问题出在哪里

생성일: 2025년 5월 5일

답변에 사용된 모델 O3 by Chat01

질문

mmexport1746485099931.jpg

mmexport1746485097797.jpg

问题出在哪里

Python

python
from PIL import Image
import matplotlib.pyplot as plt

img1 = Image.open('/mnt/data/mmexport1746485099931.jpg')
plt.imshow(img1); plt.axis('off')

(-0.5, 1178.5, 806.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img1.crop((0,300,1100,650)))
plt.axis('off')

(-0.5, 1099.5, 349.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img1.crop((0,550,1100,900)))
plt.axis('off')

(-0.5, 1099.5, 349.5, -0.5)

Python

python
img2 = Image.open('/mnt/data/mmexport1746485097797.jpg')
plt.imshow(img2); plt.axis('off')

(-0.5, 1919.5, 1079.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img2.crop((200,100,1100,650)))
plt.axis('off')

(-0.5, 899.5, 549.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img2.crop((200,550,1100,1100)))
plt.axis('off')

(-0.5, 899.5, 549.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img2.crop((550,450,1100,1050)))
plt.axis('off')

(-0.5, 549.5, 599.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img2.crop((550,800,1100,1400)))
plt.axis('off')

(-0.5, 549.5, 599.5, -0.5)

Python

python
plt.figure(figsize=(6,8))
plt.imshow(img2.crop((200,350,900,1100)))
plt.axis('off')

(-0.5, 699.5, 749.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img2.crop((200,0,900,300)))
plt.axis('off')

(-0.5, 699.5, 299.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img1.crop((0,0,1100,300)))
plt.axis('off')

(-0.5, 1099.5, 299.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img1.crop((0,800,1100,1200)))
plt.axis('off')

(-0.5, 1099.5, 399.5, -0.5)

Python

python
plt.figure(figsize=(6,4))
plt.imshow(img2.crop((0,0,1500,250)))
plt.axis('off')

(-0.5, 1499.5, 249.5, -0.5)

Python

python
plt.figure(figsize=(10,3))
plt.imshow(img2.crop((300,0,1300,250)))
plt.axis('off')

(-0.5, 999.5, 249.5, -0.5)

Python

python
plt.figure(figsize=(10,3))
plt.imshow(img2.crop((900,0,1400,250)))
plt.axis('off')

(-0.5, 499.5, 249.5, -0.5)

Python

python
plt.figure(figsize=(10,3))
plt.imshow(img2.crop((200,230,800,500)))
plt.axis('off')

(-0.5, 599.5, 269.5, -0.5)

Python

python
plt.figure(figsize=(10,3))
plt.imshow(img2.crop((650,230,1250,500)))
plt.axis('off')

(-0.5, 599.5, 269.5, -0.5)

Python

python
plt.figure(figsize=(10,3))
plt.imshow(img2.crop((1000,200,1500,350)))
plt.axis('off')

(-0.5, 499.5, 149.5, -0.5)

Python

python
plt.figure(figsize=(10,3))
plt.imshow(img2.crop((700,250,1300,550)))
plt.axis('off')

(-0.5, 599.5, 299.5, -0.5)

Python

python
import sympy as sp
θ = sp.symbols('θ', real=True)
expr = (sp.cos(θ) + sp.sin(θ))* sp.cos(θ)**3
res = sp.integrate(expr, (θ, -sp.pi/2, sp.pi/2))
res_simplified = sp.simplify(res)
res_simplified

3*pi/8

Python

python
plt.figure(figsize=(8,3))
plt.imshow(img2.crop((950,500,1500,800)))
plt.axis('off')

(-0.5, 549.5, 299.5, -0.5)

Python

python
plt.figure(figsize=(8,3))
plt.imshow(img2.crop((950,500,1500,800)))
plt.axis('off')

(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)

Python

python
plt.figure(figsize=(8,3))
plt.imshow(img2.crop((950,500,1500,800)))
plt.axis('off')

(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)



(-0.5, 549.5, 299.5, -0.5)

Python

python
plt.figure(figsize=(8,3))
plt.imshow(img2.crop((950,500,1500,800)))
plt.axis('off')

(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)



(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)

Python

python
plt.figure(figsize=(8,3))
plt.imshow(img2.crop((950,500,1500,800)))
plt.axis('off')

(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)



(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)



(-0.5, 549.5, 299.5, -0.5)

