4.1 Definition and Visualization of Vector Fields

📐 Definition: Vector Field
A function that associates one vector to each point in space is called a vector field: $$\mathbf{F}(\mathbf{r}) = (F_x(x,y,z), F_y(x,y,z), F_z(x,y,z))$$ Examples: velocity field of fluid, electric field, magnetic field, etc.

💻 Code Example 1: Visualization of 2D Vector Field

import numpy as np import matplotlib.pyplot as plt # 2D vector field definition: F(x,y) = (-y, x) (rotational field) def vector_field(x, y): """Rotating vector field""" Fx = -y Fy = x return Fx, Fy # Create grid x = np.linspace(-3, 3, 15) y = np.linspace(-3, 3, 15) X, Y = np.meshgrid(x, y) Fx, Fy = vector_field(X, Y) # Visualize vector field with quiver plot plt.figure(figsize=(10, 8)) plt.quiver(X, Y, Fx, Fy, np.sqrt(Fx2 + Fy2), cmap=‘viridis’) plt.colorbar(label=‘Vector magnitude’) plt.xlabel(‘x’) plt.ylabel(‘y’) plt.title(‘Rotational Vector Field F = (-y, x)’) plt.axis(‘equal’) plt.grid(True, alpha=0.3) plt.show() # Draw streamlines x_fine = np.linspace(-3, 3, 100) y_fine = np.linspace(-3, 3, 100) X_fine, Y_fine = np.meshgrid(x_fine, y_fine) Fx_fine, Fy_fine = vector_field(X_fine, Y_fine) plt.figure(figsize=(10, 8)) plt.streamplot(X_fine, Y_fine, Fx_fine, Fy_fine, density=1.5, color=‘blue’, linewidth=1) plt.xlabel(‘x’) plt.ylabel(‘y’) plt.title(‘Streamlines of Vector Field’) plt.axis(‘equal’) plt.grid(True, alpha=0.3) plt.show()

4.2 Gradient (gradient, grad)

📐 Definition: Gradient
The gradient of scalar field φ is: $$\nabla \phi = \text{grad},\phi = \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}\right)$$ The gradient vector points in the direction where φ increases most rapidly.

💻 Code Example 2: Calculation and Visualization of Gradient Vector Field

def scalar_field(x, y): """Scalar field: φ(x,y) = x² + y²""" return x2 + y2 def gradient_field(x, y): """Gradient: ∇φ = (2x, 2y)""" grad_x = 2x grad_y = 2y return grad_x, grad_y # Visualization x = np.linspace(-2, 2, 20) y = np.linspace(-2, 2, 20) X, Y = np.meshgrid(x, y) phi = scalar_field(X, Y) grad_x, grad_y = gradient_field(X, Y) fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6)) # Left plot: Contour lines of scalar field contour = ax1.contourf(X, Y, phi, levels=20, cmap=‘viridis’) fig.colorbar(contour, ax=ax1, label=‘φ(x,y)’) ax1.set_xlabel(‘x’) ax1.set_ylabel(‘y’) ax1.set_title(‘Scalar Field φ = x² + y²’) ax1.axis(‘equal’) # Right plot: Gradient vector field ax2.contour(X, Y, phi, levels=10, colors=‘gray’, alpha=0.3) ax2.quiver(X, Y, grad_x, grad_y, color=‘red’) ax2.set_xlabel(‘x’) ax2.set_ylabel(‘y’) ax2.set_title(‘Gradient Vector Field ∇φ = (2x, 2y)’) ax2.axis(‘equal’) plt.tight_layout() plt.show()

4.3 Divergence (divergence, div)

📐 Definition: Divergence
The divergence of vector field F is: $$\text{div},\mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$ Divergence represents the strength of the “source” of the vector field at that point.

💻 Code Example 3: Calculation of Divergence

def divergence_numerical(Fx, Fy, x, y, h=1e-5): """Numerical calculation of divergence: div F = ∂Fx/∂x + ∂Fy/∂y""" dFx_dx = (Fx(x+h, y) - Fx(x-h, y)) / (2h) dFy_dy = (Fy(x, y+h) - Fy(x, y-h)) / (2h) return dFx_dx + dFy_dy # Example 1: Vector field with positive divergence (diverging field) def Fx_diverging(x, y): return x def Fy_diverging(x, y): return y # Example 2: Vector field with zero divergence (rotational field) def Fx_rotating(x, y): return -y def Fy_rotating(x, y): return x # Calculate divergence x0, y0 = 1, 1 div_diverging = divergence_numerical(Fx_diverging, Fy_diverging, x0, y0) div_rotating = divergence_numerical(Fx_rotating, Fy_rotating, x0, y0) print(f”Divergence of diverging field F=(x,y): div F = {div_diverging:.6f} (analytical solution: 2)”) print(f”Divergence of rotational field F=(-y,x): div F = {div_rotating:.6f} (analytical solution: 0)“)

4.4 Curl (rotation, curl)

📐 Definition: Curl
The curl of 3D vector field F is: $$\text{rot},\mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}$$

💻 Code Example 4: Curl in 2D (Scalar)

In 2D, curl is a scalar value: rot F = ∂Fy/∂x - ∂Fx/∂y def curl_2d(Fx, Fy, x, y, h=1e-5): """Numerical calculation of curl in 2D""" dFy_dx = (Fy(x+h, y) - Fy(x-h, y)) / (2h) dFx_dy = (Fx(x, y+h) - Fx(x, y-h)) / (2h) return dFy_dx - dFx_dy # Calculate curl of rotational field curl_rotating = curl_2d(Fx_rotating, Fy_rotating, x0, y0) print(f”\nCurl of rotational field F=(-y,x): rot F = {curl_rotating:.6f} (analytical solution: 2)”) # Calculate curl of diverging field curl_diverging = curl_2d(Fx_diverging, Fy_diverging, x0, y0) print(f”Curl of diverging field F=(x,y): rot F = {curl_diverging:.6f} (analytical solution: 0)“)

4.5 Laplacian (Laplacian, Δ)

📐 Definition: Laplacian
The Laplacian of scalar field φ is: $$\Delta \phi = \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}$$ Appears in many physical laws such as heat equation and wave equation.

