Key takeaways
- Divergence in 2D: ∇·F = ∂P/∂x + ∂Q/∂y, where F = (P, Q).
- Positive means a source (spreading out); negative means a sink (gathering in).
- Divergence is one number at each point, not a vector.
- Pair P with x and Q with y. The mixed pairs belong to curl.
In this guide
What is divergence?
A vector field puts an arrow at every point, such as the velocity of water on a pond. Now draw a tiny circle around one point and watch the flow crossing its edge.
- If more water leaves the circle than enters, the point acts like a source. Divergence is positive.
- If more enters than leaves, the point is a sink. Divergence is negative.
- If in and out balance, divergence is zero. The flow is just passing through.
Why does divergence matter?
Divergence is how physics writes down the idea 'nothing is created or lost'. Water can't be squashed, so a water flow with no taps or drains has divergence zero everywhere. If the maths gives a non-zero divergence, something is being added or removed there.
In electricity, divergence finds where electric charge sits: field lines spread out from positive charge and gather into negative charge. This is Gauss's law, one of the four Maxwell's equations.
Divergence answers one simple question at every point: is stuff coming out here, or going in?
The formula
| Symbol | What it means | Example |
|---|---|---|
| F = (P, Q) | The vector field. Here P = xⁿ and Q = yᵐ | (x², y²) |
| ∂P/∂x | How fast the x-part grows as you move in x | 2x = 2 |
| ∂Q/∂y | How fast the y-part grows as you move in y | 2y = 6 |
| (x₀, y₀) | The point where you measure | (1, 3) |
| ∇·F | The divergence (say "del dot F") | 8 |
This calculator uses the field F = (xⁿ, yᵐ). Then ∂P/∂x = n·xⁿ⁻¹ and ∂Q/∂y = m·yᵐ⁻¹, and you choose n, m and the point (x₀, y₀).
How to calculate divergence step by step
- 1Identify P and Q. P is the x-component, Q is the y-component.
- 2Find ∂P/∂x. Differentiate P in x, treating y as a constant. See the partial derivative calculator.
- 3Find ∂Q/∂y. Differentiate Q in y, treating x as a constant.
- 4Add and evaluate. Add the two and put in the point (x₀, y₀).
Try your own field below. Choose the powers and a point, and the calculator shows both derivatives and their sum.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Expecting a vector. Divergence is one number at each point, not an arrow.
- Confusing it with the gradient. The gradient turns a number-valued function into a vector; divergence turns a vector field into a number.
- Forgetting to evaluate at the point. The divergence usually changes from place to place.
- Subtracting instead of adding. Divergence is a sum.
Tips and tricks
- Spin without spread: F = (−y, x) swirls in circles and has divergence 0.
- 3D version: add a third term,
∂R/∂z, for fields F = (P, Q, R). - Sign check: in F = (xⁿ, yᵐ) with positive x and y, a positive power always gives positive divergence.
Where you'll use it in real life
- Fluid flow: zero divergence means an incompressible liquid like water.
- Electricity: Gauss's law links divergence of the electric field to charge.
- Weather: maps of where air spreads out (divergence) or piles up (convergence).
- Heat flow: positive divergence of heat flow marks a heat source, like a heater.
Quick summary
Divergence ∇·F = ∂P/∂x + ∂Q/∂y measures how much a flow spreads out from a point: positive for a source, negative for a sink, zero for balanced flow. Differentiate each part in its own variable, add and evaluate. Use the calculator above for any field (xⁿ, yᵐ).
Worked example
A model of air flow on a map uses F = (x², y²). At the point (1, 3), is air spreading out or gathering in, and how strongly? Enter n = 2, m = 2, x₀ = 1 and y₀ = 3.
- 01Vector field F = (P, Q) = (x^2, y^2)
- 02∂P/∂x = 2x^1 → at x=1: 2
- 03∂Q/∂y = 2y^1 → at y=3: 6
- 04∇·F = ∂P/∂x + ∂Q/∂y
- 05= 2 + 6
- 06= 8
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does zero divergence mean?
The flow neither spreads out nor gathers at that point. A field with zero divergence everywhere is called incompressible or solenoidal, like the flow of water.
What is the difference between divergence and curl?
Divergence measures spreading out (source or sink). Curl measures spinning. In 2D, divergence is ∂P/∂x + ∂Q/∂y, while curl is ∂Q/∂x − ∂P/∂y.
What is the divergence of F = (x², y³) at (1, 2)?
∂P/∂x = 2x = 2 and ∂Q/∂y = 3y² = 12, so the divergence is 2 + 12 = 14.
Is divergence a vector or a scalar?
A scalar. It gives one number at each point, which tells you how strongly the field spreads out or gathers in there.