Python

python
plt.figure(figsize=(8,3))
plt.imshow(img2.crop((950,500,1500,800)))
plt.axis('off')

(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)



(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)



(-0.5, 549.5, 299.5, -0.5)

(-0.5, 549.5, 299.5, -0.5)

text

![](https://files.openai01.net/python-generations/files/file-XiAtepLxQM7BvuhFp5jGMs)

Python

python
plt.figure(figsize=(8,3))
plt.imshow(img2.crop((950,800,1500,1150)))
plt.axis('off')

(-0.5, 549.5, 349.5, -0.5)

Python

python
θ=sp.symbols('θ', real=True)
expr=sp.sin(θ)*sp.cos(θ)**3
res=sp.integrate(expr, (θ, -sp.pi/2, sp.pi/2))
res

0

Python

python
plt.figure(figsize=(10,4))
plt.imshow(img2.crop((900,150,1500,300)))
plt.axis('off')

(-0.5, 599.5, 149.5, -0.5)

Python

python
plt.figure(figsize=(10,6))
plt.imshow(img2.crop((500,0,900,250)))
plt.axis('off')

(-0.5, 399.5, 249.5, -0.5)

Python

python
plt.figure(figsize=(10,6))
plt.imshow(img2.crop((500,200,900,500)))
plt.axis('off')

(-0.5, 399.5, 299.5, -0.5)

Python

python
plt.figure(figsize=(10,6))
plt.imshow(img2.crop((800,0,1200,500)))
plt.axis('off')

(-0.5, 399.5, 499.5, -0.5)

Python

python
plt.figure(figsize=(10,6))
plt.imshow(img2.crop((800,450,1200,900)))
plt.axis('off')

(-0.5, 399.5, 449.5, -0.5)

Python

python
plt.figure(figsize=(10,6))
plt.imshow(img2.crop((800,800,1200,1050)))
plt.axis('off')

(-0.5, 399.5, 249.5, -0.5)

Python

python
plt.figure(figsize=(10,6))
plt.imshow(img2.crop((1200,500,1600,900)))
plt.axis('off')

(-0.5, 399.5, 399.5, -0.5)

Python

python
plt.figure(figsize=(6,3))
plt.imshow(img1.crop((0,320,1100,550)))
plt.axis('off')

(-0.5, 1099.5, 229.5, -0.5)

Python

python
plt.figure(figsize=(6,3))
plt.imshow(img1.crop((200,380,800,550)))
plt.axis('off')

(-0.5, 599.5, 169.5, -0.5)

Python

python
plt.figure(figsize=(8,6))
plt.imshow(img2.crop((300,400,700,700)))
plt.axis('off')

(-0.5, 399.5, 299.5, -0.5)

Python

python
plt.figure(figsize=(8,6))
plt.imshow(img2.crop((300,650,700,1100)))
plt.axis('off')

(-0.5, 399.5, 449.5, -0.5)

Python

python
plt.figure(figsize=(8,4))
plt.imshow(img2.crop((200,100,500,300)))
plt.axis('off')

(-0.5, 299.5, 199.5, -0.5)

Python

python
plt.figure(figsize=(8,4))
plt.imshow(img2.crop((450,100,850,300)))
plt.axis('off')

(-0.5, 399.5, 199.5, -0.5)

Python

python
plt.figure(figsize=(8,4))
plt.imshow(img2.crop((850,100,1200,300)))
plt.axis('off')

(-0.5, 349.5, 199.5, -0.5)

Python

python
plt.figure(figsize=(8,4))
plt.imshow(img2.crop((1100,100,1500,300)))
plt.axis('off')

(-0.5, 399.5, 199.5, -0.5)

사고

It seems like the user may have used Gauss or the divergence theorem incorrectly by introducing a factor of $\frac{1}{2}$ in the boundary integral. The formula they started with seems similar to Stokes' theorem, but the factor of $\frac{1}{2}$ isn't typically there for this kind of integration. It looks like they miscopied portions of the formula. Then, they seem to have switched the integral components, ending with an integral of $-\iint y \, dx \, dy$ , which wasn't quite aligned with their earlier set-up.

It looks like the user missed a key term in their calculation, which caused them to miscompute several factors and get the wrong answer. They incorrectly evaluated the term $(\nabla \times F) \cdot n$ , missed a projection factor, and mishandled the surface area element ratio. The user’s formula had an unnecessary factor of $1/2$ , and they also lost several components in the determinant, reducing the integrand incorrectly. Additionally, they didn't properly transition from the surface element $dS$ , missing an extra factor $a$ , leading to an $a^2$ instead of $a^3$ . I’ll explain these issues step-by-step.

