Stokes Theorem

Recall Green’s Theorem for calculating line integrals Suppose C is a piecewise smooth closed curve that is the boundary of an open region R in the plane and oriented so that the region is on the left as we move around the curve. Let be a smooth vector field on an open region containing R and C. Then Green’s theorem only applies to 2 dimensional vector fields and regions in the 2D plane Stokes theorem will generalize this to 3 dimensions

Given S, the parabaloid and a path C at its base in the xy plane traversed counterclockwise If is a unit normal vector with an outward orientation and is a vector field with continuous partial derivatives in some region containing S, then Stokes’ Theorem says – C is the boundary of S and is assumed to be piecewise smooth – For this to hold C must be traversed so that S is always on the left Stokes Theorem

Since we need a normal vector, if our surface is given bythen we can use the cross product to get a normal vector and we have If our surface is a function 2 or 3 variables, g(x,y,z) we can use the gradient to get a normal vector and we have Let’s verify Stoke’s Theorem for S and the vector field – We will calculate the line integral both ways

Stokes Theorem The advantage of Stokes’ Theorem is S can be any surface as long as its boundary is given by C This can be used to simplify the surface integral and create a simpler calculation of a line integral in a 3D vector field