Math Review Session
8.02 Math (P)Review: Outline
Vectors
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
Dot (Scalar) Product
• Magnitude and Direction
• How Parallel? How much is r along s?
θ
s
PMR- 3
Derivatives
How does function change with position?
f ( x)
f ( x)
Note: If r, s parallel
df
r×s = 0
dx
• Direction Perpendicular to both r, s
Which perpendicular? Into or out of page?
Use a right hand rule. There are many versions.
x=a
x
Think of this as height of mountain vs position
PMR- 4
1
r⋅s = 0
dW = F ⋅ ds
PMR- 2
• Think about scalar functions in 1D:
r × s = r ( s sin θ )
Note: If r, s perpendicular
s
• Ex: Work from force. How much does
force push along direction of motion?
Review: 1D Calculus
• How Perpendicular?
s sin θ
r cos θ
• Unit vector just direction vector:
r
r = rrˆ ⇒ rˆ =
Length = 1
r
Cross (Vector) Product
r
θ
r = xˆi + yˆj + zkˆ = xxˆ + yyˆ + zzˆ
PMR- 1
r ⋅ s = s ( r cos θ )
r
• Typically written using unit vectors:
PMR- 5
f '(a ) =
df
dx
= slope
x=a
x
Rate of change of f at x = a ?
PMR- 6
Math Review Session
By the way… Taylor Series
• Approximate function? Copy derivatives!
• Approximate function? Copy derivatives!
• Approximate function? Copy derivatives!
f(x)=sin(2πx)
What is f(x) near x=0.35?
1.0
0.5
0.0
-0.5
T0 ( x) = f (0.35)
0.5
0.0
-0.5
-1.0
0.00
What is f(x) near x=0.35?
1.0
0.50
0.75
X 1.00
0.00
What is f(x) near x=0.35?
1.0
T1 ( x) = f (0.35)
0.5
0.0
+ f '(0.35) ( x − 0.35 )
-0.5
-1.0
0.25
f(x)=sin(2πx)
By the way… Taylor Series
f(x)=sin(2πx)
By the way… Taylor Series
-1.0
0.25
0.50
0.75
X 1.00
0.00
PMR- 7
0.25
0.50
0.75
X 1.00
PMR- 8
PMR- 9
By the way… Taylor Series
• Approximate function? Copy derivatives!
• Approximate function? Copy derivatives!
• Approximate function? Copy derivatives!
T2 ( x) = f (0.35)
0.5
0.0
+ f '(0.35) ( x − 0.35 )
-0.5
-1.0
0.00
f(x)=sin(2πx)
What is f(x) near x=0.35?
1.0
0.25
0.50
0.75
X 1.00
+ 12 f ''(0.35) ( x − 0.35 )
1.0
What is f(x) near x=0.35?
T10 ( x)
0.5
T2 ( x) = f (0.35)
0.0
+ f '(0.35) ( x − 0.35 )
-0.5
2
-1.0
0.00
0.25
0.50
0.75
X 1.00
N
TN ( x) = ∑
PMR- 10
2
i =0
Most Common: 1st Order
1.0
T1 ( x) = f (a ) +
0.5
0.0
f '(a) ( x − a )
-0.5
+ 12 f ''(0.35) ( x − 0.35 ) …
2
f (i ) ( a )( x − a )
i!
f(x)=sin(2πx)
By the way… Taylor Series
f(x)=sin(2πx)
By the way… Taylor Series
-1.0
0.00
0.25
0.50
0.75
X 1.00
• Look out for “approximate” or “when x is
small” or “small angle” or “close to” …
i
PMR- 11
PMR- 12
Math Review Session
Integration
Scalar Functions in 2D
Sum function while walking along axis
• Function is height of mountain:
b
∫ f ( x) dx = ?
f ( x)
z = F ( x, y )
Move to More Dimensions
a
Z
We’ll start in 2D
x=a
x=b
Geometry: Find Area
x
Also: Sum Contributions
PMR- 13
Partial Derivatives
Gradient
How does function change with position?
In which direction are we moving?
Z
∂F
≈0
∂y
Y
3
∂F
>0
∂x
X
Y
PMR- 14
X
PMR- 15
Gradient
What is fastest way up the mountain?
Gradient tells you direction to move:
∂ ˆ ∂ ˆ ∂
∂F ˆ ∂F
+ j +k
∇ ≡ ˆi
+j
∇ F = ˆi
∂
∂y
∂z
x
∂x
∂y
Z
PMR- 16
Y
X
∂xF > 0
∂yF ≈ 0
PMR- 17
∂xF ≈ 0
∂yF > 0
PMR- 18
Math Review Session
Line Integral
2D Integration
N-D Integration in General
Sum function while walking under surface
along given curve
Sum function while walking under surface
Now think “contribution” from each piece
∫ f ( x, y ) ds =
∫∫ F ( x, y ) dA
C
∫∫
Find area of surface?
dA
Surface
Surface
Volume of object?
∫∫∫ dV
Mass Density
Object
Mass of object?
∫∫∫ dM = ∫∫∫ ρ dV
Object
Just like 1D integral, except now not just along x
Just Geometry: Finding Volume Under Surface
PMR- 19
PMR- 20
Can’t Easily Draw Multidimensional
Vector Functions
Object
IDEA: Break object into small pieces, visit
each, asking “What is contribution?”
PMR- 21
Integrating Vector Functions
Two types of questions generally asked:
In 2D various representations:
1) Integral of vector function yielding vector
You Now Know It All
Ex.: Mass Distribution
Small Extension to
Vector Functions
IDEA: Use Components - Just like scalar
Vector Field Diagram
PMR- 22
4
dM
rˆ
r2
object
g = −G ∫∫∫
“Grass Seeds” / “Iron Filings”
PMR- 23
∫∫ F(r )dA =
ˆi F (r )dA + ˆj F (r )dA + kˆ F (r )dA
∫∫
∫∫
∫∫
x
y
z
PMR- 24
Math Review Session
Integrating Vector Functions
Integrating Vector Functions
Two types of questions generally asked:
2) Integral of vector function yielding scalar
Line Integral Ex.: Work
W =∫
Curve
Differentials
One last example: Flux
Q: How much does field E penetrate the surface?
F ⋅ ds
People often ask, what is dA? dV? ds?
Depends on the geometry
Read Review B: Coordinate Systems
One Important Geometry Fact
IDEA: While walking along the curve how
much of the function lies along our path
Flux Φ E =
Differentials
E ⋅ dA
L = Rθ
PMR- 26
Differentials
Rectangular Coordinates
dV = dx dy dz
dV = ρ dϕ d ρ dz
dA = dx dy
dA = dx dz
dA = dy dz
dA = ρ dϕ dz
dA = ρ dϕ d ρ
dA = d ρ dz
Spherical Coordinates
r sin θ
dV = r sin θ dϕ rdθ dr
dA = r sin θ dϕ rdθ
Draw picture and think!
PMR- 28
PMR- 27
Differentials
Cylindrical Coordinates
Draw picture and think!
5
∫∫
Surface
PMR- 25
θ R
Draw picture and think!
PMR- 29
PMR- 30
Math Review Session
8.02 Math Review
Vectors:
Dot Product: How parallel?
Cross Product: How perpendicular?
Derivatives:
Rate of change (slope) of function
Gradient tells you how to go up fast
Integrals:
Visit each piece and ask contribution
PMR- 31
6