1. 8.02 Math (P)Review: Outline
8.02 Math (P)Review Session
Week 01, Day 2.5
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. 8.02 Math (P)Review Session
Week 01, Day 2.5
Vectors
Magnitude and Direction
Typically written using unit vectors:
Unit vector just direction vector:
Length = 1

3. 8.02 Math (P)Review Session
Week 01, Day 2.5
Dot (Scalar) Product
How Parallel? How much is r along s?
Ex: Work from force. How much does force push along direction of motion?
Note: If r, s perpendicular

4. Cross (Vector) Product
8.02 Math (P)Review Session
Week 01, Day 2.5
Cross (Vector) Product
How Perpendicular?
Direction Perpendicular to both r, s
Note: If r, s parallel
Which perpendicular? Into or out of page?
Use a right hand rule. There are many versions.

5. 8.02 Math (P)Review Session
Week 01, Day 2.5
Review: 1D Calculus
Think about scalar functions in 1D:
Think of this as height of mountain vs position

6. 8.02 Math (P)Review Session
Week 01, Day 2.5
Derivatives
How does function change with position?

7. By the way… Taylor Series
8.02 Math (P)Review Session
Week 01, Day 2.5
By the way… Taylor Series
Approximate function? Copy derivatives!
What is f(x) near x=0.35?

8. By the way… Taylor Series
8.02 Math (P)Review Session
Week 01, Day 2.5
By the way… Taylor Series
Approximate function? Copy derivatives!
What is f(x) near x=0.35?

9. By the way… Taylor Series
8.02 Math (P)Review Session
Week 01, Day 2.5
By the way… Taylor Series
Approximate function? Copy derivatives!
What is f(x) near x=0.35?

10. By the way… Taylor Series
8.02 Math (P)Review Session
Week 01, Day 2.5
By the way… Taylor Series
Approximate function? Copy derivatives!
What is f(x) near x=0.35?

11. By the way… Taylor Series
8.02 Math (P)Review Session
Week 01, Day 2.5
By the way… Taylor Series
Approximate function? Copy derivatives!
What is f(x) near x=0.35?

12. By the way… Taylor Series
8.02 Math (P)Review Session
Week 01, Day 2.5
By the way… Taylor Series
Approximate function? Copy derivatives!
Look out for “approximate” or “when x is small” or “small angle” or “close to” …
Most Common: 1st Order

13. 8.02 Math (P)Review Session
Week 01, Day 2.5
Integration
Sum function while walking along axis
Geometry: Find Area
Also: Sum Contributions

14. Move to More Dimensions
8.02 Math (P)Review Session
Week 01, Day 2.5
Move to More Dimensions
We’ll start in 2D

15. 8.02 Math (P)Review Session
Week 01, Day 2.5
Scalar Functions in 2D
Function is height of mountain: X Y Z

16. 8.02 Math (P)Review Session
Week 01, Day 2.5
Partial Derivatives
How does function change with position?
In which direction are we moving? X Y Z

17. 8.02 Math (P)Review Session
Week 01, Day 2.5
Gradient
What is fastest way up the mountain? Z Y X

18. 8.02 Math (P)Review Session
Week 01, Day 2.5
Gradient
Gradient tells you direction to move:

19. 8.02 Math (P)Review Session
Week 01, Day 2.5
Line Integral
Sum function while walking under surface
along given curve
Just like 1D integral, except now not just along x

20. 8.02 Math (P)Review Session
Week 01, Day 2.5
2D Integration
Sum function while walking under surface
Just Geometry: Finding Volume Under Surface

21. N-D Integration in General
8.02 Math (P)Review Session
Week 01, Day 2.5
N-D Integration in General
Now think “contribution” from each piece
Find area of surface?
Volume of object?
Mass Density
Mass of object?
IDEA: Break object into small pieces, visit each, asking “What is contribution?”

22. 8.02 Math (P)Review Session Small Extension to Vector Functions
Week 01, Day 2.5
You Now Know It All
Small Extension to
Vector Functions

23. Can’t Easily Draw Multidimensional Vector Functions
8.02 Math (P)Review Session
Week 01, Day 2.5
Can’t Easily Draw Multidimensional Vector Functions
In 2D various representations:
“Grass Seeds” / “Iron Filings”
Vector Field Diagram

24. Integrating Vector Functions
8.02 Math (P)Review Session
Week 01, Day 2.5
Integrating Vector Functions
Two types of questions generally asked:
1) Integral of vector function yielding vector
Ex.: Mass Distribution
IDEA: Use Components - Just like scalar

25. Integrating Vector Functions
8.02 Math (P)Review Session
Week 01, Day 2.5
Integrating Vector Functions
Two types of questions generally asked:
2) Integral of vector function yielding scalar
Line Integral Ex.: Work
IDEA: While walking along the curve how much of the function lies along our path

26. Integrating Vector Functions
8.02 Math (P)Review Session
Week 01, Day 2.5
Integrating Vector Functions
One last example: Flux
Q: How much does field E penetrate the surface?

27. 8.02 Math (P)Review Session
Week 01, Day 2.5
Differentials
People often ask, what is dA? dV? ds?
Depends on the geometry
Read Review B: Coordinate Systems
One Important Geometry Fact

28. 8.02 Math (P)Review Session
Week 01, Day 2.5
Differentials
Rectangular Coordinates
Draw picture and think!

29. 8.02 Math (P)Review Session
Week 01, Day 2.5
Differentials
Cylindrical Coordinates
Draw picture and think!

30. 8.02 Math (P)Review Session
Week 01, Day 2.5
Differentials
Spherical Coordinates
Draw picture and think!

31. 8.02 Math (P)Review Session
Week 01, Day 2.5
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