Accepted in the Master of Arts in Mathematics program

In summary, the conversation mainly discusses the prerequisites and recommended textbooks for the differential geometry class, including topics that should be reviewed beforehand such as multivariable differentiation and integration. The conversation also briefly mentions other relevant concepts such as the Jacobian and Hessian matrices.
  • #1
Shackleford
1,656
2
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.
 
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  • #3
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
 
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  • #4
Shackleford said:
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.
 
  • #5
jedishrfu said:
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.

micromass said:
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.

micromass said:
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.
 
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  • #6
Shackleford said:
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
 
  • #8
micromass said:

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.
 

1. What are the admission requirements for the Master of Arts in Mathematics program?

The admission requirements for the Master of Arts in Mathematics program vary depending on the university, but generally, applicants must have a bachelor's degree in mathematics or a related field with a minimum GPA of 3.0. Some programs may also require applicants to have completed specific undergraduate courses in mathematics.

2. What kind of career opportunities are available for graduates of the Master of Arts in Mathematics program?

Graduates of the Master of Arts in Mathematics program have a wide range of career opportunities available to them. These can include roles in academia, research, government agencies, and industries such as finance, technology, and data analysis. Some common job titles for graduates include mathematician, data analyst, actuary, and statistician.

3. How long does it take to complete the Master of Arts in Mathematics program?

The length of the Master of Arts in Mathematics program can vary depending on the university and whether the student is attending full-time or part-time. Generally, the program takes 1-2 years to complete. However, some schools may offer accelerated programs that can be completed in as little as one year.

4. Can I pursue a Master of Arts in Mathematics if my undergraduate degree is not in mathematics?

While many students who pursue a Master of Arts in Mathematics have a bachelor's degree in mathematics, it is not always a requirement. Some universities may accept students with a strong background in a related field, such as physics or engineering. However, these students may be required to take additional prerequisite courses before beginning the program.

5. Is a thesis or final project required for the Master of Arts in Mathematics program?

The requirements for a thesis or final project vary by university and program. Some schools may require students to complete a thesis, while others may offer the option to complete a capstone project or take additional coursework. It is important to research the specific program requirements before applying to ensure it aligns with your academic and career goals.

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