[THOUGHT] Let me work through this problem step by step.

First, let me understand the surface S. The parametrization is:

• x(θ,φ) = (cos φ + 2)cos θ
• y(θ,φ) = (cos φ + 2)sin θ
• z(θ,φ) = sin φ

This is a torus with major radius 2 and minor radius 1.

Let me solve each part:

Part I: Unit normal vector at P P = (1/√2, 1/√2, 0)

First, I need to find θ and φ at point P.

• z = sin φ = 0 → φ = 0 or π
• If φ = 0: cos φ + 2 = 3, so x = 3cos θ, y = 3sin θ
• x = y = 1/√2 means 3cos θ = 3sin θ = 1/√2, so cos θ = sin θ = 1/(3√2)
• But cos²θ + sin²θ = 1/(18) + 1/(18) = 1/9 ≠ 1

Let me reconsider. If φ = π: cos φ + 2 = 1, so x = cos θ, y = sin θ

• x = y = 1/√2 means cos θ = sin θ = 1/√2, so θ = π/4

So at P: θ = π/4, φ = π

Part II: Area of S in region W W is x² + y² ≤ 4. On the torus, x² + y² = (cos φ + 2)². So (cos φ + 2)² ≤ 4 means cos φ + 2 ≤ 2 (since cos φ + 2 > 0), so cos φ ≤ 0. This means π/2 ≤ φ ≤ 3π/2.

Part III: Volume of V ∩ W

Part IV: Curvature at Q Q = (0, 2, 1). Setting θ = φ:

• x = (cos θ + 2)cos θ
• y = (cos θ + 2)sin θ
• z = sin θ

At Q: z = sin θ = 1 → θ = π/2 Check: x = (0 + 2)·0 = 0 ✓, y = (0 + 2)·1 = 2 ✓

Let me compute all of this with Python. [/THOUGHT] [PYTHON] import numpy as np from sympy import *

Define symbols

theta, phi, t = symbols('theta phi t', real=True)

Parametrization of the torus

x_param = (cos(phi) + 2) * cos(theta) y_param = (cos(phi) + 2) * sin(theta) z_param = sin(phi)

print("Surface parametrization:") print(f"x = {x_param}") print(f"y = {y_param}") print(f"z = {z_param}")

Part I: Find theta and phi at P = (1/sqrt(2), 1/sqrt(2), 0)

z = sin(phi) = 0 => phi = 0 or pi

If phi = 0: r = cos(0) + 2 = 3, so x = 3cos(theta), y = 3sin(theta)

x = y = 1/sqrt(2) => 3cos(theta) = 1/sqrt(2) => cos(theta) = 1/(3sqrt(2))

But then cos^2 + sin^2 != 1 if cos = sin = 1/(3*sqrt(2))

If phi = pi: r = cos(pi) + 2 = 1, so x = cos(theta), y = sin(theta)

x = y = 1/sqrt(2) => cos(theta) = sin(theta) = 1/sqrt(2) => theta = pi/4

theta_P = pi/4 phi_P = pi

Verify

x_P = (cos(phi_P) + 2) * cos(theta_P) y_P = (cos(phi_P) + 2) * sin(theta_P) z_P = sin(phi_P) print(f"\nPart I: At P, theta = pi/4, phi = pi") print(f"x = {simplify(x_P)}, y = {simplify(y_P)}, z = {simplify(z_P)}")

Compute partial derivatives

r_theta = Matrix([diff(x_param, theta), diff(y_param, theta), diff(z_param, theta)]) r_phi = Matrix([diff(x_param, phi), diff(y_param, phi), diff(z_param, phi)])

print(f"\nr_theta = {r_theta.T}") print(f"r_phi = {r_phi.T}")

Normal vector N = r_theta × r_phi

N = r_theta.cross(r_phi) print(f"\nN = r_theta × r_phi = {N.T}")

Evaluate at P

N_P = N.subs([(theta, theta_P), (phi, phi_P)]) N_P = simplify(N_P) print(f"\nN at P = {N_P.T}")

Magnitude

N_P_mag = sqrt(N_P.dot(N_P)) N_P_mag = simplify(N_P_mag) print(f"|N at P| = {N_P_mag}")

