives!
• Look out for “approximate” or “when x is small” or “small angle” or “close to” …
( )1( ) ( )
'( )T x f a
f a x a= +
−
Most Common: 1st Order
Exam 1 Review Session Week 05, Day 2
Class 13 3
13PMR-
IntegrationSum function while walking along axis
( )f x
xx a=
( ) ?b
a
f x dx =∫
x b=Geometry: Find Area Also: Sum Contributions
14PMR-
Move to More Dimensions
We’ll start in 2D
15PMR-
Scalar Functions in 2D• Function is height of mountain:
XY
Z
( ),z F x y=
16PMR-
Partial DerivativesHow does function change with position?In which direction are we moving?
XY
Z
0Fx
∂>
∂0F
y∂
≈∂
17PMR-
GradientWhat is fastest way up the mountain?
XY
Z
18PMR-
0xF∂ ≈
GradientGradient tells you direction to move:
ˆ ˆF FFx y
∂ ∂∇ = +
∂ ∂i j ˆ ˆ ˆ
x y z∂ ∂ ∂
∇ ≡ +∂ ∂ ∂
i j + k
0xF∂ >0yF∂ ≈ 0yF∂ >
Exam 1 Review Session Week 05, Day 2
Class 13 4
19PMR-
Line IntegralSum function while walking under surface
along given curve
Just like 1D integral, except now not just along x
( ),C
f x y ds =∫
20PMR-
2D IntegrationSum function while walking under surface
Just Geometry: Finding Volume Under Surface
( ),Surface
F x y dA∫∫
21PMR-
N-D Integration in GeneralNow think “contribution” from each piece
Surface
dA∫∫
Object
dV∫∫∫
Mass of object?Object Object
dM dVρ=∫∫∫ ∫∫∫
Volume of object?
Find area of surface?
Mass Density
IDEA: Break object into small pieces, visit each, asking “What is contribution?”