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