Understanding Stokes Theorem
Stokes theorem links a surface integral to a boundary line integral. It says that total curl through a surface equals circulation around its edge. This calculator helps you compare both sides with the same vector field. It is useful for vector calculus, physics, and engineering checks.
Why This Calculator Helps
Manual Stokes theorem work can become long. You must find curl, choose a normal, set limits, and follow boundary orientation. Small sign errors are common. This tool keeps those steps visible. It uses numerical integration, so it can handle many smooth fields and simple surfaces. You can change grid density to improve accuracy. It also shows how geometry affects circulation. Practice needs less guessing during review and exams.
Surface Choices
The calculator supports rectangles and disks projected on the xy-plane. Each surface may lie on a plane z equals ax plus by plus c. A flat horizontal region is made by setting a and b to zero. Upward orientation uses the normal vector from the surface parameterization. Downward orientation reverses the sign and boundary direction.
Field Inputs
Enter vector field components P, Q, and R. Use x, y, and z as variables. Standard functions such as sin, cos, sqrt, exp, and log are accepted. Use multiplication signs between factors. For example, write 2*x, not 2x. The calculator estimates partial derivatives with a central difference step.
Accuracy Notes
The surface integral uses midpoint sampling over the selected region. The boundary integral also uses midpoint sampling along the edge. Higher interval counts usually improve agreement, but they take more processing. A tiny difference between both results is normal. Large differences usually point to low grid counts, sharp fields, invalid formulas, or mismatched orientation.
Practical Uses
Students can test homework examples before writing final steps. Teachers can prepare examples with visible numerical checks. Engineers can inspect circulation ideas for rotational fields. The CSV export stores the numerical summary. The PDF button saves a compact report for records or class notes.
Best Practice
Start with a simple field and a known surface. Review the curl components. Then increase the grid count. Compare the surface integral with the boundary integral. If the values agree closely, your setup is likely consistent.