“I really have to work hard, I haven’t touched my books for almost half a semester. If I don’t start now, it’ll be too late.” Tossing his bath towel over his shoulder, William Carter stood up from his chair.

Seeing that William Carter was about to take a shower, David Foster suddenly spoke up, “I have a problem here that I can’t solve. Can you help me take a look?”

Oh, that’s rare.

There’s actually a problem you can’t solve?

“Let me see it.”

William Carter reached out his hand. David Foster pushed up his glasses, handed over the exercise book, and pointed at the circled problem, saying, “It’s this one.”

“An integration problem? This shouldn’t be too hard…” After glancing at the problem, William Carter was surprised to find it was a type he hadn’t seen before. His interest was piqued, and he wasn’t in a hurry to shower anymore. He sat back down with the exercise book, picked up his pen, and started scribbling on the scratch paper.

If it were before, any problem that could stump David Foster would definitely be unsolvable for him. But for some reason, at this moment, he didn’t have the slightest thought of “impossible.”

Watching William Carter calculate for a while, David Foster felt relieved, secretly mocking him in his heart: a slacker is always a slacker, no matter how much you pretend, you’re still a slacker.

As for this problem, it was true he couldn’t solve it, but actually, there was a reference answer, and it even included a detailed solution. The reason he asked William Carter about this problem wasn’t out of genuine curiosity, but just to “gauge the competition.”

Even if he had to ask about a math problem, he wouldn’t go to William Carter, the class slacker.

Thinking this, David Foster said, “Why don’t you copy the problem and work on it? I’ll try to figure out the next one.”

The implication being: you probably can’t solve it anyway, so I won’t waste your time.

However, William Carter’s response was completely unexpected to David Foster.

“…No need, I’ve solved it.”

S-solved it?!

David Foster’s eyes widened, his eyeballs nearly popping out in shock.

“Yeah, you heard right.” Twirling the pen in his hand, William Carter explained the equations on the scratch paper, “It’s a classic double integral problem. The first step is to convert the rectangular coordinates into polar coordinates to start the calculation. Because of the symmetry of the interval, after calculation, this part of the equation can be simplified to cotx²…”

“In the end, it becomes the integral of cscx²! Then substitute into the original function!” David Foster’s pupils contracted slightly, sharply seeing through the core idea of the solution. The following steps were as smooth as water flowing downstream—just keep calculating, and there’s no difficulty at all.

Damn, why didn’t I think of that step…

“Bingo! That’s correct.” William Carter nodded with a smile, thinking to himself, “This kid is teachable.”

“Thanks… Let me borrow your scratch paper, I’ll take it back and study it.”

“Go ahead, you’re welcome!” William Carter waved his hand generously, stood up, and walked toward the bathroom.

Taking William Carter’s scratch paper back to his seat, David Foster pushed up his glasses, stared at the problem, frowned slightly, and fell into deep thought.

Although the solution was novel, the process was written clearly, and the knowledge points involved weren’t beyond the syllabus. Even without William Carter’s explanation, he could easily understand it.

But what he couldn’t figure out was, how did that guy William Carter come up with it?

And in such a short time…

Could it be that he’d done similar problems before?

David Foster thought about it and figured that was the only possibility. Otherwise, he really couldn’t imagine how this slacker, who was always busy with part-time jobs, could outpace him—the class’s acknowledged top student—when it came to solving problems.

Turning to the back of the exercise book, David Foster looked at the answer and solution, and was stunned, muttering to himself.

“Impossible…”

The answer was completely correct… but that wasn’t the point.

The point was, William Carter’s solution was actually simpler than the standard answer! The answer didn’t even consider the unorthodox approach of converting rectangular coordinates to polar coordinates, but instead directly applied formulas to split the integral twice, which made the calculations look unusually complicated.

And that was one of the methods he had considered…

How could this be!

David Foster bit his lip.

He suddenly started to doubt his life a little.

Chapter 8: The Optimal Inverse Theory of Linear Operators and Linear Functionals

Ever since he got the system, William Carter felt his life had become incredibly organized. Early in the morning, he carried his second-hand laptop to the library. For him, this experience was completely unprecedented.

As usual, in his old spot, William Carter opened his laptop, found a power outlet by the wall, plugged it in, thought for a while, and then typed a few big words into a Word document.

[The Optimal Inverse Theory of Linear Operators and Linear Functionals]

[Abstract: This paper studies several types of inverse problems for linear functionals and linear operators when all or partial information is given. It introduces the basic results of the optimal inverse theory, especially the construction of optimal inverse methods…]

He came up with this topic last night while lying in bed flipping through his notes. Their mathematical analysis teacher, Professor Turner, had casually mentioned it when talking about the Fourier inversion formula—one of the fields in contemporary mathematics that isn’t exactly popular, but is quite cutting-edge.

Combining what he found online, William Carter drafted this topic.

As for how to solve this problem?

Hmm…