Divergence Calculator: Measuring the Flow of Vector Fields
In vector calculus, **divergence** measures how much a vector field "spreads out" or "converges" at a given point. It's a scalar value that tells you whether you're at a source (positive divergence), a sink (negative divergence), or neither (zero divergence).
What Is Divergence?
Given a vector field $\vec{F} = (F_x, F_y, F_z)$, the divergence is:
$$\text{div} \vec{F} = \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$
The symbol $\nabla \cdot$ is read as "del dot" or "nabla dot."
2D Example
For a 2D vector field $\vec{F} = (x, y)$:
$$\text{div} \vec{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} = 1 + 1 = 2$$
This field has constant positive divergence—it represents an expanding flow.
Physical Interpretation
Positive Divergence (Source)
Imagine placing a tiny box at a point where div $\vec{F} > 0$. More "stuff" flows out of the box than flows in. This indicates a **source**—like air leaving a balloon or water from a sprinkler.
Negative Divergence (Sink)
If div $\vec{F} < 0$, more flow enters the box than leaves. This is a **sink**—like water draining into a hole or air being sucked into a vacuum.
Zero Divergence (Incompressible)
If div $\vec{F} = 0$, the flow is **incompressible**. What flows in equals what flows out. Examples: steady water flow, electromagnetic fields in a vacuum.
Applications
Fluid Dynamics
Divergence measures whether fluid is being created or destroyed at a point. In incompressible fluids (like water), div $\vec{v} = 0$ everywhere.
Electromagnetism
**Gauss's Law** can be written using divergence:
$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$
Where $\vec{E}$ is the electric field and $\rho$ is charge density. Positive charges are sources; negative charges are sinks.
Heat Transfer
The divergence of heat flux tells you whether heat is being generated or absorbed at a point.
Divergence Theorem
One of the most important results in vector calculus relates divergence to flux:
$$\int\int\int_V (\nabla \cdot \vec{F}) \, dV = \oint\oint_S \vec{F} \cdot \hat{n} \, dS$$
The total divergence inside a volume equals the net flux through its surface.
Conclusion
Divergence is a fundamental concept in physics and engineering. Use our **Divergence Calculator** to compute partial derivatives and understand field behavior at any point.