Top Student at Their Peak-Chapter 221 - 107: Presenting Roses, Hands Retain Fragrance

Yuan Zhengxin was very satisfied with Qiao Yu’s performance.

It couldn’t even be described as just very satisfied... The old man regretted, regretted that Qiao Yu wasn’t his own grandson!

How wonderful it would be if this child were his own grandson.

Especially at this moment, as the sun was setting in the west, a ray of sunlight leaked through the window, bathing the child who was fully absorbed, making Qiao Yu resemble an angel coated in light.

The more he looked, the more he adored him.

To what extent did he adore him?

Let’s put it this way: the old man got up to go to the bathroom. When he returned, he made some noise, alerting Qiao Yu. Qiao Yu raised his head blankly, his eyes unfocused, glanced in the direction of the noise, then immediately lowered his head to continue writing and drawing on the manuscript paper.

In fact, it didn’t affect him at all, but for the rest of the time, the old man didn’t dare to even drink water, fearing he might have to go to the bathroom again and disturb Qiao Yu’s thinking. He even softened his breathing a little.

Time quietly passed as the old man gazed at Qiao Yu from time to time, and finally, the young boy next to him stretched out a big lazy stretch.

Only the old man’s kind gaze turned solemn because of this stretch.

"How was the effect of self-study here this afternoon?" the old man asked seriously.

Qiao Yu nodded vigorously, not caring about the old man’s seriousness at all, and replied loudly: "Reporting, Master Grandpa, it’s very good! I’m not bragging, I feel like I’m about to fully grasp Nidham’s ’Visualization of Differential Geometry and Forms’!"

"Oh? You haven’t asked me a single question, so I have to test you." The old man said seriously.

"No problem. You can test me on anything from this book except the last bit of content!" Qiao Yu confidently stood up and handed the book to the old man beside him.

Yuan Zhengxin took the book Qiao Yu handed over, flipped through it casually, then set it aside and asked, "Nidham believes that geometric intuition can influence the study and research of differential geometry. How do you view his opinion?"

Qiao Yu answered loudly: "I believe that Nidham’s opinion is correct. First of all, geometric intuition is a concept emphasized in Nidham’s book, referring to the ability to understand geometric objects and their properties through visual and spatial perception.

This intuition helps us form concrete spatial images when faced with abstract mathematical concepts, making it easier to understand complex structures and relationships. Especially when dealing with complex manifolds and surfaces, maintaining a focus on geometric intuition is crucial.

Through geometric intuition, researchers can propose and verify hypotheses more effectively, thereby making progress in theoretical derivations and applications. Let me give an example..."

Qiao Yu spoke with a powerful, clear, and steady voice. His answers were well-organized, logically distinct, with points, analysis, and examples. Yuan Zhengxin roughly understood why this child could get so many perfect scores and why Zhang Shuwen concluded that this child had remarkable talent after just one glance.

In this way, in a quiet yet classical villa, the elderly and the youth, one asking, one answering, seemed to traverse a hundred years of time, much like in ancient private schools where students faced their teachers.

The old man slightly nodded, finally showing a smile, and said: "One last question, how do you understand the relationship between the tangent bundle and tangent space on a manifold, and discuss the impact of the geometric structure of the tangent bundle on the manifold."

Without hesitation, Qiao Yu answered loudly: "The tangent bundle is the collection of all tangent spaces on a manifold, often denoted by TM, where each tangent space TpM corresponds to a point p on the manifold M. The tangent space is a vector space, while the tangent bundle combines all these tangent spaces.

The geometric structure of the tangent bundle affects the manifold by allowing us to define vector fields and differential forms. By studying the tangent bundle, we can analyze the local and global properties of the manifold. For example, we can discuss smooth maps, the topological structure of manifolds, and how to define geometric quantities on manifolds.

Oh, and the tangent bundle also plays a crucial role in defining connections and calculating curvatures, thereby helping us understand the curvature properties of the manifold..."

The old man asked a total of six questions, covering all aspects of the sections of the book that Qiao Yu had read. frёewebnoѵel.ƈo๓

Qiao Yu’s answers left him no room for criticism, really, every detail was practically perfect.

"Great! Well, it’s just past five, there’s still some time before dinner, do you dare to challenge a practice test on differential geometry I have here?"

"No problem at all! Master Grandpa, if you want to evaluate me, just bring it on!" Qiao Yu still appeared full of confidence.

"So confident? Looks like I have to pick a slightly harder one." With that, Yuan Zhengxin stood up, took out a folder from the bookshelf behind him, and selected a test paper to hand over.

Qiao Yu took the test paper, but didn’t start right away. Instead, he first read through the paper from beginning to end, then made a distressed face and said: "Master Grandpa, I can do the test paper, but can I make a small request?"

"Oh? What request?" the old man patiently asked.

Qiao Yu, holding the paper, said: "You see, some questions on this paper aren’t difficult per se, but if I use differential geometry methods, they’re too cumbersome, like this third question. To find the curvature and torsion of a parametric curve at point P, if I use differential geometry, I have to calculate derivatives first, then solve using curvature and torsion formulas.

But if I use algebra methods, a direct parameter substitution method would do, like x=t, y=t^2, z=t^3, leveraging this relation to directly deduce the geometric properties of the curve, then substitute into the geometric formula, saving the time of calculating the cross product."