Calculator Inputs
The calculator models an affine vector field over a planar surface z = s0 + sx·x + sy·y on a rectangular x-y region.
Plotly Surface and Vector Field View
The surface is the selected plane patch. Line segments show sampled vector directions. The separate line from the center shows the chosen oriented surface normal.
Example Data Table
| Point | x | y | z | P | Q | R | F · n dS integrand |
|---|---|---|---|---|---|---|---|
| S1 | 0.3 | -0.25 | 1.2125 | 3.0625 | -1.4125 | 3.3625 | 1.4781 |
| S2 | 1 | -0.25 | 1.5625 | 4.8125 | -1.0625 | 2.3125 | -0.3594 |
| S3 | 1.7 | -0.25 | 1.9125 | 6.5625 | -0.7125 | 1.2625 | -2.1969 |
| S4 | 0.3 | 1.25 | 0.8375 | 1.1875 | 1.9625 | 4.4875 | 4.3844 |
| S5 | 1 | 1.25 | 1.1875 | 2.9375 | 2.3125 | 3.4375 | 2.5469 |
| S6 | 1.7 | 1.25 | 1.5375 | 4.6875 | 2.6625 | 2.3875 | 0.7094 |
These rows are representative sample points on the chosen surface patch.
Formula Used
Vector field: F(x, y, z) = (P, Q, R)
Affine components: P = p0 + p1x + p2y + p3z, Q = q0 + q1x + q2y + q3z, R = r0 + r1x + r2y + r3z
Surface: z = s0 + sx·x + sy·y over xmin ≤ x ≤ xmax and ymin ≤ y ≤ ymax
Upward oriented surface element: dS⃗ = (-sx, -sy, 1) dx dy
Flux integral: ∬S F · n dS = ∬R F(x, y, z(x, y)) · (-sx, -sy, 1) dx dy
Downward orientation: multiply the normal vector by -1
Because both the field and the surface are affine, the integrand is linear in x and y. The calculator therefore reports an exact rectangular integral and a midpoint check.
How to Use This Calculator
- Enter the affine coefficients for P, Q, and R.
- Define the plane using s0, sx, and sy.
- Set the x and y bounds for the rectangular patch.
- Choose upward or downward orientation.
- Adjust midpoint cells for a tighter numerical check.
- Press Calculate Integral to show the result above the form.
- Review the chart, summary metrics, and sample table.
- Download CSV or PDF when you need a report.
Frequently Asked Questions
1. What does this calculator measure?
It computes flux, the net flow of a vector field through a chosen planar surface patch. Positive values align with the chosen orientation. Negative values oppose it.
2. Which surfaces are supported here?
This version supports planar graph surfaces written as z = s0 + sx·x + sy·y over a rectangle. That keeps the model exact and easy to visualize.
3. What type of vector field can I enter?
You can enter an affine vector field. Each component may contain a constant term and separate x, y, and z coefficients.
4. Why are there exact and midpoint results?
The exact value comes from integrating the affine integrand over the rectangle. The midpoint value acts as a numerical verification and illustrates approximation behavior.
5. How does orientation affect the answer?
Reversing orientation flips the normal vector. That changes the sign of the flux while preserving the same surface geometry.
6. What does average flux density mean?
It is the exact flux divided by the surface area. Use it to compare how strongly the field crosses surfaces of different sizes.
7. Why is the plot useful?
The plot shows the plane patch, sampled field directions, and the chosen normal vector. This helps connect the numerical output with the underlying geometry.
8. Can I use decimals and negative bounds?
Yes. The inputs accept decimals, negative values, and reversed bounds. Reversed bounds are automatically reordered before the calculation runs.