Checking Stokes’ theorem on a unit disk
Stokes’ theorem connects circulation measured around a closed boundary with curl measured through the surface that boundary encloses. This calculator fixes that geometry to a useful standard case: the boundary is the unit circle in the xy-plane, the surface is its flat unit disk, and the normal points upward in the +z direction. With the geometry fixed, you supply the three component functions of a vector field F = <P, Q, R>.
The calculator produces two numerical approximations for that field. One estimates circulation around the unit circle; the other estimates upward flux of curl through the disk. For a field that is suitably smooth on and around this disk, Stokes’ theorem says the values should be close. Comparing them makes the boundary-to-interior relationship concrete rather than leaving it as an abstract identity.
Unit-circle orientation in Stokes’ theorem
For this Stokes’ theorem calculation, orientation follows the right-hand rule. The disk normal points upward, so the positive boundary direction is counterclockwise when the circle is viewed from above. Reversing the direction of travel would reverse the sign of the line integral. Apparent disagreements between two Stokes calculations are often orientation disagreements rather than differentiation errors.
For its counterclockwise boundary calculation, the page uses the following parameterization of the unit circle.
Because this boundary remains in z = 0, its line integral uses only P and Q sampled on that plane. The surface is also flat, and its upward normal is k, so the flux calculation uses the z-component of curl: ∂Q/∂x - ∂P/∂y.
Entering P, Q, and R for the vector field
The Stokes’ theorem form accepts P(x,y,z), Q(x,y,z), and R(x,y,z), the components of a three-dimensional vector field. Expressions may use x, y, and z, common functions including sin, cos, exp, and sqrt, and the constant pi. The blank fields let you choose the field rather than requiring a preset example.
For this particular unit disk, R is part of the entered field but does not change either displayed integral. On the circle, dz = 0, so the R dz term vanishes. On the disk, the relevant curl component is ∂Q/∂x - ∂P/∂y, which contains no R. The input remains visible to preserve the full vector-field notation used by Stokes’ theorem.
The calculator does not assign physical units to the field. If you use units, keep them internally consistent: circulation has field-times-length units, while curl flux has curl-times-area units. Stokes’ theorem equates those compatible quantities for the same oriented surface.
Numerical line integral and curl-flux calculation
This Stokes’ theorem checker approximates the boundary integral with many midpoint samples around the unit circle. It approximates the disk integral with a polar grid, using a small centered finite difference to estimate ∂Q/∂x and ∂P/∂y at each sample location. It is therefore a numerical comparison tool, not a symbolic integration system, and it is most reliable for smooth real-valued fields on the disk.
| Quantity | What is integrated | What it means here |
|---|---|---|
| Boundary line integral | F · dr around the unit circle | Total counterclockwise circulation on the boundary |
| Surface integral of curl | (∇ × F) · n through the unit disk | Total upward rotational flux through the interior |
| Stokes comparison | Read the two numerical estimates side by side | Smooth fields should produce nearly equal values |
Small differences between the two estimates are expected because both integrations are sampled numerically. A substantial difference can indicate a reversed orientation, a field that is not smooth on the disk, a singularity within the disk, or an expression that is undefined at one or more sampled points.
Worked Stokes’ theorem example with constant curl
A direct test field is P = -y/2, Q = x/2, and R = 0. Its relevant curl component is constant: ∂Q/∂x - ∂P/∂y = 1/2 - (-1/2) = 1. The unit disk has area π, so its upward curl flux is π.
On the boundary, set x = cosθ and y = sinθ. The line-integral integrand is P(-sinθ) + Q(cosθ), which becomes (sin²θ + cos²θ)/2 = 1/2. Integrating from 0 to 2π gives π again. Entering this field should therefore return two values close to 3.141593.
For a contrasting check, enter P = x, Q = y, and R = 0. Since ∂Q/∂x - ∂P/∂y is zero everywhere, both the curl flux and the boundary circulation should be near zero. These examples distinguish constant positive curl from a field whose local rotational contributions vanish.
Reading a Stokes’ theorem comparison result
After you click Compute, compare the line-integral estimate with the curl-flux estimate rather than expecting identical displayed digits. Close values demonstrate the theorem numerically for the entered field and fixed geometry. Opposite signs usually warrant an orientation check. A large discrepancy or NaN calls for inspecting the expressions for singularities, division by zero, or values that leave the real-number setting inside the disk.
Consistent units matter whenever the components represent a physical field. The page performs no conversion, so all component expressions must already use one coherent unit system. Smoothness, real values across the disk, and the stated counterclockwise/upward orientation are the key conditions for an interpretable comparison.
How local curl becomes boundary circulation
This calculator’s numerical method mirrors the content of Stokes’ theorem: it accumulates tangential motion along the unit-circle boundary and separately accumulates the relevant local curl over the disk. The two sums are not generic weighted totals; they represent the two distinct integrals that the theorem equates. Changing P or Q can alter both sides, while singular or non-smooth behavior can make a numerical check unreliable.
When troubleshooting a surprising result, first examine the field near the origin and around the circle, then verify the signs in P and Q. Positive values of ∂Q/∂x - ∂P/∂y tend to contribute counterclockwise circulation for this upward-oriented disk; negative values tend to contribute clockwise circulation. Contributions of different signs can cancel, so a result near zero does not require the field itself to be zero.
Practical conditions for a reliable unit-disk check
Use Stokes’ theorem expressions that remain finite and real-valued throughout the unit disk. For example, 1/(x²+y²) is singular at the origin, so it is not suitable for this full-disk numerical comparison. Discontinuities can also destabilize the finite-difference derivative estimate. Make parentheses explicit in trigonometric and exponential expressions so the parser evaluates the field you intend.
The two outputs also offer geometric intuition. A strongly positive result corresponds to net counterclockwise circulation relative to the selected orientation, while a strongly negative result corresponds to net clockwise circulation. A near-zero value can arise because positive and negative curl regions cancel over the disk. That connection between local rotation and motion around a boundary is precisely the behavior this Stokes’ theorem calculator is designed to show.