Master Stokes' theorem connecting surface integrals to line integrals through computational practice with curl calculations and circulation analysis. This comprehensive Jupyter notebook demonstrates oriented surfaces with boundary curves, curl interpretation, and applications to electromagnetic theory and fluid dynamics. CoCalc's SageMath environment offers 3D visualization of surfaces and boundaries, symbolic curl computation, and theorem verification tools, enabling students to understand the deep relationship between circulation around boundaries and curl through surfaces.

Published/Calculus Series/Advanced Calculus with SageMath/Advanced Calculus with SageMath - Chapter 7.ipynb

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Advanced Calculus with SageMath - Chapter 7

Stokes' Theorem

This notebook contains Chapter 7 from the main Advanced Calculus with SageMath notebook.

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

In [3]:

# 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!")

Out[3]:

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

Chapter 7: Stokes' Theorem

Stokes' Theorem Statement

For a surface S with boundary curve C:

\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot \vec{n} \, dS

This relates the circulation of a vector field around a closed curve to the flux of its curl through any surface bounded by that curve.

In [24]:

var('x y z u v t')
from sage.misc.latex import LatexExpr

pretty_print(LatexExpr(r"\textbf{STOKES' THEOREM}"))
pretty_print(LatexExpr(r"\text{Vector field: } \vec{F} = "), vector([y, -x, z**2]))

# curl F
F_stokes = vector([y, -x, z**2])
curl_F = vector([
    diff(F_stokes[2], y) - diff(F_stokes[1], z),
    diff(F_stokes[0], z) - diff(F_stokes[2], x),
    diff(F_stokes[1], x) - diff(F_stokes[0], y)
])
pretty_print(LatexExpr(r"\nabla \times \vec{F} = "), curl_F)

# Surface: disk z=1, 0<=u<=2, 0<=v<=2π
pretty_print(LatexExpr(r"\text{Surface: } \vec{r}(u,v)=(u\cos v,\,u\sin v,\,1),\; 0\le u\le 2,\; 0\le v\le 2\pi"))

disk_x, disk_y, disk_z = u*cos(v), u*sin(v), 1
r_u_disk = vector([cos(v), sin(v), 0])
r_v_disk = vector([-u*sin(v), u*cos(v), 0])
normal_disk = r_u_disk.cross_product(r_v_disk)
pretty_print(LatexExpr(r"\vec{n} = "), normal_disk)

curl_on_disk = curl_F.subs({x: disk_x, y: disk_y, z: disk_z})
pretty_print(LatexExpr(r"(\nabla \times \vec{F})\big|_{\text{surface}} = "), curl_on_disk)

curl_dot_normal = curl_on_disk.dot_product(normal_disk).simplify_full()
pretty_print(LatexExpr(r"(\nabla \times \vec{F}) \cdot \vec{n} = "), curl_dot_normal)

# Nested integrals
surface_integral = integrate(integrate(curl_dot_normal, v, 0, 2*pi), u, 0, 2)
pretty_print(LatexExpr(r"\iint_S (\nabla \times \vec{F})\cdot \vec{n}\,dS = " + latex(surface_integral)))

# Boundary curve
boundary_x, boundary_y, boundary_z = 2*cos(t), 2*sin(t), 1
dx_dt, dy_dt, dz_dt = diff(boundary_x, t), diff(boundary_y, t), diff(boundary_z, t)
F_on_boundary = F_stokes.subs({x: boundary_x, y: boundary_y, z: boundary_z})
pretty_print(LatexExpr(r"\vec{F}\big|_{C} = "), F_on_boundary)

dr_dt = vector([dx_dt, dy_dt, dz_dt])
line_integrand = F_on_boundary.dot_product(dr_dt).simplify_full()
pretty_print(LatexExpr(r"\vec{F}\cdot \frac{d\vec{r}}{dt} = "), line_integrand)

line_integral_stokes = integrate(line_integrand, t, 0, 2*pi)
pretty_print(LatexExpr(r"\oint_C \vec{F}\cdot d\vec{r} = " + latex(line_integral_stokes)))

# Comparison
pretty_print(LatexExpr(r"\text{Line} = " + latex(line_integral_stokes) + r",\quad \text{Surface} = " + latex(surface_integral)))
if line_integral_stokes == surface_integral:
    pretty_print(LatexExpr(r"\text{Stokes' theorem VERIFIED! }(\text{both }=-8\pi)"))
else:
    pretty_print(LatexExpr(r"\text{Results don't match - check calculations}"))

Out[24]:

$\displaystyle \textbf{STOKES' THEOREM}$

$\displaystyle \text{Vector field: } \vec{F} = \left(y,\,-x,\,z^{2}\right)$

$\displaystyle \nabla \times \vec{F} = \left(0,\,0,\,-2\right)$

$\displaystyle \text{Surface: } \vec{r}(u,v)=(u\cos v,\,u\sin v,\,1),\; 0\le u\le 2,\; 0\le v\le 2\pi$

