1PMR-

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

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

r

r 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

s

cosr

coss r 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

sinr s 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-

Derivatives

How does function change with position?

dx

df

'( ) slopex a

dff a

dx

( )f x

xx aRate of change of at ?f x a

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)

=si

n(2

x) 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)

=si

n(2

x)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)

=si

n(2

x)By the way… Taylor Series

• Approximate function? Copy derivatives!

1( ) (0.35)

'(0.35) 0.35

T 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)

=si

n(2

x)By the way… Taylor Series

• Approximate function? Copy derivatives!

2

212

( ) (0.35)

'(0.35) 0.35

''(0.35) 0.35

T x f

f 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)

=si

n(2

x)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 f

f 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)

=si

n(2

x)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

13PMR-

Integration

Sum function while walking along axis

( )f x

xx a

( ) ?b

a

f x dx

x bGeometry: 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 Derivatives

How does function change with position?

In which direction are we moving?

XY

Z

0F

x

0F

y

17PMR-

Gradient

What is fastest way up the mountain?

XY

Z

18PMR-

0xF

Gradient

Gradient tells you direction to move:

ˆ ˆF FF

x y

i j ˆ ˆ ˆ

x y z

i j +k

0xF 0yF 0yF

19PMR-

Line IntegralSum function while walking under surface

along given curve

Just like 1D integral, except now not just along x

,Cf x y ds

20PMR-

2D Integration

Sum function while walking under surface

Just Geometry: Finding Volume Under Surface

,Surface

F x y dA

21PMR-

N-D Integration in General

Now think “contribution” from each piece

Surface

dA

Object

dV

Mass of object?Object Object

dM dV

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 Functions

Two types of questions generally asked:

Ex.: Mass Distribution

1) Integral of vector function yielding vector

IDEA: Use Components - Just like scalar

2ˆ

object

dMG

r g r

( )dAF r

ˆ ˆ ˆ( ) ( ) ( )x y zF dA F dA F dA i r j r k r

25PMR-

Integrating Vector Functions

Two 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 d E A

Q: How much does field E penetrate the surface?

27PMR-

Differentials

People often ask, what is dA? dV? ds?

Depends on the geometry

Read Review B: Coordinate Systems

One Important Geometry Fact

L R

R

28PMR-

Differentials

Rectangular Coordinates

dV dx dy dz

dA dx dydA dx dzdA dy dz

Draw picture and think!

29PMR-

Differentials

Cylindrical Coordinates

dV d d dz

dA d dz dA d d dA d dz

Draw picture and think!

30PMR-

Differentials

Spherical Coordinates

sindV r d rd dr

sindA r d rd

Draw picture and think!

sinr

31PMR-

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