Core Idea
The divergence theorem links a closed surface integral to a volume integral. For volume work, choose a vector field with constant divergence. The common choice is F equals one third of position vector. Its divergence is one. Therefore, the outward flux through the boundary equals the enclosed volume. This calculator applies that idea to standard solids and custom flux data.
Why It Helps
Direct volume integration can be slow. A surface description is sometimes easier. Engineers may know boundary flux. Students may know the surface equation. The theorem converts that information into a volume estimate. It also gives a useful check against familiar formulas.
Supported Shapes
The tool includes boxes, spheres, cylinders, cones, and ellipsoids. These shapes use exact geometric formulas. The same result is then interpreted through divergence theorem flux. A custom mode accepts a measured surface integral. You may enter flux for a field with divergence one. You may also enter total r dot n flux and divide it by three.
Advanced Options
Units are handled through a selected length unit. The calculator reports the native cubic unit and a cubic meter conversion. Decimal control helps with classroom answers and report values. Extra outputs show equivalent cube edge, equivalent sphere radius, and expected flux values. These checks make errors easier to find.
Practical Use
Start with the solid type. Enter only the dimensions required by that shape. Choose a unit and the needed decimal precision. For custom data, select the flux meaning carefully. Press the calculate button. The result appears above the form, so it is easy to review before editing inputs.
Accuracy Notes
All standard shape formulas assume ideal geometry. Measured custom flux depends on sensor quality, surface orientation, and numerical integration. The surface must be closed. Normals must point outward. If the vector field does not have constant divergence, a simple division is not enough. In that case, use the full volume integral of divergence.
Best Practices
Keep consistent units for every dimension. Avoid mixing inches with feet. Round only after the final calculation. Save the CSV when comparing many solids. Use the PDF when sharing a solved example. Recheck any negative or zero dimension before trusting the answer, and review surface closure carefully.