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Master Green's theorem through computational examples connecting line integrals to double integrals over planar regions. This interactive Jupyter notebook demonstrates circulation and flux calculations, simply connected regions, and applications to area computation and conservative field verification. CoCalc's SageMath platform offers boundary curve parametrization, region visualization, and automated theorem verification tools, enabling students to explore the profound connection between boundary integrals and area integrals through immediate computational feedback and geometric visualization.

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Kernel: SageMath 10.7

Advanced Calculus with SageMath - Chapter 5

Green's Theorem and Applications

This notebook contains Chapter 5 from the main Advanced Calculus with SageMath notebook.

For the complete course, please refer to the main notebook: Advanced Calculus with SageMath.ipynb

# Comprehensive imports for advanced calculus
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import scipy.optimize as opt
import scipy.integrate as integrate
from scipy.integrate import solve_ivp, odeint
import sympy as sp
from sympy import *
from sage.all import *
import seaborn as sns

# Configure plotting
plt.style.use('seaborn-v0_8')
sns.set_palette("husl")
plt.rcParams['figure.figsize'] = (12, 8)

print("Advanced Calculus Environment Initialized")
print("Tools: SageMath, NumPy, SciPy, SymPy, Matplotlib")
print("Ready for multivariable calculus, vector analysis, and PDEs!")
Advanced Calculus Environment Initialized Tools: SageMath, NumPy, SciPy, SymPy, Matplotlib Ready for multivariable calculus, vector analysis, and PDEs!

Chapter 5: Green's Theorem and Applications

Green's Theorem Statement

For a positively oriented, simple closed curve C and the region D it encloses:

∮C(P dx+Q dy)=∬D(∂Q∂x−∂P∂y)dA\oint_C (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA

This connects line integrals around closed curves to double integrals over regions.

from IPython.display import display, Math
var('x y t R r theta')

print("GREEN'S THEOREM")
print("=" * 50)

# Vector field F = (P, Q) = (x^2 y, x y^2)
P = x^2 * y
Q = x * y^2
display(Math(r"\vec{F}=(P,Q)=(" + latex(P) + r"," + latex(Q) + r")"))

# Curl = ∂Q/∂x − ∂P/∂y
dQ_dx = diff(Q, x)
dP_dy = diff(P, y)
curl_2D = dQ_dx - dP_dy
display(Math(r"\frac{\partial Q}{\partial x}=" + latex(dQ_dx)))
display(Math(r"\frac{\partial P}{\partial y}=" + latex(dP_dy)))
display(Math(r"\mathrm{curl}=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}="
             + latex(curl_2D)))

# Region: circle of radius R
display(Math(r"D=\{(x,y): x^2+y^2\le R^2\}"))

# Method 1: Line integral
print("\nMETHOD 1: Line Integral")
x_circle = R*cos(t)
y_circle = R*sin(t)
dx_dt = diff(x_circle, t)
dy_dt = diff(y_circle, t)

P_on_circle = P.subs({x: x_circle, y: y_circle})
Q_on_circle = Q.subs({x: x_circle, y: y_circle})
integrand_line = P_on_circle*dx_dt + Q_on_circle*dy_dt
integrand_line_simplified = simplify(integrand_line)
display(Math(r"\text{Integrand}:\;"+ latex(integrand_line_simplified)))

line_integral_result = integrate(integrand_line_simplified, (t, 0, 2*pi))
display(Math(r"\text{Line integral}=\int_{0}^{2\pi} "
             + latex(integrand_line_simplified) + r"\,dt = "
             + latex(line_integral_result)))

# Method 2: Double integral (Green's Theorem)
print("\nMETHOD 2: Double Integral (Green's Theorem)")
display(Math(r"\iint_D \mathrm{curl}\, dA = " + latex(curl_2D)))

curl_polar = curl_2D.subs({x: r*cos(theta), y: r*sin(theta)})
curl_polar_simplified = simplify(curl_polar)
display(Math(r"\text{In polar: } " + latex(curl_polar_simplified)))

inner_integral = integrate(curl_polar_simplified * r, (r, 0, R))
double_integral_result = integrate(inner_integral, (theta, 0, 2*pi))
display(Math(r"\text{Double integral}=" + latex(double_integral_result)))

print("\nVERIFICATION")
display(Math(r"\text{Line integral}=" + latex(line_integral_result)
             + r",\quad \text{Double integral}=" + latex(double_integral_result)))
print("Green's theorem verified! ")
GREEN'S THEOREM ==================================================

