Struggling with Advanced Calculus and Geometry: How to Study and Improve?

[yik-boh]

I'm a college student and I'm taking 12 units of math now. I'm having quite a hard time on some topics so I really want to know how should I study for these math topics. Can you also give some other references that I can use? (like the MIT Courseware, Harvard Online Courses, etc.)Here are the list of topics under my math classes:

Math 185 - Modern Geometry

Klein's idea of Geometry
Mobius Geometry
Hyperbolic Geometry
Elliptic Geometry
Absolute Geometry
Real Projective Plane, Multidimensional projective Plane
Universal Geometry
Axiom Systems: Hilbert's and Bachmann's

Books being used: Modern Geometries The Analytic Approach by Micheal Henle

Math 171 - Advanced Calculus I (It includes Calculus topics but not the computation part but more on proving and analysis)

The Real Number System
-Supremum and Infimum of a Set
-Completeness Axiom
-The Archimedean Property
-Density of the Rationals and Irrationals
-Extended Real Number System

The Real Line
-Some Set Theory
-Open and Closed Sets
-Open Coverings, Heine-Borel Theorem
-The Bolzano-Weierstrass Theorem

Functions and Limits
-Epsilon-delta Definition of Limits
-Limit Inferior and Limit Superior

Continuity
-Definition of Continuity
-Bounded Functions, Boundedness Theorem
-Extreme Value Theorem
-Intermediate Value Theorem
-Uniform Continuity, Uniform Continuity Theorem

Integral Calculus
-Definitions
-Integral as the Area under a Curve
-Upper and Lower Integrals
-Existence of the Integral
-Function of Bounded Variation
-Riemann-Stieltjes Integral

Sequences of Real Numbers
-Limit of a Sequence, Convergence and Divergence
-Bounded and Monotonic Sequences
-Sequences of Functional Values
-A Useful Limit Theorem
-Limit Superior and Limit Inferior
-Cauchy's Convergence CriterionSequences and Series of Functions
-Pointwise Convergence
-Uniform Convergence
-Properties Preserved by Uniform Convergence
- Definition of Metric and Metric Space, Euclidean Metric, Schwarz and Triangle InequalityBook being used:

Introduction to Real Analysis by William F. Trench
A First Course in Real Analysis by M.H. Protter and C.B. MorreyMath 101 - Mathematical Analysis III (This one is heavy in Calculus. More on Vector Calculus)

Vectors and the Geometry of Space
-Vectors
-Dot Product
-Cross Product
-Equations of Lines and Planes
-Cylinders and Quadric Surfaces

Vector Functions
-Vector Functions and Space Curves
-Derivatives and Integrals of Vector Functions
-Arc Length and Curvature
-Motion in Space: Velocity and Acceleration

Partial Derivatives
-Partial Derivatives
-Tangent Planes and Linear Approximation
-Directional Derivatives and The Gradient Vector
-Lagrange Multipliers

Multiple Integrals
-Review of Double and Triple Intergals
-Triple Integrals in Cylindrical Coordinates
-Triple Integrals in Spherical Coordinates
-Change of Variables in Multiple Integrals

Vector Calculus
-Vector Fields
-Line Integrals
-The Fundamental Theorem for Line Integrals
-Green's Theorem
-Curl and Divergence
-Parametric Surfaces and their Areas
-Surface Integrals
-Stokes' Theorem
-The Divergence Theorem

Book being used: Stewart's Calculus Early Transcendentals by James StewartI still another Math but I misplaced the class syllabus and I'm going to look for it first. I'll post it here when I find it.

I'm really having a hard time in making proofs especially in Math 171. I got a 18/80 on our first long exam. :| I really want to do better in the remaining exams so I hope you can help me. Thanks!

 

[00Donut]

Well the only thing I can offer any advice on is the Advanced Calculus. I used this book
http://community.middlebury.edu/~abbott/UA/UA.html until a couple months ago to get me started on Analysis. It covers pretty much all the things you listed under analysis in a very friendly manner. I highly recommend it.

As for how to study, my only advice is to do lots of problems. Also, if you don't understand something, seek out help, but only after you've tried REALLY hard to understand it yourself first.

 

[yik-boh]

Well the only thing I can offer any advice on is the Advanced Calculus. I used this book
http://community.middlebury.edu/~abbott/UA/UA.html until a couple months ago to get me started on Analysis. It covers pretty much all the things you listed under analysis in a very friendly manner. I highly recommend it.

As for how to study, my only advice is to do lots of problems. Also, if you don't understand something, seek out help, but only after you've tried REALLY hard to understand it yourself first.

Thank you for this! :)

BUMP!