Unit normal (outward)

n_outward = N_P / N_P_mag n_outward = simplify(n_outward) print(f"\nOutward unit normal at P = {n_outward.T}")

Inward unit normal

n_inward = -n_outward n_inward = simplify(n_inward) print(f"Inward unit normal at P = {n_inward.T}")

n1, n2, n3 = n_inward[0], n_inward[1], n_inward[2] print(f"\nn1 = {n1}, n2 = {n2}, n3 = {n3}") print(f"n1 + n2 = {simplify(n1 + n2)}") [/PYTHON]

[THOUGHT] I need to continue computing the other parts. Let me run more code to get the area, volume, and curvature. [/THOUGHT] [PYTHON] import numpy as np from sympy import *

theta, phi, t = symbols('theta phi t', real=True)

Part II: Area of S in region W

W: x^2 + y^2 <= 4

On torus: x^2 + y^2 = (cos(phi) + 2)^2

So (cos(phi) + 2)^2 <= 4 => cos(phi) + 2 <= 2 => cos(phi) <= 0

This means pi/2 <= phi <= 3*pi/2

Area element dS = |r_theta × r_phi| dtheta dphi

|N| = (cos(phi) + 2) (from the cross product)

x_param = (cos(phi) + 2) * cos(theta) y_param = (cos(phi) + 2) * sin(theta) z_param = sin(phi)

r_theta = Matrix([diff(x_param, theta), diff(y_param, theta), diff(z_param, theta)]) r_phi = Matrix([diff(x_param, phi), diff(y_param, phi), diff(z_param, phi)]) N = r_theta.cross(r_phi) N_mag = sqrt(N.dot(N)) N_mag = simplify(N_mag) print(f"Area element |N| = {N_mag}")

For a torus, |N| = (cos(phi) + 2) * 1 = cos(phi) + 2 (since minor radius is 1)

Actually let me compute it properly

N = ((cos(phi)+2)*cos(phi)*cos(theta), (cos(phi)+2)*cos(phi)*sin(theta), (cos(phi)+2)*sin(phi))

|N|^2 = (cos(phi)+2)^2 * cos^2(phi) * (cos^2(theta) + sin^2(theta)) + (cos(phi)+2)^2 * sin^2(phi)

= (cos(phi)+2)^2 * (cos^2(phi) + sin^2(phi)) = (cos(phi)+2)^2

So |N| = cos(phi) + 2 (positive since cos(phi) >= -1)

Area of S in W: integrate over theta from 0 to 2pi, phi from pi/2 to 3pi/2

area_integrand = cos(phi) + 2 A_S = integrate(area_integrand, (theta, 0, 2pi), (phi, pi/2, 3pi/2)) A_S = simplify(A_S) print(f"\nPart II: Area A_S = {A_S}")

Part III: Volume of V ∩ W

V is the interior of the torus

W is x^2 + y^2 <= 4, which is a cylinder of radius 2

The torus has major radius R=2 and minor radius r=1

The cylinder x^2 + y^2 = 4 passes through the center of the torus tube

Using cylindrical coordinates (rho, theta, z):

Torus: (rho - 2)^2 + z^2 <= 1

Cylinder: rho <= 2

For rho <= 2: the torus constraint becomes (rho - 2)^2 + z^2 <= 1

Since rho <= 2, we have rho - 2 <= 0, so (rho - 2)^2 = (2 - rho)^2

z^2 <= 1 - (2 - rho)^2 = 1 - 4 + 4rho - rho^2 = -3 + 4rho - rho^2

For this to have solutions, -3 + 4*rho - rho^2 >= 0

rho^2 - 4*rho + 3 <= 0

(rho - 1)(rho - 3) <= 0

1 <= rho <= 3

Combined with rho <= 2: 1 <= rho <= 2

Volume = integral over theta from 0 to 2*pi, rho from 1 to 2, z from -sqrt(1-(2-rho)^2) to sqrt(1-(2-rho)^2)

V = integral_0^{2pi} d_theta * integral_1^2 rho * 2sqrt(1-(2-rho)^2) d_rho

rho = symbols('rho', real=True, positive=True) z_max = sqrt(1 - (2 - rho)**2)

