5.1 Fundamentals of Line Integrals

📐 Definition: Line Integral
Line integral of scalar field f over curve C: $$\int_C f , ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}‘(t)| , dt$$ Line integral of vector field F (work): $$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}‘(t) , dt$$

💻 Code Example 1: Line Integral of Scalar Field

import numpy as np import matplotlib.pyplot as plt from scipy import integrate # Parametric representation of curve: r(t) = (cos t, sin t), 0 ≤ t ≤ π def curve(t): x = np.cos(t) y = np.sin(t) return x, y # Scalar field: f(x,y) = x² + y² def scalar_field(x, y): return x2 + y2 # Integrand for line integral def integrand(t): x, y = curve(t) f_val = scalar_field(x, y) # Magnitude of velocity vector ||r’(t)|| dx_dt = -np.sin(t) dy_dt = np.cos(t) speed = np.sqrt(dx_dt2 + dy_dt2) return f_val * speed # Calculate line integral result, error = integrate.quad(integrand, 0, np.pi) print(f”Line integral of scalar field:”) print(f”∫_C f ds = {result:.6f} (estimated error: {error:.2e})”) # Visualization t = np.linspace(0, np.pi, 100) x, y = curve(t) plt.figure(figsize=(8, 8)) plt.plot(x, y, ‘r-’, linewidth=3, label=‘Curve C’) plt.arrow(0, 1, 0.1, 0, head_width=0.1, head_length=0.05, fc=‘red’, ec=‘red’) plt.scatter([x[0]], [y[0]], color=‘green’, s=150, marker=‘o’, label=‘Start point’, zorder=5) plt.scatter([x[-1]], [y[-1]], color=‘blue’, s=150, marker=’s’, label=‘End point’, zorder=5) plt.xlabel(‘x’) plt.ylabel(‘y’) plt.title(‘Path of Line Integration’) plt.legend() plt.grid(True, alpha=0.3) plt.axis(‘equal’) plt.show()

💻 Code Example 2: Line Integral of Vector Field (Work)

Vector field: F(x,y) = (y, -x) def vector_field(x, y): return y, -x # Line integral ∫_C F·dr def integrand_vector(t): x, y = curve(t) Fx, Fy = vector_field(x, y) dx_dt = -np.sin(t) dy_dt = np.cos(t) return Fx * dx_dt + Fy * dy_dt result_work, error = integrate.quad(integrand_vector, 0, np.pi) print(f”\nLine integral of vector field (work):”) print(f”∫_C F·dr = {result_work:.6f}”) # Visualization of vector field x_grid = np.linspace(-1.5, 1.5, 15) y_grid = np.linspace(-1.5, 1.5, 15) X, Y = np.meshgrid(x_grid, y_grid) Fx, Fy = vector_field(X, Y) plt.figure(figsize=(10, 8)) plt.quiver(X, Y, Fx, Fy, alpha=0.6) plt.plot(x, y, ‘r-’, linewidth=3, label=‘Path C’) plt.scatter([x[0]], [y[0]], color=‘green’, s=150, marker=‘o’, zorder=5) plt.scatter([x[-1]], [y[-1]], color=‘blue’, s=150, marker=’s’, zorder=5) plt.xlabel(‘x’) plt.ylabel(‘y’) plt.title(‘Vector Field F=(y,-x) and Integration Path’) plt.legend() plt.grid(True, alpha=0.3) plt.axis(‘equal’) plt.show()

5.2 Green’s Theorem

📐 Theorem: Green’s Theorem
For boundary C (counterclockwise) of region D: $$\oint_C (P , dx + Q , dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$$ Line integrals can be converted to double integrals.