Thought for 6m 21s

답변

我把你的笔记和标准解法一行一行对照了一下，发现两处关键疏漏导致最后的系数、被积函数都错了——结果自然就从 $-\dfrac{\pi}{4}a^{3}$ 变成了你算出来的 $-a^{2}\bigl(\tfrac{\theta}{2}+\tfrac{\sin\theta}{4}\bigr)$ 。

1 斯托克斯公式前面的 “ $\tfrac12$ ” 多写了

在公式

I=\oint_{\Gamma}(P\,dx+Q\,dy+R\,dz) =\frac12\iint_{\Sigma} \begin{vmatrix} \cos\alpha&\cos\beta&\cos\gamma\\ \partial/\partial x&\partial/\partial y&\partial/\partial z\\ P&Q&R \end{vmatrix}dS

你把系数写成了 $\tfrac12$ ，而标准公式前面其实是 1。
虽然后面你又把 $-2$ 提出来抵消了一半，但这一步还是容易让后面的换元心里没谱——建议直接照书写成

I=\iint_{\Sigma} \begin{vmatrix} \cos\alpha&\cos\beta&\cos\gamma\\ \partial/\partial x&\partial/\partial y&\partial/\partial z\\ P&Q&R \end{vmatrix}dS .

2 求行列式时漏掉了 $x\cos\beta$ 那一项

对于
$P=y^{2},\;Q=z^{2},\;R=x^{2}$
应得

\operatorname{curl}\mathbf F =(-2z,\,-2x,\,-2y).

行列式展开后

(\operatorname{curl}\mathbf F)\!\cdot\!\mathbf n =-2\bigl(z\cos\alpha+x\cos\beta+y\cos\gamma\bigr), \quad \mathbf n=\frac{(x,y,z)}{a}.

你只抄下了 $-2z\cos\alpha$ ，把 $-2x\cos\beta$ 给漏掉了（ $-2y\cos\gamma$ 似乎也在后面被误删）。
正是这一步让你最后得出的被积函数只剩一个 $y$ ，而标准答案里是 $x+y$ 。

3 把球面面积元 $dS$ 换成 $dx\,dy$ 时少乘了 $\dfrac{a}{z}$

因为外法向量的 $z$ -方向余弦是 $\cos\gamma=\dfrac{z}{a}$ ，所以

dS=\frac{dx\,dy}{\cos\gamma}=\frac{a}{z}\,dx\,dy.

标准解里正是利用这一点把

z(\,x+y\,)\frac{a}{z}\,dx\,dy

化成了 $(x+y)\,dx\,dy$ ，最终带来 $a^{3}$ 。
你把这一系数当成 1，结果只剩 $a^{2}$ 。

修正后的一步到位

行列式得到 $(\operatorname{curl}\mathbf F)\!\cdot\!\mathbf n =-\frac{2}{a}\bigl(xz+xy+yz\bigr).$
乘 $dS=a\,dx\,dy/z$ 后， $xz$ 与 $z$ 抵消， $yz$ 与 $z$ 抵消，只剩 $-2(x+y)\,dx\,dy.$
投影区域：(x^{2}+y^{2}\le a x;\Longrightarrow;0\le r\le a\cos\theta,; -\tfrac{\pi}{2}\le\theta\le\tfrac{\pi}{2}).
极坐标积分 $I=-2\int_{-\pi/2}^{\pi/2}\!\!\!\! \int_{0}^{a\cos\theta} (r\cos\theta+r\sin\theta)\,r\,dr\,d\theta =-\frac{\pi}{4}a^{3}.$

把这三个地方改正，你的推导就能和标准答案完全一致了。

질문

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사고

답변

1 斯托克斯公式前面的 “12\tfrac1221​” 多写了

2 求行列式时漏掉了 xcos⁡βx\cos\betaxcosβ 那一项

3 把球面面积元 dSdSdS 换成 dx dydx\,dydxdy 时少乘了 az\dfrac{a}{z}za​

修正后的一步到位

이 Q&A 공유하기

1 斯托克斯公式前面的 “ $\tfrac12$ ” 多写了

2 求行列式时漏掉了 $x\cos\beta$ 那一项

3 把球面面积元 $dS$ 换成 $dx\,dy$ 时少乘了 $\dfrac{a}{z}$