F⃗=(P,Q)=(x2y,xy2)\displaystyle \vec{F}=(P,Q)=( x^{2} y , x y^{2} )

∂Q∂x=y2\displaystyle \frac{\partial Q}{\partial x}= y^{2}

∂P∂y=x2\displaystyle \frac{\partial P}{\partial y}= x^{2}

curl=∂Q∂x−∂P∂y=−x2+y2\displaystyle \mathrm{curl}=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}= -x^{2} + y^{2}

D={(x,y):x2+y2≤R2}\displaystyle D=\{(x,y): x^2+y^2\le R^2\}

METHOD 1: Line Integral

Integrand:  0\displaystyle \text{Integrand}:\; 0

Line integral=∫02π0 dt=0\displaystyle \text{Line integral}=\int_{0}^{2\pi} 0 \,dt = 0

METHOD 2: Double Integral (Green's Theorem)

∬Dcurl dA=−x2+y2\displaystyle \iint_D \mathrm{curl}\, dA = -x^{2} + y^{2}

In polar: −r2cos⁡(θ)2+r2sin⁡(θ)2\displaystyle \text{In polar: } -r^{2} \cos\left(\theta\right)^{2} + r^{2} \sin\left(\theta\right)^{2}

Double integral=0\displaystyle \text{Double integral}= 0

VERIFICATION

Line integral=0,Double integral=0\displaystyle \text{Line integral}= 0 ,\quad \text{Double integral}= 0

Green's theorem verified! ✓ 
import numpy as np
import matplotlib.pyplot as plt

# Parameters
R_val = 1.0
n = 31
x = np.linspace(-1.25*R_val, 1.25*R_val, n)
y = np.linspace(-1.25*R_val, 1.25*R_val, n)
X, Y = np.meshgrid(x, y)

# Vector field F = (x^2 y, x y^2)
U = X**2 * Y
V = X * Y**2

# Curl (2D scalar): ∂Q/∂x − ∂P/∂y = y^2 − x^2
CURL = Y**2 - X**2

# Circle C and region D
tt = np.linspace(0, 2*np.pi, 400)
XC = R_val*np.cos(tt)
YC = R_val*np.sin(tt)
mask_outside = (X**2 + Y**2) > R_val**2
CURL_masked = np.ma.masked_where(mask_outside, CURL)

fig, axs = plt.subplots(1, 2, figsize=(12, 5), constrained_layout=True)

# Left: vector field + boundary C with orientation
axs[0].quiver(X, Y, U, V, color='tab:blue', alpha=0.7, pivot='mid', scale=40)
axs[0].plot(XC, YC, color='tab:orange', lw=2, label='$C: x^2+y^2=R^2$')
# Orientation arrow on C (counterclockwise)
t0 = np.pi/6
x0, y0 = R_val*np.cos(t0), R_val*np.sin(t0)
tx, ty = -R_val*np.sin(t0), R_val*np.cos(t0)
axs[0].arrow(x0, y0, 0.15*tx, 0.15*ty, width=0.005, color='tab:orange', length_includes_head=True)
axs[0].set_aspect('equal', 'box')
axs[0].set_title(r'Field $\vec F=(x^2y, xy^2)$ and $C$')
axs[0].set_xlabel('x'); axs[0].set_ylabel('y')
axs[0].legend(loc='upper right')

# Right: curl over D
levels = np.linspace(-R_val**2, R_val**2, 25)
cf = axs[1].contourf(X, Y, CURL_masked, levels=levels, cmap='coolwarm')
axs[1].plot(XC, YC, color='k', lw=1.5)
axs[1].set_aspect('equal', 'box')
axs[1].set_title(r'Curl $\nabla\times\vec F = y^2 - x^2$ over $D$')
axs[1].set_xlabel('x'); axs[1].set_ylabel('y')
cbar = fig.colorbar(cf, ax=axs[1], fraction=0.046, pad=0.04)
cbar.set_label(r'$\nabla\times\vec F$')

plt.show()
Image in a Jupyter notebook

Continuing Your Learning Journey

You've completed Green's Theorem and Applications! The concepts you've mastered here form essential building blocks for what comes next.

Ready for Surface Integrals and Flux?

In Chapter 6, we'll build upon these foundations to explore even more fascinating aspects of the subject. The knowledge you've gained here will directly apply to the advanced concepts ahead.

What's Next

Chapter 6 will expand your understanding by introducing new techniques and applications that leverage everything you've learned so far.

Continue to Chapter 6: Surface Integrals and Flux →

or

Return to Complete Course