Surface Integral (Flux) Calculator
Calculate Flux ∬S F · dS through a surface z = g(x,y).
1. Vector Field F(x, y, z)
2. Surface S: z = g(x,y)
Integration Bounds (Domain D)
Total Flux
Surface Visualization
Drag to RotateVisual: Wireframe shows the surface z = g(x,y). Red lines indicate normal vectors used for flux.
What is a Surface Integral (Flux)?
A Surface Integral measures how much of a vector field “flows” through a curved surface. This specific quantity is called Flux. Think of the vector field as water flowing and the surface as a fishing net. Flux tells you how much water passes through the net per second.
The Formula
For a surface defined by z = g(x,y), we calculate flux by projecting it onto the xy-plane (Domain D):
∬S F · dS = ∬D (-P⋅∂z/∂x – Q⋅∂z/∂y + R) dA
This formula automatically accounts for the surface’s tilt using partial derivatives.
Real World Apps
- Electromagnetism: Gauss’s Law calculates electric flux through a closed surface to find charge.
- Fluid Dynamics: Calculating the rate of flow through a pipe or turbine.
- Heat Transfer: Measuring heat energy leaving a radiator surface.
Understanding “Normal Vectors”
To calculate flux, we need to know which way the surface is facing. The Normal Vector (n) is an arrow sticking straight out (perpendicular) from the surface.
If the flow (F) is in the same direction as n, flux is positive. If they oppose each other, flux is negative. If flow skims parallel to the surface, flux is zero.
Frequently Asked Questions (FAQ)
Q: What if the surface is closed (like a sphere)?
A: This calculator works for “open” surfaces defined by functions $z=g(x,y)$. For closed surfaces, you typically split them into top/bottom parts or use the Divergence Theorem.
Q: Why is it called “Flux”?
A: It comes from the Latin fluxus meaning “flow”. Even if the field is static (like an electric field), we imagine “field lines” flowing through the surface.
Q: Can I use this for Surface Area?
A: Yes! If you set the vector field to specific values, you can calculate area, but it’s complex. Standard Surface Area formula is $\iint \sqrt{1 + (\partial z/\partial x)^2 + (\partial z/\partial y)^2} dA$. This tool calculates Vector Flux, which is slightly different.