Calculate curl of a vector field in python and plot it with matplotlib

When working with vector fields in Python, it is often necessary to calculate the curl of the field and visualize it using matplotlib. In this article, we will explore three different ways to solve this problem and determine which option is the most efficient.

Option 1: Using NumPy and Matplotlib

The first option involves using the NumPy library to calculate the curl of the vector field and then plotting it using Matplotlib. Here is the code:

import numpy as np
import matplotlib.pyplot as plt

# Define the vector field
def vector_field(x, y):
    return np.array([x**2, -y**2])

# Calculate the curl
def curl(vector_field):
    return np.gradient(vector_field[1], axis=0) - np.gradient(vector_field[0], axis=1)

# Generate grid points
x = np.linspace(-10, 10, 100)
y = np.linspace(-10, 10, 100)
X, Y = np.meshgrid(x, y)

# Calculate the curl at each point
C = curl(vector_field(X, Y))

# Plot the curl
plt.quiver(X, Y, C[0], C[1])
plt.show()

This option uses the numpy.gradient function to calculate the partial derivatives of the vector field and then subtracts them to obtain the curl. The resulting curl is then plotted using the quiver function from matplotlib.

Option 2: Using SymPy and Matplotlib

The second option involves using the SymPy library to calculate the curl of the vector field symbolically and then plotting it using Matplotlib. Here is the code:

import sympy as sp
import matplotlib.pyplot as plt

# Define the vector field
x, y = sp.symbols('x y')
vector_field = sp.Matrix([x**2, -y**2])

# Calculate the curl
curl = sp.Matrix([sp.diff(vector_field[1], x) - sp.diff(vector_field[0], y)])

# Convert the curl to a function
curl_func = sp.lambdify((x, y), curl)

# Generate grid points
x_vals = np.linspace(-10, 10, 100)
y_vals = np.linspace(-10, 10, 100)
X, Y = np.meshgrid(x_vals, y_vals)

# Calculate the curl at each point
C = curl_func(X, Y)

# Plot the curl
plt.quiver(X, Y, C[0], C[1])
plt.show()

This option uses the SymPy library to symbolically calculate the curl of the vector field. The resulting curl is then converted to a function using the lambdify function from SymPy. The curl is then evaluated at each point on the grid and plotted using the quiver function from matplotlib.

Option 3: Using SciPy and Matplotlib

The third option involves using the SciPy library to calculate the curl of the vector field numerically and then plotting it using Matplotlib. Here is the code:

import numpy as np
import matplotlib.pyplot as plt
from scipy import interpolate

# Define the vector field
def vector_field(x, y):
    return np.array([x**2, -y**2])

# Generate grid points
x = np.linspace(-10, 10, 100)
y = np.linspace(-10, 10, 100)
X, Y = np.meshgrid(x, y)

# Calculate the curl at each point
C = np.gradient(vector_field(X, Y)[1], axis=0) - np.gradient(vector_field(X, Y)[0], axis=1)

# Interpolate the curl to obtain a smooth plot
f = interpolate.interp2d(x, y, C, kind='cubic')
x_new = np.linspace(-10, 10, 1000)
y_new = np.linspace(-10, 10, 1000)
C_new = f(x_new, y_new)

# Plot the curl
plt.quiver(x_new, y_new, C_new[0], C_new[1])
plt.show()

This option uses the numpy.gradient function to calculate the partial derivatives of the vector field and then subtracts them to obtain the curl. The resulting curl is then interpolated using the interp2d function from SciPy to obtain a smooth plot. The interpolated curl is then plotted using the quiver function from matplotlib.

After comparing the three options, it is clear that Option 1, which uses NumPy and Matplotlib, is the most efficient solution. It provides accurate results and does not require any additional libraries or complex calculations. Therefore, Option 1 is the recommended approach for calculating the curl of a vector field in Python and plotting it with matplotlib.

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11 Responses

    1. I respectfully disagree. While option 3 may be efficient, I find option 2 to be more accurate and reliable for calculating and plotting the curl. Its always a matter of personal preference and specific requirements.

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