Easier to self-teach: differential geometry or complex analysis

In summary, complex analysis is a system of mathematics that deals with extending real numbers, while differential geometry is a course that is harder to learn but may be more applicable to theoretical physics. There are some applications of complex analysis in QFT, but differential geometry is more extensively used throughout physics. I would recommend taking differential geometry and self-learning complex analysis.
  • #1
jdstokes
523
1
Hi all,

I'm torn between taking complex analysis or differential geometry at the advanced third year level.

Which of these would you consider the easiest to self-learn or the least applicable to the study of theoretical physics?

I know that differential geometry shows up in general relativity but I'm not sure about relativistic quantum mechanics and other advanced theories.

Are there actually any serious uses of complex analysis other than contour integration?? It seems like complex analysis is all the hype but I've never actually seen it USED for anything other than this.

Thanks.
 
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  • #2
I can't really defend complex analysis, as in my physics courses, I've also used it only (explicitly, at least) for contour integration. Anyway, though I have only tasted a little of differential geometry, I think it is hard enough that you will want to follow a course on it. Complex analysis on the other hand, is mainly about a systematic build-up and running through lots of proofs, which I think you can also do yourself.

Though you can probably better take a course instead of self-study, if you really want to thoroughly understand a subject, I tend to recommend doing differential geometry and self-learn complex analysis, which was easier in my opinion (though my exam grade shows it can still be trickier than you think :tongue2:)
 
  • #3
jdstokes said:
Hi all,

I'm torn between taking complex analysis or differential geometry at the advanced third year level.

Which of these would you consider the easiest to self-learn or the least applicable to the study of theoretical physics?

I know that differential geometry shows up in general relativity but I'm not sure about relativistic quantum mechanics and other advanced theories.

Are there actually any serious uses of complex analysis other than contour integration?? It seems like complex analysis is all the hype but I've never actually seen it USED for anything other than this.

Thanks.

I am just starting to sink complex analysis in. It is pretty amazing to extend the real numbers. I'd say its worth all the hype. It systematically extends our system but gives non trivial results. I wonder if the complex system is it or does it lie inside an even larger system?
 
  • #4
I've taken classes in both courses, and to me complex analysis was a lot easier to learn than differential geometry. As to which will benefit you more, I would suggest that, whilst complex analysis may be useful later on, you will definitely need differential geometry. Therefore, I would advise you to take DG and self-learn complex analysis.
 
  • #5
The system we have now, the complex one, has been proven by mathematicians to be "it" no need for more generalization
 
  • #6
nealh149 said:
The system we have now, the complex one, has been proven by mathematicians to be "it" no need for more generalization

Huh? :confused:
 
  • #7
^ me too...
 
Last edited:
  • #8
I think he means that the complex numbers are closed, unlike the sets within the complex numbers.
 
  • #9
Hi,

I agree with the previous posters. My brief exposure to differential geometry was frightening, while I found complex analysis to be natural, fascinatingly beautiful and relatively easier. It is something everyone should learn simply to appreciate the beauty of pure mathematics. I think there are some applications of complex analysis in QFT, like the Feynman path integral formulation, but differential geometry is used more extensively throughout physics. Though GTR is the place it is traditionally used, I believe it is also be in the coordinate free formulation of classical mechanics based on calculus on manifolds. This is all second-hand knowledge though.

Molu
 

1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces using calculus and linear algebra. It is used to understand the geometric properties of objects in space and their relationships with each other.

2. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It involves the use of calculus and algebra to analyze the properties of complex functions and their behavior.

3. Which is easier to self-teach: differential geometry or complex analysis?

This is subjective and depends on the individual's background and understanding of mathematics. However, many people find complex analysis easier to self-teach because it is more closely related to traditional calculus and algebra, while differential geometry involves more abstract concepts and requires a strong understanding of vector calculus and linear algebra.

4. What are the applications of differential geometry?

Differential geometry has many applications in various fields, including physics, engineering, and computer graphics. It is used to study the properties of curves and surfaces in space, which has practical applications in designing and analyzing structures and objects in real-world scenarios.

5. How can I self-teach differential geometry or complex analysis?

There are many resources available for self-teaching these subjects, such as textbooks, online courses, and video tutorials. It is recommended to have a strong foundation in calculus and linear algebra before attempting to learn these subjects. It is also helpful to practice solving problems and to seek guidance from experts or study groups if needed.

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