Vector Calculus: Fields & Paths

Vector calculus forms the backbone of physical sciences and advanced engineering, providing the mathematical language to describe fluid flow, electromagnetism, and gravitational potential. In this lesson, we move beyond the differential calculus of single variables into the rich interplay between vector fields and the geometry of paths in ${\mathbb{R}}^n$.

1. Formal Definition of Vector Fields

Let $\Omega \subseteq {\mathbb{R}}^n$ be an open set. A vector field $\mathbf{F}$ on $\Omega$ is a function that assigns to each point $\mathbf{x}\in \Omega$ a vector $\mathbf{F}(\mathbf{x})\in {\mathbb{R}}^n$: $\mathbf{F}:\Omega \to {\mathbb{R}}^n.$

In the language of differential geometry, a vector field is a section of the tangent bundle $T\Omega$. For ${\mathbb{R}}^n$, we typically write: $\mathbf{F}(x_1,\dots ,x_n)={\sum }_{i=1}^n{F}_i(x_1,\dots ,x_n){\mathbf{e}}_i,$ where ${\mathbf{e}}_i$ are the standard basis vectors. We say $\mathbf{F}$ is of class $C^k$ if each component function $F_i$ is $k$-times continuously differentiable.

Visualization and Flow

A vector field can be visualized as a “force field” where the vector at each point indicates direction and magnitude. If $\mathbf{F}$ represents a velocity field of a fluid, a “streamline” is a curve $\gamma (t)$ such that ${\gamma }^{\prime }(t)=\mathbf{F}(\gamma (t))$. This links vector fields to the theory of ordinary differential equations (ODEs).

2. Line Integrals

Line integrals generalize the concept of integration to higher dimensions by allowing us to integrate functions along curves (trajectories).

2.1 Line Integrals of the First Kind (Scalar Fields)

The line integral of a scalar field $f:\Omega \to \mathbb{R}$ along a smooth curve $C$ (parametrized by $\mathbf{r}(t),a\le t\le b$) with respect to arc length is defined as:

${\int }_{C}fds={\int }_a^{b}f(\mathbf{r}(t))\Vert {\mathbf{r}}^{\prime }(t)\Vert dt.$

This integral is independent of the parametrization of $C$, provided the orientation remains consistent. It is used for calculating mass from linear density or the area of “fences” built over the curve.

2.2 Line Integrals of the Second Kind (Vector Fields)

The line integral of a vector field $\mathbf{F}$ along $C$ measures the accumulation of the field’s component tangent to the path. It is defined as:

${\int }_C\mathbf{F}\cdot d\mathbf{r}={\int }_a^b\mathbf{F}(\mathbf{r}(t))\cdot {\mathbf{r}}^{\prime }(t)dt.$

In physics, if $\mathbf{F}$ is a force field, this integral represents the work done by the field on a particle moving along $C$.

3. Conservative Vector Fields and Potentials

A vector field $\mathbf{F}$ is called conservative (or a gradient field) if there exists a scalar potential $\varphi :\Omega \to \mathbb{R}$ such that: $\mathbf{F}=\nabla \varphi$

The Fundamental Theorem for Line Integrals

If $\mathbf{F}=\nabla \varphi$ is a conservative field on a region $\Omega$, then for any piecewise smooth curve $C$ starting at $\mathbf{a}$ and ending at $\mathbf{b}$ in $\Omega$: ${\int }_C\mathbf{F}\cdot d\mathbf{r}=\varphi (\mathbf{b})-\varphi (\mathbf{a})$

This implies two critical properties of conservative fields:

1. Path Independence: The integral depends only on the endpoints.
2. Vanishing Loop Integrals: For any closed curve $C$ ($\mathbf{a}=\mathbf{b}$), ${\oint }_C\mathbf{F}\cdot d\mathbf{r}=0$.

4. Divergence and Curl

To analyze the local behavior of vector fields, we define two fundamental operators using the Nabla ($\nabla$) operator.

