Proofs In Advance Calculus

[NukeEng101]

I'm a sophomore at Rensselaer Polytechnic Institute and I'm taking MATH 4600 which is Advanced Calculus. I love the class and it is very interesting, we're taking what we learned in Multivariable Calculus, but just at a much higher level. However, my teacher does a lot of proofs behind why it's true and a lot of theory. I am having trouble understanding the proofs of the implicit theorem, the inversion theorem, and even just partial derivatives.

I know how to do the problems with actual examples, it's just the theory that's a little weak and I really want to improve on it to fully appreciate it a lot more. Thanks to anyone who could help give me pointers to understand the proofs a lot more!

 

[Robert1986]

Read more proofs.

 

[mathwonk]

give us more detail so we are not left with just the option of giving you the full proof of the inverse function theorem. or read spivak, calculus on manifolds.

 

[A. Bahat]

And make sure you try to see the "big picture" behind a proof. This makes all of the details usually easy. Ask yourself why each step is needed; what's the idea behind it? For example, the inverse function theorem is basically a corollary of the contraction mapping theorem. [You could take this line of thought even further. The contraction mapping theorem applies in any metric space, not just Euclidean space. By considering function spaces, we get the existence/uniqueness theorem for ODEs. Or by considering arbitrary Banach spaces we can generalize the inverse function theorem, necessary for infinite-dimensional settings like functional analysis. So it's really important that you get the fundamental idea behind the proof to be able to apply it in new situations like these!]

 

[blue_raver22]

As a fellow scientist, I understand the importance of understanding the underlying theory and proofs behind mathematical concepts. It is great to hear that you are enjoying your advanced calculus class and are eager to improve your understanding of the proofs.

First of all, don't be discouraged if you are having trouble understanding the proofs. Advanced calculus can be challenging and it takes time and practice to fully grasp the concepts. One way to improve your understanding is to actively engage with the material. This could include taking detailed notes during lectures, asking questions during class, and working through practice problems on your own.

In addition, I would recommend seeking out additional resources such as textbooks, online lectures, or tutoring services. These resources can provide alternative explanations and examples that may help you better understand the proofs.

Another helpful strategy is to break down the proofs into smaller, more manageable parts and focus on understanding each step before moving on to the next. It can also be helpful to make connections between the proofs and the concepts you learned in multivariable calculus. This can help you see the bigger picture and understand the relevance of the proofs.

Lastly, don't hesitate to reach out to your teacher for additional help or clarification. They are there to support your learning and can provide valuable insights and explanations.

Keep up the hard work and don't get discouraged. With dedication and persistence, I am sure you will improve your understanding of the proofs in advance calculus. Good luck!