Solve sphere flux with guided calculus steps. Enter coefficients, radius, and center for accurate analysis. Review formulas, graphs, exports, examples, and practical learning notes.
Supported field form: F(x, y, z) = <a₁x + a₂y + a₃z + a₄, b₁x + b₂y + b₃z + b₄, c₁x + c₂y + c₃z + c₄>
Calculate Flux to see the result above the form.This tool is ideal for vector calculus practice because it keeps the theorem step visible and easy to verify.
| Example | Vector Field | Sphere Center | Radius | Divergence | Flux |
|---|---|---|---|---|---|
| Sample 1 | <2x + y + 3, -x + 4y + 2z, 5x + y - 2z + 1> | (0, 0, 0) | 2 | 4 | 134.041286 |
| Sample 2 | <3x - y, 2x + y + z, z + 5> | (1, -1, 2) | 3 | 5 | 565.486678 |
| Sample 3 | <x + 7, y - 4, z + 9> | (2, 2, 2) | 1.5 | 3 | 42.411501 |
The divergence theorem links volume behavior to surface flow. It compares total outward flux through a closed surface with divergence gathered inside it. A sphere is a clean testing shape. Its symmetry simplifies many steps. This calculator uses that idea directly. You enter affine vector field coefficients and sphere data. The tool then finds divergence, sphere volume, and total outward flux. It also shows the field form clearly. That helps students connect symbols, geometry, and final values without losing the theorem logic.
A sphere has constant curvature and a simple volume formula. That makes flux problems easier. For affine vector fields, divergence is constant everywhere. This matters a lot. When divergence stays constant, integration over the sphere interior becomes direct. You only need divergence multiplied by volume. The center of the sphere does not change that conclusion. Only the radius changes the volume term. Larger spheres collect more total flux. The growth is cubic. That is why the graph on this page curves upward.
Start by entering the sphere center and radius. Then fill the coefficients for each vector component. The first row controls the x component. The second row controls the y component. The third row controls the z component. Constant terms are allowed. They shift the field but do not affect divergence. After submission, review the result card first. It appears above the form. Then inspect the formula section and graph. Export the results if needed. The sample table below also helps you verify ideas with a worked example.
The key result is total outward flux. A positive value means net flow leaves the sphere. A negative value means net flow enters it. A zero value means the source and sink effect balances overall. Surface area is shown for context. Average flux density is also included. That value spreads total flux across the sphere area. Use it for quick comparison between different radii. This calculator is useful for classwork, revision, and checking manual solutions. It also works for spheres not centered at the origin in practice.
It computes outward flux through a sphere using the divergence theorem for an affine vector field. It also reports divergence, volume, surface area, and average flux density. The result assumes outward orientation, which is standard for closed surfaces.
This page supports affine fields of the form Ax + By + Cz + constant in each component. That includes linear terms, mixed terms, and constants. The divergence is constant, so the theorem becomes simple and fast to apply.
Constant terms change the field expression, but they do not change divergence. Because this calculator uses divergence times volume, constants do not affect the final flux through a closed sphere for the supported field type.
For the supported affine field, the sphere center does not change divergence. So the total flux depends on divergence and sphere volume only. The center is still included to define the sphere clearly and present the geometry accurately.
Radius controls volume, and volume grows with the cube of radius. Because flux equals divergence times volume here, even small radius increases can produce much larger flux values. The graph helps you see that nonlinear growth quickly.
Yes. The calculator reports outward flux. For inward orientation, multiply the shown flux by negative one. The magnitude stays the same, but the sign reverses because the normal vector points in the opposite direction.
Not exactly. It is a divergence theorem solver specialized for spheres and affine fields. Instead of integrating over the surface directly, it integrates constant divergence over the enclosed volume. That is usually faster and cleaner.
First compare the divergence value. Then verify the sphere volume. Finally multiply both numbers and confirm the sign. If your manual answer differs, check the diagonal coefficients, radius entry, and whether you used outward or inward orientation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.