How this calculation works
Construct symbolic field partial derivatives for curl and surface derivatives for its oriented area vector. Compute the surface flux by tensor-product quadrature and the four lifted boundary-edge integrals independently. Compare both sides and a refined run, then inspect the rotatable surface, oriented boundary and sampled curl vectors.
∬S (curl F)·dS = ∮∂S F·dr, with graph-surface dS=(−gₓ,−gᵧ,1) dx dy for upward orientationA worked example
For F=(−y,x,0), curl F=(0,0,2). On z=1−x²−y² above the unit rectangle x,y∈[−0.5,0.5], upward curl flux is two. Independently integrating F·dr around the counterclockwise lifted boundary also gives two. Downward orientation changes both signs.
Good to know
Graph surfaces over increasing rectangular xy domains only. Bounded symbolic grammar and finite Simpson samples do not prove the smoothness, domain or orientation hypotheses of Stokes’ theorem. Required sample values must be finite. Numerical agreement and refinement differences are not proof or certified error bounds. The surface and curl picture is sampled and orthographically projected.
Reference: OpenStax Calculus: Stokes theorem (opens in a new tab) ↗
Compare a surface integral with its oriented boundary
Stokes’ theorem relates the flux of curl through an oriented surface to the circulation of the original vector field around its consistently oriented boundary. The two quantities use different integrands and different domains. This page calculates them independently rather than obtaining the second result by copying the first.
The supported surface is a graph z=g(x,y) above a rectangle in the input plane. Field components P,Q,R can depend on x,y,z, while g depends on x,y. The field, surface and the derivatives required by the numerical samples must be defined. The underlying theorem has smoothness and neighborhood assumptions that finite sample success alone does not establish.
Build curl and the graph-surface area vector
For F=(P,Q,R), curl F has components (Rᵧ−Q_z,P_z−Rₓ,Qₓ−Pᵧ). These are constructed from symbolic partial derivatives and evaluated on the graph surface. They describe local circulation density in the field, not the boundary work integrand itself.
Parameterizing the graph by (x,y,g(x,y)) gives the upward oriented area vector (−gₓ,−gᵧ,1) dx dy. The surface-side scalar integrand is therefore curl F evaluated at z=g, dotted with that area vector. Choosing downward orientation negates the area vector and the signed flux. The steps retain the derivative expressions so that the chosen orientation is explicit.
Walk around the lifted rectangle in the matching direction
For an upward normal, the rectangle is traversed counterclockwise as viewed in the xy plane. Each of its four straight input-plane edges is lifted to z=g(x,y). Along that lifted edge, the velocity includes z′=gₓx′+gᵧy′, so the R component of the original field can contribute to circulation even though the input boundary looks planar.
The boundary integral is calculated separately on each edge using F·r′. The table keeps the four contributions rather than presenting an unexplained total. Downward orientation reverses the boundary direction as well as the surface normal; both signed integrals should reverse together when the theorem’s hypotheses hold. The SVG boundary arrow follows the selected traversal.
Follow the original curved-surface example
Take F=(−y,x,0). Differentiation gives curl F=(0,0,2). For z=1−x²−y², the upward area vector is (2x,2y,1) dx dy, whose dot product with curl is simply two. The rectangle from −0.5 to 0.5 in both x and y has area one, so the curl flux is two.
The boundary calculation substitutes the four lifted edges into −y dx+x dy. The height function does not contribute through R here because R=0. The four edge contributions sum to two with counterclockwise orientation. This specific example explains the matching numbers without suggesting that height or the third field component can be ignored for every other input.
Use a different surface only under the right conditions
Stokes’ theorem can let suitable spanning surfaces share the same circulation when their oriented boundary is the same and the field satisfies the theorem’s hypotheses on an appropriate neighborhood. That is a mathematical statement with conditions, not a license to move a surface through singularities or change its boundary unnoticed.
A numerical display of close values does not verify those conditions. A field with an excluded point, an undefined derivative between nodes or an unsuitable surface can defeat a naive application even if a particular finite grid happens to avoid the problem. The calculator rejects known invalid required samples but does not infer a complete domain proof from the ones that remain.
Read refinement and the rotatable picture as supporting evidence
The surface side uses tensor-product composite Simpson quadrature, and each boundary edge uses its own scalar Simpson integral. A second run doubles the interval count in each coordinate and edge parameter. The reported resolution differences show how those numerical results changed, rather than labeling either run an exact value or a certified error estimate.
The rotatable wireframe shows the selected graph surface, the oriented boundary and a finite set of curl arrows. One common display factor scales the curl vectors. They are local field samples, not individual flux contributions or integrated trajectories, and projection can shorten an arrow that points toward the viewer. Use pointer dragging or the labeled view sliders to inspect the geometry.
The vector line integral calculator for parametric curves covers a separate planar path-entry task. The directional derivative calculator studies a scalar surface slope in a chosen input direction. Keep their different meanings separate when comparing numerical results, and retain the formulas, rectangle, orientation and resolution with a copied scenario or exported SVG.
A few questions, answered
Are the two sides calculated independently?
Yes. Curl flux uses a two-dimensional surface integral, while circulation uses four separately parameterized boundary-edge integrals of the original field.
Why does normal orientation change both signs?
Reversing the normal requires the opposite boundary traversal under the right-hand orientation rule.
Does a small difference prove Stokes’ theorem applies?
No. It is numerical agreement under sampled evaluation; the theorem’s smoothness and domain hypotheses still require analysis.
Can I enter a nonpolynomial surface?
Yes, within the bounded real arithmetic grammar and required differentiability/domain conditions.
Are the blue arrows flux contributions?
No. They display sampled curl vectors with one common length factor; the flux also depends on the oriented surface area vector.