Master vector calculus fundamentals through interactive SageMath computations covering vector fields, gradients, and directional derivatives. This comprehensive Jupyter notebook introduces 3D coordinate systems, vector operations, and field visualizations with applications to fluid dynamics and electromagnetic fields. CoCalc's cloud-based platform provides instant access to symbolic computation tools and dynamic 3D plotting capabilities, enabling students to explore gradient fields, calculate divergence and curl, and understand vector calculus concepts through hands-on experimentation without software installation.

Published/Calculus Series/Advanced Calculus with SageMath/Advanced Calculus with SageMath - Chapter 1.ipynb

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Advanced Calculus with SageMath - Chapter 1

Historical Context and Mathematical Foundations

This notebook contains Chapter 1 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 1: Historical Context and Mathematical Foundations

The Evolution of Calculus

Advanced calculus emerged from the need to understand phenomena in multiple dimensions:

1734: Leonhard Euler develops partial derivatives
1762: Joseph-Louis Lagrange introduces the method of Lagrange multipliers
1828: George Green publishes Green's theorem
1854: George Gabriel Stokes formulates Stokes' theorem
1867: William Thomson (Lord Kelvin) and others develop vector calculus
1887: Oliver Heaviside creates modern vector notation

Mathematical Prerequisites Review

Before diving into multivariable calculus, let's review essential concepts:

In [4]:

# Mathematical foundations using Sage vectors and functions
x, y, z, t = var('x y z t')

print("MATHEMATICAL FOUNDATIONS")
print("=" * 50)

# Vector operations
v1 = vector([3, -2, 1])
v2 = vector([1, 4, -2])

print("Vector v₁:", v1)
print("Vector v₂:", v2)
print("Dot product v₁ · v₂:", v1.dot_product(v2))
print("Cross product v₁ × v₂:", v1.cross_product(v2))
print("Magnitude |v₁|:", v1.norm())

print("\nCOORDINATE SYSTEMS")
print("Cartesian to cylindrical: (x,y,z) → (ρ,φ,z)")
print("ρ = √(x² + y²), φ = arctan(y/x), z = z")

# Example coordinate transformation
cart_point = [3, 4, 5]
rho = sqrt(cart_point[0]**2 + cart_point[1]**2)
phi = arctan2(cart_point[1], cart_point[0])
print(f"Point (3,4,5) → (ρ={float(rho):.2f}, φ={float(phi):.2f}, z=5)")

Out[4]:

MATHEMATICAL FOUNDATIONS
==================================================
Vector v₁: (3, -2, 1)
Vector v₂: (1, 4, -2)
Dot product v₁ · v₂: -7
Cross product v₁ × v₂: (0, 7, 14)
Magnitude |v₁|: sqrt(14)

COORDINATE SYSTEMS
Cartesian to cylindrical: (x,y,z) → (ρ,φ,z)
ρ = √(x² + y²), φ = arctan(y/x), z = z
Point (3,4,5) → (ρ=5.00, φ=0.93, z=5)

Continuing Your Learning Journey

You've completed Historical Context and Mathematical Foundations! The concepts you've mastered here form essential building blocks for what comes next.

Ready for Multivariable Functions and Partial Derivatives?

In Chapter 2, 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 2 will expand your understanding by introducing new techniques and applications that leverage everything you've learned so far.

Continue to Chapter 2: Multivariable Functions and Partial Derivatives →

Return to Complete Course