Calculator Inputs
Supported functions: sin, cos, tan, sqrt, abs, exp, log, min, max, and powers using ^.
Example Data Table
| Example | Vector field | Surface | Expected trend |
|---|---|---|---|
| 1 | F = (x, y, z) | Sphere radius 2, full outward surface | Flux ≈ 32π ≈ 100.53 |
| 2 | F = (0, 0, 5) | Plane z = 2 over [-1,1] × [-1,1] | Flux = 20 with positive z normal |
| 3 | F = (x, y, 1) | Graph z = x² + y² over [-1,1] × [-1,1] | Positive upward flux is expected |
| 4 | F = (x, y, 0) | Cylinder radius 2, z from 0 to 3 | Positive outward flux is expected |
Formula Used
General flux integral: ∬S F · n dS
For a parameterized surface r(u,v), the calculator evaluates ∬ F(r(u,v)) · (ru × rv) du dv with midpoint sampling.
Coordinate plane patch: use a constant unit normal along x, y, or z.
Example for z = c: flux = ∬ F(x,y,c) · (0,0,±1) dA.
Graph surface: for z = g(x,y), the oriented area vector is (−gx, −gy, 1).
Downward orientation multiplies that vector by −1.
Sphere patch: x = r sinφ cosθ, y = r sinφ sinθ, z = r cosφ.
The outward area vector density is r² sinφ · ûradial.
Cylinder side: x = r cosθ, y = r sinθ, z = z.
The outward area vector density is (r cosθ, r sinθ, 0).
How to Use This Calculator
- Select the surface type you want to integrate across.
- Enter the vector field components P, Q, and R.
- Fill the geometric inputs for the chosen surface.
- Choose orientation carefully because sign depends on the normal.
- Increase cell counts for smoother numerical estimates.
- Press calculate to display the result above the form.
- Review the sample plot and example table for validation.
- Download CSV or PDF after a successful calculation.
FAQs
1. What does the calculator actually compute?
It estimates the surface flux integral of a vector field through an oriented surface. The result measures how strongly the field passes through the chosen surface in the selected normal direction.
2. Why can the answer be negative?
A negative flux means the field points mostly opposite the chosen normal direction. Switching the orientation from outward to inward, or upward to downward, flips the sign.
3. How accurate is the numerical result?
The calculator uses midpoint integration over a grid. Accuracy improves when you increase both cell counts, especially for curved surfaces or rapidly changing vector fields.
4. Which expressions can I type?
You can use numbers, x, y, z, theta, phi, powers with ^, and common functions like sin, cos, tan, sqrt, abs, exp, log, min, and max.
5. When should I use the graph surface option?
Use it when your surface can be written as z = g(x,y) over a rectangle. The calculator estimates partial derivatives numerically, then builds the oriented area vector automatically.
6. Does the cylinder option include top and bottom caps?
No. The cylinder mode integrates only over the curved side surface. If you need a closed cylinder, compute the side and both caps separately, then add the three flux values.
7. How can I check whether my setup is reasonable?
Start with a known example from the example table, compare the estimate with the expected trend, then gradually replace the field or surface with your own inputs.
8. Why is orientation so important?
Flux depends on the normal vector. The same field and surface can produce equal magnitudes with opposite signs if the chosen normal direction is reversed.