$\displaystyle \vec{n} = \left(0,\,0,\,u \cos\left(v\right)^{2} + u \sin\left(v\right)^{2}\right)$

$\displaystyle (\nabla \times \vec{F})\big|_{\text{surface}} = \left(0,\,0,\,-2\right)$

$\displaystyle (\nabla \times \vec{F}) \cdot \vec{n} = -2 \, u$

$\displaystyle \iint_S (\nabla \times \vec{F})\cdot \vec{n}\,dS = -8 \, \pi$

$\displaystyle \vec{F}\big|_{C} = \left(2 \, \sin\left(t\right),\,-2 \, \cos\left(t\right),\,1\right)$

$\displaystyle \vec{F}\cdot \frac{d\vec{r}}{dt} = -4$

$\displaystyle \oint_C \vec{F}\cdot d\vec{r} = -8 \, \pi$

$\displaystyle \text{Line} = -8 \, \pi ,\quad \text{Surface} = -8 \, \pi$

$\displaystyle \text{Stokes' theorem VERIFIED! }(\text{both }=-8\pi)$

In [4]:

# Fixed: use symmetric TwoSlopeNorm around 0 to avoid vmin<vcenter<vmax violation
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib import colors as mcolors

# Disk parameterization
n_u, n_v = 80, 180
U = np.linspace(0, 2.0, n_u)
V = np.linspace(0, 2*np.pi, n_v)
U, V = np.meshgrid(U, V)

X = U * np.cos(V)
Y = U * np.sin(V)
Z = np.ones_like(X)

# Curl(F) = (0,0,-2); normal (unnormalized) = (0,0,U) => density = -2U
density = -2.0 * U
cmap = cm.coolwarm
M = float(np.max(np.abs(density))) or 1.0  # safe guard if all zeros
norm = mcolors.TwoSlopeNorm(vmin=-M, vcenter=0.0, vmax=M)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

# Surface colored by curl · n density
ax.plot_surface(X, Y, Z, facecolors=cmap(norm(density)), rstride=1, cstride=1,
                edgecolor='none', shade=False, alpha=0.9)

# Boundary curve (circle) and its positive orientation arrows
tb = np.linspace(0, 2*np.pi, 200)
xb, yb, zb = 2*np.cos(tb), 2*np.sin(tb), np.ones_like(tb)
ax.plot(xb, yb, zb, color='k', lw=2)

t_ar = np.linspace(0, 2*np.pi, 24, endpoint=False)
ax.quiver(2*np.cos(t_ar), 2*np.sin(t_ar), np.ones_like(t_ar),
          -np.sin(t_ar), np.cos(t_ar), 0*t_ar,
          length=0.6, normalize=True, color='gold', linewidths=1)

# Vector field F(x,y,1) = (y,-x,1) sampled on the disk
r_samples = np.linspace(0.4, 2.0, 6)
th_samples = np.linspace(0, 2*np.pi, 16, endpoint=False)
RR, TT = np.meshgrid(r_samples, th_samples)
xs = (RR*np.cos(TT)).ravel()
ys = (RR*np.sin(TT)).ravel()
zs = np.ones_like(xs)
Fx, Fy, Fz = ys, -xs, np.ones_like(xs)
ax.quiver(xs, ys, zs, Fx, Fy, Fz, length=0.5, normalize=True, color='dimgray', linewidths=0.8)

# Upward normal at the center
ax.quiver(0, 0, 1, 0, 0, 1, length=0.8, color='limegreen')

# Colorbar for density
mappable = cm.ScalarMappable(norm=norm, cmap=cmap)
mappable.set_array(density)
cbar = fig.colorbar(mappable, ax=ax, shrink=0.6, pad=0.08)
cbar.set_label("(curl F) · n density")

# Aesthetics
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.set_title("Stokes' Theorem: flux of curl through disk equals circulation on boundary")
ax.set_xlim(-2.5, 2.5)
ax.set_ylim(-2.5, 2.5)
ax.set_zlim(0.4, 2.0)
ax.set_box_aspect((1, 1, 0.35))
ax.view_init(elev=25, azim=-60)

plt.show()

Out[4]:

Continuing Your Learning Journey

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

Ready for Real-World Applications and Modeling?

In Chapter 8, 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 8 will expand your understanding by introducing new techniques and applications that leverage everything you've learned so far.

Continue to Chapter 8: Real-World Applications and Modeling →

Return to Complete Course

Advanced Calculus with SageMath - Chapter 7

Stokes' Theorem

Chapter 7: Stokes' Theorem

Stokes' Theorem Statement

Continuing Your Learning Journey

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