Calculator inputs
Example data table
| r | Ar | Aφ | Az | ∂Ar/∂r | ∂Aφ/∂φ | ∂Az/∂z | Divergence |
|---|---|---|---|---|---|---|---|
| 2.0 | 4.0 | 1.5 | 3.0 | 1.2 | 0.8 | -0.3 | 3.3 |
For this example, divergence = 1.2 + 4/2 + 0.8/2 - 0.3 = 3.3.
Formula used
For a vector field F = Ar er + Aφ eφ + Az ez in cylindrical coordinates, the divergence is:
∇·F = (1/r) ∂(rAr)/∂r + (1/r) ∂Aφ/∂φ + ∂Az/∂z
Equivalent form: ∇·F = ∂Ar/∂r + Ar/r + (1/r)(∂Aφ/∂φ) + ∂Az/∂z
The calculator uses the equivalent expanded form so each contribution can be shown separately and interpreted easily.
How to use this calculator
- Enter the radial position r, ensuring it is not zero.
- Provide the vector components Ar, Aφ, and Az at the evaluation point.
- Enter the partial derivatives ∂Ar/∂r, ∂Aφ/∂φ, and ∂Az/∂z.
- Choose how many decimal places you want in the output.
- Press Calculate divergence to show the result above the form.
- Review the term breakdown, field magnitude, classification, and Plotly graph.
- Use the CSV or PDF buttons to export the displayed results.
Frequently asked questions
1. What does divergence represent in cylindrical coordinates?
It measures local net outflow of a vector field around a point. Positive values indicate source-like behavior, negative values indicate sink-like behavior, and values near zero suggest balanced flow.
2. Why can’t r be zero in this calculator?
The cylindrical divergence formula includes division by r. At r = 0, that expression becomes singular unless the field is treated with a special limiting argument outside this calculator.
3. Why is Aφ itself not used directly in the total?
Divergence depends on how components vary, not only on their values. For the azimuthal part, the formula uses ∂Aφ/∂φ divided by r rather than Aφ by itself.
4. What is the meaning of the radial term?
The radial contribution combines direct radial change and geometric scaling. That is why the expanded form includes both ∂Ar/∂r and Ar/r in one combined radial term.
5. Can this calculator be used for physics problems?
Yes. It is useful in electromagnetics, fluid flow, heat transfer, and vector calculus exercises wherever a field is expressed in cylindrical coordinates and evaluated at a point.
6. What does the chart show?
The chart shows how divergence changes as r varies while the entered field values and derivatives remain fixed. It helps visualize the effect of the 1/r terms.
7. Is the displayed field magnitude part of divergence?
No. Field magnitude is an extra reference value only. Divergence is calculated from the component derivatives and geometric terms, not from the magnitude alone.
8. When is the divergence approximately zero?
It is approximately zero when radial, azimuthal, and axial contributions balance one another. In that case, there is no net local flux expansion or contraction.