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問題

mmexport1746485099931.jpgmmexport1746485099931.jpg
mmexport1746485097797.jpgmmexport1746485097797.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)

思考

It seems like the user may have used Gauss or the divergence theorem incorrectly by introducing a factor of 12\frac{1}{2} in the boundary integral. The formula they started with seems similar to Stokes' theorem, but the factor of 12\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 −∬y dx dy-\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 (∇×F)⋅n(\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 1/21/2, and they also lost several components in the determinant, reducing the integrand incorrectly. Additionally, they didn't properly transition from the surface element dSdS, missing an extra factor aa, leading to an a2a^2 instead of a3a^3. I’ll explain these issues step-by-step.

Thought for 6m 21s

回答

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 −π4a3-\dfrac{\pi}{4}a^{3} 变成了你算出来的 −a2(θ2+sin⁡θ4)-a^{2}\bigl(\tfrac{\theta}{2}+\tfrac{\sin\theta}{4}\bigr)。

1 斯托克斯公式前面的 “12\tfrac12” 多写了

在公式

I=∮Γ(P dx+Q dy+R dz)=12∬Σ∣cos⁡αcos⁡βcos⁡γ∂/∂x∂/∂y∂/∂zPQR∣dSI=\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 12\tfrac12，而标准公式前面其实是 1。
虽然后面你又把 −2-2 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=∬Σ∣cos⁡αcos⁡βcos⁡γ∂/∂x∂/∂y∂/∂zPQR∣dS.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 xcos⁡βx\cos\beta 那一项

对于
P=y2,  Q=z2,  R=x2P=y^{2},\;Q=z^{2},\;R=x^{2}
应得

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

行列式展开后

(curl⁡F) ⁣⋅ ⁣n=−2(zcos⁡α+xcos⁡β+ycos⁡γ),n=(x,y,z)a.(\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 −2zcos⁡α-2z\cos\alpha，把 −2xcos⁡β-2x\cos\beta 给漏掉了（−2ycos⁡γ-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 yy，而标准答案里是 x+yx+y。

3 把球面面积元 dSdS 换成 dx dydx\,dy 时少乘了 az\dfrac{a}{z}

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

dS=dx dycos⁡γ=az dx dy.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 )az dx dyz(\,x+y\,)\frac{a}{z}\,dx\,dy

化成了 (x+y) dx dy(x+y)\,dx\,dy，最终带来 a3a^{3}。
你把这一系数当成 1，结果只剩 a2a^{2}。

修正后的一步到位

  1. 行列式得到 (curl⁡F) ⁣⋅ ⁣n=−2a(xz+xy+yz).(\operatorname{curl}\mathbf F)\!\cdot\!\mathbf n =-\frac{2}{a}\bigl(xz+xy+yz\bigr).
  2. 乘 dS=a dx dy/zdS=a\,dx\,dy/z 后，xzxz 与 zz 抵消，yzyz 与 zz 抵消，只剩 −2(x+y) dx dy.-2(x+y)\,dx\,dy.
  3. 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}).
  4. 极坐标积分 I=−2∫−π/2π/2 ⁣ ⁣ ⁣ ⁣∫0acos⁡θ(rcos⁡θ+rsin⁡θ) r dr dθ=−π4a3.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。

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