Learn line integrals and work calculations through interactive SageMath computations covering path integrals, conservative vector fields, and fundamental theorem for line integrals. This comprehensive Jupyter notebook explores parametric curves, arc length calculations, work done by force fields, and circulation problems with applications in physics and engineering. CoCalc provides dynamic visualization of vector fields along curves, symbolic path integration tools, and potential function calculations, allowing students to understand line integral concepts through hands-on computational exploration.

Published/Calculus Series/Advanced Calculus with SageMath/Advanced Calculus with SageMath - Chapter 4.ipynb

¹⁸ views
ubuntu2404

Kernel: SageMath 10.7

Advanced Calculus with SageMath - Chapter 4

Vector Fields and Line Integrals

This notebook contains Chapter 4 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 4: Vector Fields and Line Integrals

Understanding Vector Fields

A vector field assigns a vector to each point in space. Examples:

Gravitational field: $\vec{F} = -GMm/r^2 \hat{r}$
Electric field: $\vec{E} = kq/r^2 \hat{r}$
Fluid velocity: $\vec{v}(x,y,z)$

Line Integrals and Work

The line integral $\int_C \vec{F} \cdot d\vec{r}$ represents work done by force field $\vec{F}$ along curve C:

\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \frac{d\vec{r}}{dt} dt

In [11]:

from sage.misc.latex import LatexExpr
var('t')  # ensure t is defined

print("VECTOR FIELDS AND LINE INTEGRALS")
print("=" * 50)

# Define a vector field F = (y, -x)
F_x, F_y = y, -x
F_field = vector([F_x, F_y])
show(LatexExpr(r"\vec{F}(x,y) = " + latex(F_field)))

# Parametric curve: r(t) = (2cos t, 2sin t), 0 ≤ t ≤ 2π
r_x, r_y = 2*cos(t), 2*sin(t)
r_curve = vector([r_x, r_y])

print("\nPARAMETRIC CURVE")
show(LatexExpr(r"\vec{r}(t) = " + latex(r_curve) + r",\quad t\in[0,2\pi]"))

# Compute dr/dt
dr_dt = vector([diff(r_x, t), diff(r_y, t)])
show(LatexExpr(r"\frac{d\vec{r}}{dt} = " + latex(dr_dt)))

# Substitute parametric equations into vector field
F_on_curve = vector([F_x.subs({x: r_x, y: r_y}), F_y.subs({x: r_x, y: r_y})])

print("\nVECTOR FIELD ON CURVE")
show(LatexExpr(r"\vec{F}(\vec{r}(t)) = " + latex(F_on_curve)))

# Compute F · dr/dt
dot_product = F_on_curve.dot_product(dr_dt)
dot_product_simplified = simplify(dot_product)

print("\nDOT PRODUCT")
show(LatexExpr(r"\vec{F}\cdot \frac{d\vec{r}}{dt} = " + latex(dot_product_simplified)))

# Compute line integral
line_integral = integrate(dot_product_simplified, (t, 0, 2*pi))

print("\nLINE INTEGRAL")
show(LatexExpr(r"\int_C \vec{F}\cdot d\vec{r} = " + latex(line_integral)))

print("\nPhysical interpretation: Work done by the vector field along the curve")

Out[11]:

VECTOR FIELDS AND LINE INTEGRALS
==================================================

$\displaystyle \vec{F}(x,y) = \left(y,\,-x\right)$

PARAMETRIC CURVE

$\displaystyle \vec{r}(t) = \left(2 \, \cos\left(t\right),\,2 \, \sin\left(t\right)\right) ,\quad t\in[0,2\pi]$

$\displaystyle \frac{d\vec{r}}{dt} = \left(-2 \, \sin\left(t\right),\,2 \, \cos\left(t\right)\right)$

VECTOR FIELD ON CURVE

$\displaystyle \vec{F}(\vec{r}(t)) = \left(2 \, \sin\left(t\right),\,-2 \, \cos\left(t\right)\right)$

DOT PRODUCT

$\displaystyle \vec{F}\cdot \frac{d\vec{r}}{dt} = -4 \, \cos\left(t\right)^{2} - 4 \, \sin\left(t\right)^{2}$

LINE INTEGRAL

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

Physical interpretation: Work done by the vector field along the curve

In [6]:

# Visualization: vector field F=(y, -x) and circular path r(t) with tangent vs. F(r(t))
var('x y t')
R = 2

p_field = plot_vector_field((y, -x), (x, -3, 3), (y, -3, 3), color='lightgray')
p_curve = parametric_plot((R*cos(t), R*sin(t)), (t, 0, 2*pi), color='crimson', thickness=3)

t0 = pi/4
p0 = vector([R*cos(t0), R*sin(t0)]).n()
dr_dt0 = vector([-R*sin(t0), R*cos(t0)])
v_tan = (dr_dt0/dr_dt0.norm())*0.9
v_F = vector([p0[1], -p0[0]]); v_F = (v_F/v_F.norm())*0.9

g = (p_field + p_curve
     + point(tuple(p0), size=30, color='black')
     + arrow(tuple(p0), tuple(p0 + v_tan), color='blue', arrowsize=4, thickness=3)
     + arrow(tuple(p0), tuple(p0 + v_F), color='green', arrowsize=4, thickness=3)
     + text(r"$\frac{d\vec{r}}{dt}$", tuple(p0 + v_tan + vector([0.15, 0.05])), color='blue', fontsize=11)
     + text(r"$\vec{F}(\vec{r}(t))$", tuple(p0 + v_F + vector([0.15, -0.05])), color='green', fontsize=11))

show(g, aspect_ratio=1, frame=True,
     title='Vector field F=(y,-x) with circular path r(t): tangent vs. F(r(t))')

Out[6]:

Continuing Your Learning Journey

You've completed Vector Fields and Line Integrals! The concepts you've mastered here form essential building blocks for what comes next.

Ready for Green's Theorem and Applications?

In Chapter 5, 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 5 will expand your understanding by introducing new techniques and applications that leverage everything you've learned so far.

Continue to Chapter 5: Green's Theorem and Applications →

Return to Complete Course