cross (vector) product review: 1d calculus derivatives
TRANSCRIPT
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?”
22PMR-
You Now Know It All
Small Extension to Vector Functions
23PMR-
Can’t Easily Draw Multidimensional Vector Functions
In 2D various representations:
Vector Field Diagram“Grass Seeds” / “Iron Filings”
24PMR-
Integrating Vector FunctionsTwo types of questions generally asked:
Ex.: Mass Distribution
1) Integral of vector function yielding vector
IDEA: Use Components - Just like scalar
2ˆ
object
dMGr
= − ∫∫∫g r
( )dA =∫∫F r
ˆ ˆ ˆ( ) ( ) ( )x y zF dA F dA F dA+ +∫∫ ∫∫ ∫∫i r j r k r
Exam 1 Review Session Week 05, Day 2
Class 13 5
25PMR-
Integrating Vector FunctionsTwo types of questions generally asked:
Line Integral Ex.: Work
2) Integral of vector function yielding scalar
IDEA: While walking along the curve how much of the function lies along our path
CurveW d= ⋅∫ F s
26PMR-
Integrating Vector FunctionsOne last example: Flux
Surface
Flux E dΦ = ⋅∫∫ E A
Q: How much does field E penetrate the surface?
27PMR-
DifferentialsPeople often ask, what is dA? dV? ds?Depends on the geometry Read Review B: Coordinate Systems
One Important Geometry Fact
L Rθ=
θ R
28PMR-
DifferentialsRectangular Coordinates
dV dx dy dz=
dA dx dy=dA dx dz=dA dy dz=
Draw picture and think!
29PMR-
DifferentialsCylindrical Coordinates
dV d d dzρ ϕ ρ=
dA d dzρ ϕ=dA d dρ ϕ ρ=dA d dzρ=
Draw picture and think!
30PMR-
DifferentialsSpherical Coordinates
sindV r d rd drθ ϕ θ=
sindA r d rdθ ϕ θ=
Draw picture and think!
sinr θ
Exam 1 Review Session Week 05, Day 2
Class 13 6
31PMR-
8.02 Math Review
Vectors:Dot Product: How parallel?Cross Product: How perpendicular?
Derivatives:Rate of change (slope) of functionGradient tells you how to go up fast
Integrals:Visit each piece and ask contribution