💻 Code Example 5: Calculation and Application of Laplacian

def laplacian_2d(phi, x, y, h=1e-4): """Numerical calculation of 2D Laplacian""" lap = (phi(x+h, y) + phi(x-h, y) + phi(x, y+h) + phi(x, y-h) - 4*phi(x, y)) / h2 return lap # Test function: φ(x,y) = x² + y² phi = lambda x, y: x2 + y2 lap = laplacian_2d(phi, 1, 1) print(f”\nLaplacian of φ = x² + y²: Δφ = {lap:.6f} (analytical solution: 4)”) # Solution of Laplace equation Δφ = 0 (harmonic function) phi_harmonic = lambda x, y: x2 - y**2 lap_harmonic = laplacian_2d(phi_harmonic, 1, 1) print(f”Laplacian of φ = x² - y²: Δφ = {lap_harmonic:.6f} (analytical solution: 0)“)

4.6 Conservative Fields and Potential Functions

📐 Theorem: Condition for Conservative Field
The condition for vector field F to be conservative (scalar potential φ exists) is: $$\text{rot},\mathbf{F} = \mathbf{0}$$ In this case, F can be expressed as F = grad φ.

💻 Code Example 6: Determination of Conservative Field and Calculation of Potential Function

import sympy as sp x, y = sp.symbols(‘x y’) # Vector field F = (2xy, x² + 2y) Fx_sym = 2xy Fy_sym = x**2 + 2*y # Calculate curl curl_z = sp.diff(Fy_sym, x) - sp.diff(Fx_sym, y) print(“Determination of conservative field for vector field F = (2xy, x² + 2y):”) print(f”rot F = ∂Fy/∂x - ∂Fx/∂y = {curl_z}”) if curl_z == 0: print(”→ Since rot F = 0, this is a conservative field\n”) # Calculate potential function # Find φ such that ∂φ/∂x = Fx, ∂φ/∂y = Fy phi = sp.integrate(Fx_sym, x) # Integrate with respect to x print(f”∫ Fx dx = {phi} + g(y)”) # Determine function g(y) of y dPhi_dy = sp.diff(phi, y) g_prime = Fy_sym - dPhi_dy g = sp.integrate(g_prime, y) phi_final = phi + g print(f”Potential function: φ = {phi_final}”) # Verification grad_phi_x = sp.diff(phi_final, x) grad_phi_y = sp.diff(phi_final, y) print(f”\nVerification:”) print(f”∂φ/∂x = {grad_phi_x} = Fx ✓”) print(f”∂φ/∂y = {grad_phi_y} = Fy ✓”)

💻 Code Example 7: Application to Materials Science (Diffusion Flux)

Fick’s first law: J = -D ∇C (diffusion flux) # Diffusion flux occurs due to concentration gradient def concentration(x, y): """Concentration distribution C(x,y)""" return np.exp(-(x2 + y2)) # Calculate gradient (concentration gradient) x = np.linspace(-2, 2, 20) y = np.linspace(-2, 2, 20) X, Y = np.meshgrid(x, y) # Calculate gradient by numerical differentiation h = x[1] - x[0] C = concentration(X, Y) dC_dx = np.gradient(C, h, axis=1) dC_dy = np.gradient(C, h, axis=0) # Diffusion flux: J = -D ∇C D = 1.0 # Diffusion coefficient Jx = -D * dC_dx Jy = -D * dC_dy fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6)) # Left plot: Concentration distribution contour = ax1.contourf(X, Y, C, levels=15, cmap=‘viridis’) fig.colorbar(contour, ax=ax1, label=‘Concentration C’) ax1.set_xlabel(‘x’) ax1.set_ylabel(‘y’) ax1.set_title(‘Concentration Distribution C(x,y) = exp(-(x²+y²))’) ax1.axis(‘equal’) # Right plot: Diffusion flux vector ax2.contour(X, Y, C, levels=10, colors=‘gray’, alpha=0.3) ax2.quiver(X, Y, Jx, Jy, color=‘red’, alpha=0.7) ax2.set_xlabel(‘x’) ax2.set_ylabel(‘y’) ax2.set_title(‘Diffusion Flux J = -D∇C’) ax2.axis(‘equal’) plt.tight_layout() plt.show() # Divergence div J (net inflow/outflow) div_J = np.gradient(Jx, h, axis=1) + np.gradient(Jy, h, axis=0) print(f”\nDivergence at center: div J = {div_J[len(y)//2, len(x)//2]:.6f}”) print(“(negative → inflow, positive → outflow)“)

Summary

Vector fields associate vectors to each point in space, representing fluid velocity, electric fields, etc.
Gradient (grad) is a vector field pointing in the direction of steepest increase of a scalar field
Divergence (div) represents the “source” of a vector field, curl represents the “vortex”
Laplacian is an important operator describing diffusion and wave phenomena
In conservative fields, curl is zero and a potential function exists

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