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

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

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

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.

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

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

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?

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?

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?

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?

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?

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

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

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

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

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

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

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

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

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

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?”

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

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

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

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

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?

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

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

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

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

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