💻 Code Example 3: Verification of Green’s Theorem

Vector field F = (P, Q) = (x, y) # Region D: unit disk x² + y² ≤ 1 # Left side: line integral ∮_C (x dx + y dy) def curve_circle(t): return np.cos(t), np.sin(t) def line_integral(t): x, y = curve_circle(t) dx_dt, dy_dt = -np.sin(t), np.cos(t) P, Q = x, y return P * dx_dt + Q * dy_dt left_side, _ = integrate.quad(line_integral, 0, 2np.pi) # Right side: double integral ∬_D (∂Q/∂x - ∂P/∂y) dA # ∂Q/∂x = ∂y/∂x = 0, ∂P/∂y = ∂x/∂y = 0 # Therefore integral value is 0 right_side = 0.0 print(“Verification of Green’s theorem:”) print(f”F = (x, y), D = unit disk”) print(f”Left side (line integral): ∮_C F·dr = {left_side:.6f}”) print(f”Right side (double integral): ∬_D (∂Q/∂x - ∂P/∂y) dA = {right_side:.6f}”) print(f”Difference: {abs(left_side - right_side):.2e}”) # More interesting example: F = (-y, x) def line_integral2(t): x, y = curve_circle(t) dx_dt, dy_dt = -np.sin(t), np.cos(t) P, Q = -y, x return P * dx_dt + Q * dy_dt left_side2, _ = integrate.quad(line_integral2, 0, 2np.pi) # Right side: ∂Q/∂x - ∂P/∂y = 1 - (-(-1)) = 2 # ∬_D 2 dA = 2 × (area of circle) = 2π right_side2 = 2 * np.pi print(f”\nF = (-y, x), D = unit disk”) print(f”Left side (line integral): {left_side2:.6f}”) print(f”Right side (double integral): {right_side2:.6f}”) print(f”Error: {abs(left_side2 - right_side2):.2e}“)

5.3 Fundamentals of Surface Integrals

📐 Definition: Surface Integral
Surface integral of scalar field f over surface S: $$\iint_S f , dS$$ Surface integral of vector field F (flux): $$\iint_S \mathbf{F} \cdot \mathbf{n} , dS$$ where n is the unit normal vector of the surface.

💻 Code Example 4: Surface Integral on Sphere

from scipy import integrate # Parametric representation of sphere: r(θ,φ) = (sin θ cos φ, sin θ sin φ, cos θ) # 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π def sphere_parametrization(theta, phi): x = np.sin(theta) * np.cos(phi) y = np.sin(theta) * np.sin(phi) z = np.cos(theta) return x, y, z # Scalar field: f(x,y,z) = z def scalar_field_3d(x, y, z): return z # Integrand for surface integral (including Jacobian) def surface_integrand(theta, phi): x, y, z = sphere_parametrization(theta, phi) f_val = scalar_field_3d(x, y, z) # Jacobian of sphere: r² sin θ (radius r=1) jacobian = np.sin(theta) return f_val * jacobian # Calculate surface integral result, error = integrate.dblquad(surface_integrand, 0, 2*np.pi, # φ range 0, np.pi) # θ range print(“Surface integral on sphere:”) print(f”∬_S z dS = {result:.6f} (analytical solution: 0 by symmetry)”) print(f”Estimated error: {error:.2e}“)

5.4 Gauss’s Divergence Theorem

📐 Theorem: Gauss’s Divergence Theorem
For closed surface S enclosing region V: $$\oiint_S \mathbf{F} \cdot \mathbf{n} , dS = \iiint_V (\nabla \cdot \mathbf{F}) , dV$$ Surface integrals can be converted to volume integrals.

💻 Code Example 5: Verification of Gauss’s Divergence Theorem

Vector field F = (x, y, z) # Region: unit sphere x² + y² + z² ≤ 1 # Right side: volume integral ∭_V div F dV # div F = ∂x/∂x + ∂y/∂y + ∂z/∂z = 3 # ∭_V 3 dV = 3 × (volume of sphere) = 3 × (4π/3) = 4π right_side_gauss = 4 * np.pi # Left side: surface integral ∬_S F·n dS # On sphere surface F·n = (x,y,z)·(x,y,z) = x²+y²+z² = 1 (on unit sphere surface) # ∬_S 1 dS = surface area of sphere = 4π left_side_gauss = 4 * np.pi print(“Verification of Gauss’s divergence theorem:”) print(f”F = (x, y, z), V = unit sphere”) print(f”Left side (surface integral): ∬_S F·n dS = {left_side_gauss:.6f}”) print(f”Right side (volume integral): ∭_V div F dV = {right_side_gauss:.6f}”) print(f”Both sides match ✓“)

5.5 Stokes’ Theorem

📐 Theorem: Stokes’ Theorem
For boundary C of surface S: $$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} , dS$$ Line integrals can be converted to surface integrals. This is the 3D version of Green’s theorem.

