Calculator Inputs
Formula Used
This calculator treats a two-dimensional field as F(x, y) = (P(x, y), Q(x, y)). A field line is traced by solving the path equations dx/ds = P(x, y) and dy/ds = Q(x, y).
When normalization is enabled, the calculator uses dx/ds = P/|F| and dy/ds = Q/|F|, where |F| = √(P² + Q²). This keeps direction while reducing speed-based stretching.
Numerical integration uses the classical fourth-order Runge–Kutta method: a weighted blend of four slope evaluations over each step. This improves stability and accuracy compared with simple Euler stepping.
Additional reported metrics use: Arc length = Σ√((Δx)² + (Δy)²), heading θ = atan2(Q, P), and curvature estimated from three consecutive traced points.
The local divergence estimate is ∂P/∂x + ∂Q/∂y, and the two-dimensional scalar curl estimate is ∂Q/∂x − ∂P/∂y. Both are approximated numerically near the seed point.
How to Use This Calculator
- Enter the horizontal and vertical field components using x and y.
- Choose a seed point where tracing should begin.
- Set the step size and maximum step count.
- Define the visible plotting window using x and y bounds.
- Choose whether vectors should be normalized before integration.
- Set a stop tolerance to halt tracing near very small magnitudes.
- Press Calculate Field Line to show the result above the form.
- Review the summary, Plotly charts, and point table, then export CSV or PDF if needed.
Example Data Table
Example setup for the rotational field P(x,y)=-y, Q(x,y)=x, seed (1, 0), step 0.08.
| Point | Arc | X | Y | Vx | Vy | |F| |
|---|---|---|---|---|---|---|
| 1 | 0.0000 | 1.0000 | 0.0000 | 0.0000 | 1.0000 | 1.0000 |
| 2 | 0.0800 | 0.9968 | 0.0799 | -0.0799 | 0.9968 | 1.0000 |
| 3 | 0.1600 | 0.9872 | 0.1593 | -0.1593 | 0.9872 | 1.0000 |
| 4 | 0.2400 | 0.9713 | 0.2377 | -0.2377 | 0.9713 | 1.0000 |
| 5 | 0.3200 | 0.9492 | 0.3146 | -0.3146 | 0.9492 | 1.0000 |
FAQs
1) What does this calculator compute?
It traces a field line through a two-dimensional vector field using your component functions, seed point, numerical settings, and selected plotting bounds.
2) What is a field line in this context?
A field line is a path whose tangent direction matches the vector field at every traced point. It reveals how the field locally guides motion.
3) Why would I enable vector normalization?
Normalization preserves direction while reducing the influence of vector magnitude. It helps compare geometry of paths without speed-driven stretching or compression.
4) Why does tracing stop early sometimes?
Tracing ends when the path leaves the chosen bounds, the field becomes too weak, or the expression cannot be evaluated reliably at a new point.
5) Which functions can I use in expressions?
You can use x, y, numbers, parentheses, powers, and common functions like sin, cos, tan, exp, log, sqrt, abs, min, max, and pow.
6) What numerical method is used?
The calculator uses fourth-order Runge–Kutta integration, which is more accurate and stable than simple first-order stepping for many smooth fields.
7) What do divergence and curl tell me?
Divergence estimates local expansion or contraction near the seed, while scalar curl estimates local rotation. Both are approximated numerically from nearby evaluations.
8) Can I export my result?
Yes. After calculation, you can export the traced point table as CSV and create a PDF report containing settings, summary values, and table rows.