Calculator inputs
Use explicit multiplication like x*y and functions such as sin(x), exp(z), or sqrt(abs(x)).
Example data table
These examples help verify the calculator with known vector fields and expected local behavior.
| Vector field F(x,y,z) | Point | Expected divergence | Expected curl | Comment |
|---|---|---|---|---|
| (x², y², z²) | (1, 1, 1) | 6 | (0, 0, 0) | Pure source-like expansion with no rotation. |
| (-y, x, 0) | (2, 3, 0) | 0 | (0, 0, 2) | Rigid planar rotation with zero net outflow. |
| (y*z, x*z, x*y) | (1, 2, 3) | 0 | (0, 0, 0) | Symmetric mixed field at this point. |
| (x*y, y*z, z*x) | (1, 2, 3) | 6 | (-3, -3, -1) | Mixed outflow and rotational behavior together. |
Formula used
For a vector field F = (Fx, Fy, Fz), the calculator estimates first partial derivatives with the central difference rule.
Central difference derivative
∂f/∂x ≈ [f(x+h,y,z) − f(x−h,y,z)] / (2h)
Divergence
div F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
Curl
curl F = (∂Fz/∂y − ∂Fy/∂z, ∂Fx/∂z − ∂Fz/∂x, ∂Fy/∂x − ∂Fx/∂y)
Divergence measures local source or sink strength. Curl measures local rotation. The magnitude of curl summarizes the overall rotational intensity near the selected point.
How to use this calculator
- Enter the three vector field components using x, y, and z.
- Provide the evaluation point coordinates.
- Choose a positive step size h for the numerical derivative estimate.
- Set the graph span and the number of sample points.
- Press the calculate button to display divergence, curl components, and the graph.
- Use the export buttons to save the current summary as CSV or PDF.
8 FAQs
1) What does divergence tell me?
Divergence measures net local outflow. Positive values suggest a source-like region, negative values suggest a sink-like region, and zero suggests balanced inflow and outflow nearby.
2) What does curl represent?
Curl represents local spinning tendency in the vector field. Its three components describe rotational behavior about the x, y, and z axes.
3) Why does the calculator use a step size?
The step size controls the central difference estimate. A very large step can blur local behavior, while a very tiny step can amplify floating-point noise.
4) Which functions can I enter?
You can use x, y, z, parentheses, powers, and common functions such as sin, cos, tan, sqrt, abs, exp, ln, and log.
5) Does zero curl mean no motion?
No. Zero curl only means no local rotation. The field can still have strong directional flow or nonzero divergence at the same point.
6) Can divergence and curl both be nonzero?
Yes. A field can expand or compress locally while also rotating. The mixed example table shows that both behaviors may appear together.
7) Why does the graph vary only along x?
The graph samples one axis to keep the display readable. It holds y and z fixed and shows how divergence and curl magnitude change around the chosen x location.
8) Is this a symbolic calculator?
No. It is a numerical calculator. It estimates derivatives at a chosen point, which makes it flexible for many practical expressions without symbolic algebra libraries.