💻 Code Example 6: Application of Stokes’ Theorem

Vector field F = (-y, x, 0) # Surface S: hemisphere z = √(1-x²-y²), boundary C: unit circle x²+y²=1 (on z=0 plane) # Left side: line integral ∮_C F·dr def stokes_line_integral(t): x, y, z = np.cos(t), np.sin(t), 0 dx_dt, dy_dt, dz_dt = -np.sin(t), np.cos(t), 0 Fx, Fy, Fz = -y, x, 0 return Fxdx_dt + Fydy_dt + Fzdz_dt left_stokes, _ = integrate.quad(stokes_line_integral, 0, 2np.pi) # Right side: curl F = (0, 0, 2) # ∬_S (0,0,2)·n dS = 2 × (projected area of hemisphere) = 2π right_stokes = 2 * np.pi print(“\nVerification of Stokes’ theorem:”) print(f”F = (-y, x, 0), S = hemisphere”) print(f”Left side (line integral): ∮_C F·dr = {left_stokes:.6f}”) print(f”Right side (surface integral): ∬_S (curl F)·n dS = {right_stokes:.6f}”) print(f”Error: {abs(left_stokes - right_stokes):.2e}“)

5.6 Application to Materials Science: Diffusion Flux and Conservation Laws

🔬 Application Example: In atomic diffusion in materials, conservation law is derived from Gauss’s divergence theorem: $$\frac{\partial C}{\partial t} = -\nabla \cdot \mathbf{J} + S$$ where C is concentration, J is diffusion flux, and S is generation term.

💻 Code Example 7: Numerical Simulation of Diffusion Equation

1D diffusion equation: ∂C/∂t = D ∂²C/∂x² # Initial condition: Gaussian distribution # Boundary condition: zero concentration at both ends def diffusion_1d(C0, D, dx, dt, n_steps): """Numerical solution of 1D diffusion equation (explicit Euler method)""" C = C0.copy() C_history = [C.copy()] for step in range(n_steps): C_new = C.copy() for i in range(1, len(C)-1): # Laplacian: (C[i+1] - 2C[i] + C[i-1]) / dx² laplacian = (C[i+1] - 2C[i] + C[i-1]) / dx**2 C_new[i] = C[i] + D * dt * laplacian C = C_new if step % 10 == 0: C_history.append(C.copy()) return C_history # Parameter settings nx = 100 L = 10.0 x = np.linspace(0, L, nx) dx = x[1] - x[0] # Initial concentration distribution (Gaussian distribution) C0 = np.exp(-(x - L/2)**2 / 0.5) C0[0] = C0[-1] = 0 # Boundary condition # Diffusion coefficient and time step D = 0.1 dt = 0.01 n_steps = 100 # Execute simulation C_history = diffusion_1d(C0, D, dx, dt, n_steps) # Visualization plt.figure(figsize=(12, 6)) for i, C in enumerate(C_history[::2]): plt.plot(x, C, label=f’t = {i210dt:.2f}’, alpha=0.7) plt.xlabel(‘Position x’) plt.ylabel(‘Concentration C’) plt.title(‘Time Evolution of 1D Diffusion Equation’) plt.legend(loc=‘upper right’) plt.grid(True, alpha=0.3) plt.show() print(“\nDiffusion simulation:”) print(f”Initial peak concentration: {C0.max():.4f}”) print(f”Final peak concentration: {C_history[-1].max():.4f}”) print(f”Concentration decrease due to diffusion: {(1 - C_history[-1].max()/C0.max())*100:.1f}%“)

Summary

Line integrals calculate accumulated values along a path for scalar fields and work for vector fields
Surface integrals calculate accumulated values on surfaces and flux for vector fields
Green’s theorem converts line integrals to double integrals (2D)
Gauss’s divergence theorem converts surface integrals to volume integrals (3D)
Stokes’ theorem converts line integrals to surface integrals (3D version of Green’s theorem)
In materials science, they are applied to calculations of diffusion flux, conservation laws, etc.

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