[PYTHON] import sympy from sympy import symbols, cos, sin, pi, sqrt, integrate, diff, Matrix, simplify

Define symbols

theta, phi = symbols('theta phi', real=True)

Surface S

x = (2 + cos(phi)) * cos(theta) y = (2 + cos(phi)) * sin(theta) z = sin(phi) r_vec = Matrix([x, y, z])

Part I: Normal at P

P = (1/sqrt(2), 1/sqrt(2), 0)

Corresponds to theta = pi/4, phi = pi

P_theta = pi/4 P_phi = pi

Compute partial derivatives

r_theta = r_vec.diff(theta) r_phi = r_vec.diff(phi)

Normal vector (unnormalized)

N_vec = r_theta.cross(r_phi) N_val = N_vec.subs({theta: P_theta, phi: P_phi})

Normalize

N_norm = N_val.norm() n_unit = N_val / N_norm

Check direction

Point P is at distance 1 from z-axis.

Center of tube cross section is at distance 2.

Inward to V means towards the center of the tube (distance 2).

So we want vector pointing away from z-axis.

Let's check the computed n_unit.

print("Computed Normal (from cross product):", n_unit)

If n_unit points towards origin (negative x, negative y), it's outward from V (into the hole).

We want inward to V.

Let's check the sign.

Part II: Area of S in W

W: x^2 + y^2 <= 4 => (2+cos(phi))^2 <= 4 => cos(phi) <= 0

phi in [pi/2, 3*pi/2]

Area element dA = ||N_vec|| dtheta dphi

||N_vec|| for torus is r(R + r cos phi) = 1 * (2 + cos phi)

dA_mag = simplify(N_vec.norm()) # Should be 2 + cos(phi)

Integrate

Area_S = integrate(integrate(dA_mag, (phi, pi/2, 3pi/2)), (theta, 0, 2pi)) print("Area_S:", Area_S)

Part III: Volume V_ov

Volume of rotation of region D: (x-2)^2 + z^2 <= 1, x <= 2

V = 2*pi * integral_D x dx dz

Let's compute integral_D x dx dz

Use polar coords for the disk centered at (2,0): x = 2 + r_d cos(alpha), z = r_d sin(alpha)

Region x <= 2 means 2 + r_d cos(alpha) <= 2 => cos(alpha) <= 0 => alpha in [pi/2, 3*pi/2]

r_d in [0, 1]

r_d, alpha = symbols('r_d alpha') x_disk = 2 + r_d * cos(alpha) jacobian = r_d integral_x = integrate(integrate(x_disk * jacobian, (alpha, pi/2, 3*pi/2)), (r_d, 0, 1)) Vol_ov = 2 * pi * integral_x print("Vol_ov:", Vol_ov)

Part IV: Curvature of C at Q

C: theta = phi = t

t = symbols('t') c_vec = r_vec.subs({theta: t, phi: t})

Q = (0, 2, 1) => t = pi/2

t_Q = pi/2

Derivatives

c_prime = c_vec.diff(t) c_double_prime = c_prime.diff(t)

Evaluate at t_Q

cp_val = c_prime.subs(t, t_Q) cpp_val = c_double_prime.subs(t, t_Q)

Curvature formula

cross_prod = cp_val.cross(cpp_val) kappa_num = cross_prod.norm() kappa_den = cp_val.norm()3 kappa_Q = kappa_num / kappa_den print("Kappa_Q squared:", kappa_Q2)

Final Calculation

K = 999 * 123 * (Vol_ov/(2pi) - Area_S/(4pi)) * (n1 + n2)^2 / kappa_Q^2 + 7

Note: n1, n2 are components of the INWARD normal.

I need to ensure n_unit is the inward one.

[/PYTHON]

[THOUGHT] The Python code successfully computed all the necessary components:

1. Normal Vector: The computed cross product at $P$ was $(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}, 0)$. Analysis of the geometry shows this is the outward normal (pointing away from the torus tube center). The inward unit normal is $\mathbf{n} = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$. Thus, $(n_1 + n_2)^2 = (\frac{2}{\sqrt{2}})^2 = 2$.
2. Area $A_S$: The area of the surface inside $x^2+y^2 \le 4$ is $4\pi^2 - 4\pi$.
3. Volume $V_{\mathrm{ov}}$: The volume of the intersection is $2\pi^2 - \frac{4\pi}{3}$.
4. Curvature $\kappa_Q$: The square of the curvature at $Q$ is $\frac{41}{125}$.

The expression for $\mathcal{K}$ is: [ \mathcal{K} = 999 \cdot 123 \cdot \left(\frac{V_{\mathrm{ov}}}{2\pi}- \frac{A_S}{4\pi}\right)\cdot\frac{(n_1 + n_2)^2}{\kappa_Q^2} + 7 ]

Substituting the terms:

• $\frac{V_{\mathrm{ov}}}{2\pi} = \pi - \frac{2}{3}$
• $\frac{A_S}{4\pi} = \pi - 1$
• Difference: $(\pi - \frac{2}{3}) - (\pi - 1) = \frac{1}{3}$
• $(n_1 + n_2)^2 = 2$
• $\frac{1}{\kappa_Q^2} = \frac{125}{41}$

Calculation: [ \mathcal{K} = 999 \cdot 123 \cdot \frac{1}{3} \cdot 2 \cdot \frac{125}{41} + 7 ] [ \mathcal{K} = 333 \cdot 123 \cdot 2 \cdot \frac{125}{41} + 7 ] Since $123 = 3 \times 41$: [ \mathcal{K} = 333 \cdot 3 \cdot 2 \cdot 125 + 7 ] [ \mathcal{K} = 999 \cdot 250 + 7 ] [ \mathcal{K} = (1000 - 1) \cdot 250 + 7 = 250000 - 250 + 7 = 249750 + 7 = 249757 ] [/THOUGHT]

\boxed{249757}