Divergence Theorem Verification
Verify that Volume Integral of Divergence equals Net Flux through Surface.
1. Vector Field F(x, y, z)
2. Volume V (Rectangular Box)
∠(Div F) dV
∯ F · n dS
3D Visualization
Drag to RotateDivergence Theorem: The stuff created inside (Volume Integral) must exit through the walls (Flux).
What is the Divergence Theorem?
Also known as Gauss’s Theorem, this is one of the most important theorems in vector calculus. It connects the flow of a vector field through a closed surface to the behavior of the field inside the volume.
The Equation
âˆV (∇ · F) dV = ∯S F · n dS
Total Sources Inside = Total Flow Out
The “Water Hose” Analogy
Imagine a box submerged in water.
- Flux (Right Side): This measures the net amount of water leaving the box through its walls. If more leaves than enters, net flux is positive.
- Divergence (Left Side): This measures if water is being “created” inside the box (like a hidden hose turned on). Positive divergence means a point is a “source”.
The theorem simply says: If water is flowing out of the box (Flux), there must be a source creating water inside (Divergence). Mass is conserved!
Frequently Asked Questions (FAQ)
Q: Does shape matter?
A: The theorem holds for any closed shape (sphere, cube, blob), as long as the surface encloses the volume completely. This tool uses a box to make the math easier to visualize.
Q: What if Divergence is zero?
A: If $\nabla \cdot F = 0$ everywhere (like magnetic fields), then the total flux through any closed surface is zero. What goes in must come out. This is known as a “Solenoidal” field.
Q: Why 6 faces?
A: A rectangular box has 6 flat sides (Top, Bottom, Left, Right, Front, Back). To calculate the total surface integral $\oint_S$, we calculate the flux through each of these 6 rectangles and add them up.