Volume element: rho * 2 * z_max

vol_integrand = rho * 2 * z_max vol_inner = integrate(vol_integrand, (rho, 1, 2)) vol_inner = simplify(vol_inner) print(f"Inner integral (rho): {vol_inner}")

V_ov = 2 * pi * vol_inner V_ov = simplify(V_ov) print(f"\nPart III: Volume V_ov = {V_ov}")

Part IV: Curvature at Q

Curve C: theta = phi, so parameter t = theta = phi

x(t) = (cos(t) + 2)*cos(t)

y(t) = (cos(t) + 2)*sin(t)

z(t) = sin(t)

x_t = (cos(t) + 2) * cos(t) y_t = (cos(t) + 2) * sin(t) z_t = sin(t)

At Q = (0, 2, 1): z = sin(t) = 1 => t = pi/2

t_Q = pi/2 print(f"\nAt Q, t = pi/2") print(f"x = {simplify(x_t.subs(t, t_Q))}, y = {simplify(y_t.subs(t, t_Q))}, z = {simplify(z_t.subs(t, t_Q))}")

First derivative

dx_dt = diff(x_t, t) dy_dt = diff(y_t, t) dz_dt = diff(z_t, t) c_prime = Matrix([dx_dt, dy_dt, dz_dt]) print(f"\nc'(t) = {c_prime.T}")

Second derivative

d2x_dt2 = diff(dx_dt, t) d2y_dt2 = diff(dy_dt, t) d2z_dt2 = diff(dz_dt, t) c_double_prime = Matrix([d2x_dt2, d2y_dt2, d2z_dt2]) print(f"c''(t) = {c_double_prime.T}")

Evaluate at t = pi/2

c_prime_Q = c_prime.subs(t, t_Q) c_prime_Q = simplify(c_prime_Q) c_double_prime_Q = c_double_prime.subs(t, t_Q) c_double_prime_Q = simplify(c_double_prime_Q)

print(f"\nc'(pi/2) = {c_prime_Q.T}") print(f"c''(pi/2) = {c_double_prime_Q.T}")

Cross product

cross_Q = c_prime_Q.cross(c_double_prime_Q) cross_Q = simplify(cross_Q) print(f"\nc' × c'' at Q = {cross_Q.T}")

Magnitudes

cross_mag = sqrt(cross_Q.dot(cross_Q)) cross_mag = simplify(cross_mag) c_prime_mag = sqrt(c_prime_Q.dot(c_prime_Q)) c_prime_mag = simplify(c_prime_mag)

print(f"|c' × c''| = {cross_mag}") print(f"|c'| = {c_prime_mag}")

Curvature

kappa_Q = cross_mag / (c_prime_mag**3) kappa_Q = simplify(kappa_Q) print(f"\nPart IV: Curvature kappa_Q = {kappa_Q}")

Now compute K

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

n1_plus_n2 = sqrt(2) term1 = V_ov / (2pi) term2 = A_S / (4pi) diff_term = term1 - term2 print(f"\nV_ov/(2pi) = {simplify(term1)}") print(f"A_S/(4pi) = {simplify(term2)}") print(f"V_ov/(2pi) - A_S/(4pi) = {simplify(diff_term)}")

factor = (n1_plus_n22) / (kappa_Q2) factor = simplify(factor) print(f"(n1+n2)^2 / kappa_Q^2 = {factor}")

K = 999 * 123 * diff_term * factor + 7 K = simplify(K) print(f"\nK = {K}") print(f"K (numerical) = {float(K)}") [/PYTHON]

[THOUGHT] The Python calculation gives us all the components:

• Part I: Inward unit normal at P: n = (√2/2, √2/2, 0), so n1 + n2 = √2
• Part II: Area A_S = 4π(π - 1)
• Part III: Volume V_ov = 2π(3π - 2)/3
• Part IV: Curvature κ_Q = √205/25

Computing K:

• V_ov/(2π) - A_S/(4π) = (π - 2/3) - (π - 1) = 1/3
• (n1 + n2)²/κ_Q² = 2/(205/625) = 1250/205 = 250/41
• K = 999 × 123 × (1/3) × (250/41) + 7 = 249757

The answer is 249757. [/THOUGHT]

\boxed{249757}