# MATH V3386- Final Exam Guide - Comprehensive Notes for the exam ( 109 pages long!)

110 views109 pages

Published on 30 Nov 2017

School

Department

Course

Professor

Columbia

MATH V3386

FINAL EXAM

STUDY GUIDE

Differential Geometry --Gemomety comes from geos--> soil land or earth, metricein--> to rule

So geometry means to rule the world, kinda

The first deviation from plane geometry was geometry on a curved surface

Now the question is where do the derivates come in?

dx and dy (in a plane)

So we take a tangent and define its parts as

Starting from the center of a sphere, we pick

θ=longitude

And

ψ=latitude

Some use

φ=π/2-ψ=co-latitude

Tangent vector has coordiantes

Introduction and Outline

Tuesday, September 5, 2017

11:42 AM

September 5th Page 1

find more resources at oneclass.com

find more resources at oneclass.com

September 5th Page 2

find more resources at oneclass.com

find more resources at oneclass.com

## Document Summary

Differential geometry --gemomety comes from geos--> soil land or earth, metricein--> to rule. So geometry means to rule the world, kinda. The first deviation from plane geometry was geometry on a curved surface. So we take a tangent and define its parts as dx and dy (in a plane) Starting from the center of a sphere, we pick. A euclidean set is a sub set of a euclidean space. An implicit representation of a set will tell you if a point lies in a set, but not which points explicitly. A map f:a->b from a to b is a rule which for each x in a assigns the point y=f(x) in b. F is one-to-one (injective) if f(x1)=f(x2) iff x1=x2. F is onto (surjective) if for every y in b, there exists at least one x in a such that f(x)=y. Vector valued functions map 1d on many dimensions. Function of vector variable 2+d->1d (also functions of several variables)