4.1 Divergence (Source Density)

The divergence of $\mathbf{F}=⟨P,Q,R⟩$ in ${\mathbb{R}}^3$ is the scalar field: $\text{div }\mathbf{F}=\nabla \cdot \mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$ It measures the net flow of the field from a point (the “outwardness”). In fluid dynamics, a divergence-free field ($\nabla \cdot \mathbf{F}=0$) is called incompressible.

4.2 Curl (Rotation Density)

The curl of $\mathbf{F}$ in ${\mathbb{R}}^3$ is the vector field: $\text{curl }\mathbf{F}=\nabla \times \mathbf{F}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|$ The curl measures the infinitesimal rotation of the field. A field with $\nabla \times \mathbf{F}=0$ is called irrotational.

5. Topology and Poincaré’s Lemma

A common misconception is that all irrotational fields are conservative. While $\nabla \times (\nabla \varphi )=0$ is always true, the converse ($\nabla \times \mathbf{F}=0⟹\mathbf{F}=\nabla \varphi$) requires the domain $\Omega$ to be simply connected.

Poincaré’s Lemma

In a simply connected domain (a domain where every loop can be continuously contracted to a point), a smooth vector field $\mathbf{F}$ is conservative if and only if it is irrotational.

Example of failure: The “vortex field” $\mathbf{F}=⟨\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}⟩$ on ${\mathbb{R}}^2\setminus \{(0,0)\}$ is irrotational everywhere (its curl is zero), but its integral around a circle enclosing the origin is $2\pi \ne 0$. Thus, it is not conservative. The puncture at the origin makes the domain not simply connected.

6. Green’s Theorem

Green’s Theorem provides a bridge between the line integral around a simple closed curve and the double integral over the plane region it encloses.

Let $C$ be a positively oriented, piecewise smooth, simple closed curve in ${\mathbb{R}}^2$, and let $D$ be the region bounded by $C$. If $\mathbf{F}=⟨P,Q⟩$ has continuous partial derivatives on an open region containing $D$, then: ${\oint }_C(Pdx+Qdy)={\iint }_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA$

Note that $\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$ is simply the $k$-component of the curl of a 2D vector field.

7. Python Implementation: Numerical Line Integrals

Below is a Python script that visualizes a non-conservative vector field (a vortex) and calculates the work done along a circular path numerically.

import numpy as np
import matplotlib.pyplot as plt

# Define the vector field F(x, y) = (-y, x) / (x^2 + y^2)
def F(x, y):
    r2 = x**2 + y**2
    return -y/r2, x/r2

# 1. Visualize the field
x = np.linspace(-2, 2, 20)
y = np.linspace(-2, 2, 20)
X, Y = np.meshgrid(x, y)
U, V = F(X, Y)

plt.figure(figsize=(8, 8))
plt.quiver(X, Y, U, V, color='r')
plt.title("Vortex Vector Field $\\mathbf{F} = (-y, x) / r^2$")

# 2. Numerical Line Integral along unit circle
# r(t) = (cos(t), sin(t)), r'(t) = (-sin(t), cos(t))
t = np.linspace(0.001, 2*np.pi, 1000) # avoid exact zero
dt = t[1] - t[0]

rx = np.cos(t)
ry = np.sin(t)
drx = -np.sin(t)
dry = np.cos(t)

# Calculate F(r(t)) . r'(t)
fx, fy = F(rx, ry)
integrand = fx * drx + fy * dry
integral = np.sum(integrand) * dt

print(f"Numerical Line Integral: {integral:.4f}")
print(f"Analytical Result (2*pi): {2*np.pi:.4f}")

plt.plot(rx, ry, 'b-', label='Path C (Unit Circle)')
plt.legend()
plt.show()

This lesson provides the groundwork for Stokes’ Theorem and the Divergence Theorem, which generalize these results to higher-dimensional surfaces and manifolds.