Accepted in the Master of Arts in Mathematics program

[Shackleford]

It's mainly designed for those who want to teach at a junior college or whatever. I plan to stay in oil and gas, and I'm going to take:

MATH 5350: INTRODUCTION TO DIFFERENTIAL GEOMETRY. Prerequisites: Three semesters of calculus, or consent of instructor. Multi-variable calculus, linear algebra, and ordinary differential equations are used to study the geometry of curves and surfaces in 3-space. Topics include: Curves in the plane and in 3-space, curvature, Frenet frame, surfaces in 3-space, the first and second fundamental form, curvature of surfaces, Gauss’s theorem egregium, and minimal surfaces.

MATH 5333: ANALYSIS. Prerequisites: Three semesters of calculus, or consent of instructor. A survey of the concepts of limit, continuity, differentiation and integration for functions of one variable and functions of several variables; selected applications are used to motivate and to illustrate the concepts.

I'm not concerned about analysis, because I did well in undergrad and have a strong interest in it. However, I took Cal III in 2008, so I'm a bit rusty in it. I do have my calculus textbook. What, specifically, should I review to prepare for the differential geometry class? Moreover, there is no assigned textbook; the professor will provide the notes. What's a good differential geometry textbook that I can use to supplement the class? Thanks.

 

[jedishrfu]

the wiki article references several books, Spivak comes up a lot so his book is probably pretty good.

http://en.wikipedia.org/wiki/Differential_geometry

Also MIT has a set of video lectures that you can watch online

http://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008/

 

[micromass]

Spivak is good, but it consists out of $5$ volumes, and the first volume is not relevant to the contents of your course. So you'll have a lot of reading to do to actually get to what your course covers.

The standard reference is Do Carmo, this is a very good and leisurely book. https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20

A very good book and quite easy is Pressley: https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20 It also contains the things you need to study without much extra fluff.

Another of my favorite is Bar, but this does have a very severe lack of exercises. https://www.amazon.com/dp/B00AKE1X8E/?tag=pfamazon01-20

 

[micromass]

What, specifically, should I review to prepare for the differential geometry class?

Especially multivariable differentiation is important here. Know what partial derivatives are. You should know the chain rule cold. Know theorems like when you can switch partial derivatives, etc. Be sure to know what a Jacobian and a Hessian is. Knowing how the general derivative is a linear transformation is good. If you have seen the inverse and implicit function theorems, then be sure to understand those well. The multivariable Taylor theorem can pop up occasionally.

Theory like multivariable integration and vector calculus seem to be less important for the course you're going to take. But you should definitely be able to calculate (multivariable) integrals and derivatives.

Series are not important. Sequences are also not important.

If you mention a topic here, I can tell you whether it's worth revising or not.

 

[Shackleford]

the wiki article references several books, Spivak comes up a lot so his book is probably pretty good.

http://en.wikipedia.org/wiki/Differential_geometry

Also MIT has a set of video lectures that you can watch online

http://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008/

Thanks. The Spivak collection appears to be a bit much for me, though. I forgot about the MIT online courses! Heh.

Spivak is good, but it consists out of $5$ volumes, and the first volume is not relevant to the contents of your course. So you'll have a lot of reading to do to actually get to what your course covers.

The standard reference is Do Carmo, this is a very good and leisurely book. https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20

A very good book and quite easy is Pressley: https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20 It also contains the things you need to study without much extra fluff.

Another of my favorite is Bar, but this does have a very severe lack of exercises. https://www.amazon.com/dp/B00AKE1X8E/?tag=pfamazon01-20

Thanks. I think the Pressley textbook is most appropriate for me, as the class is probably just a very basic introduction to the material. I suspect that most of the designated classes for this program are this way, which is precisely what I want. I don't intend on getting a Ph.D in mathematics or doing research. However, I will personally enjoy learning more mathematics beyond my undergraduate education.

Especially multivariable differentiation is important here. Know what partial derivatives are. You should know the chain rule cold. Know theorems like when you can switch partial derivatives, etc. Be sure to know what a Jacobian and a Hessian is. Knowing how the general derivative is a linear transformation is good. If you have seen the inverse and implicit function theorems, then be sure to understand those well. The multivariable Taylor theorem can pop up occasionally.

Theory like multivariable integration and vector calculus seem to be less important for the course you're going to take. But you should definitely be able to calculate (multivariable) integrals and derivatives.

Series are not important. Sequences are also not important.

If you mention a topic here, I can tell you whether it's worth revising or not.

Great. I actually did well in my Cal III and Vector Analysis courses, so most of the aforementioned topics are already familiar to me. I'm not familiar with the Hessian, though.

 

[PNutMargarine]

Great. I actually did well in my Cal III and Vector Analysis courses, so most of the aforementioned topics are already familiar to me. I'm not familiar with the Hessian, though.

It's just the determinant of the Jacobian

 

[micromass]

It's just the determinant of the Jacobian

No, it's not.

http://en.wikipedia.org/wiki/Hessian_matrix

 

[Shackleford]

No, it's not.

http://en.wikipedia.org/wiki/Hessian_matrix

I looked up the Jacobian in my calculus textbook, and it is simple when making a transformation with a single integral, e.g. x = g(u), dx=g'(u)dx. I tried to figure it out for the case involving two integrals - x=g(u,v) and y=h(u,v) - but couldn't make sense of it. The notation for the Jacobian is a bit strange: partial(x,y)/partial(u,v). It makes more sense in my vector analysis textbook where they derived the Jacobian for a linear transformation.