∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 6 Theory

Vector Fields, Line Integrals, Green's, Stokes' & Divergence Theorems

Circulation, curl, outward flux, and the 3D Fundamental Theorems

Watch live particles flow through a 3D vector field F(x,y,z) = ⟨P, Q, R⟩ and visualize how microscopic curl (∇ × F) and divergence (∇ · F) accumulate into boundary circulation and flux.

1. Green's, Stokes' & Gauss's Divergence Theorems

Every major theorem of vector calculus is a higher-dimensional statement of the Fundamental Theorem of Calculus: integrating a microscopic derivative (gradient, curl, or divergence) over the interior of a domain equals the net boundary measurement.

∭E(∇⋅F) dV=∯∂EF⋅dS\iiint_E (\nabla \cdot \mathbf{F})\,dV = \oiint_{\partial E} \mathbf{F} \cdot d\mathbf{S}

Quick Reference & Cheat Sheet

Divergence & Curl Operators
∇⋅F=Px+Qy+Rz,∇×F=⟨Ry−Qz,Pz−Rx,Qx−Py⟩\nabla \cdot \mathbf{F} = P_x + Q_y + R_z, \quad \nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle
Divergence measures net outflow per unit volume; Curl measures local rotational axis and speed.