Stokes' Theorem Calculator

Calculate and verify Stokes' Theorem for vector fields and surfaces.

Vector Field Input

Enter the components of your vector field F(x,y,z) = [P, Q, R]:

Surface and Curve

Select a predefined surface and its boundary curve:

Surface Description:

Unit disk in the xy-plane centered at the origin with radius 1. The boundary is a circle traversed counterclockwise.

About Stokes' Theorem

What is Stokes' Theorem?

Stokes' Theorem relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface. It's a fundamental theorem in vector calculus with applications in physics and engineering.

How to Use This Calculator

  1. Enter the components of your vector field F(x,y,z) = [P, Q, R]
  2. Select a predefined surface and its boundary
  3. Adjust any parameters for the selected surface
  4. Click "Calculate" to compute both sides of Stokes' Theorem
  5. View the results and verification

Input Format

Enter expressions using standard mathematical notation. Use * for multiplication, ^ for exponents, and variables x, y, and z.

Examples:

  • y (y variable)
  • -x (negative x)
  • x*y (x times y)
  • y^2 (y squared)
  • sin(x) + cos(y) (trigonometric functions)
  • exp(x) (exponential function)

Example Vector Fields

Example 1: Simple Rotational Field

  • P(x,y,z): y
  • Q(x,y,z): -x
  • R(x,y,z): 0
  • Expected result: -2Ï€ ≈ -6.283