Divergence and Stokes Theorem

Objectives

In this lab you will explore how Mathematica can be used to work with divergence and curl.

Initialization

First some setup so we can do the vector stuff:

      Needs["Calculus`VectorAnalysis`"]

This loads the routines for divergence (Div) and curl (Curl) among other things.

Divergence

Here's a problem from the book (number 6 on page 1027): Use the Divergence theorem to evaluate

if and S: boundary of cube , , . Here's how we solve it:

       F={x^2 z, -y, xyz}
       divF = Div[F]
       Integrate[Integrate[Integrate[divF,{x,0,a}],{y,0,a}],{z,0,a}]

which should give an answer of .

Stokes Theorem

Here's a problem from the book (number 8 on page 1033): Verify Stokes Theorem where and S: , . First the surfaces integral

       F={z-y,x-z,x-y}
       curlF = Curl[F]
       S=z+x^2+y^2-4;
       n=Grad[S]
       curlFdotn=Simplify[curlF.n/.{z->4-x^2-y^2}]
       Integrate[Integrate[curlFdotn,{x,-Sqrt[4-y^2],Sqrt[4-y^2]}],{y,-2,2}]

Now the line integral

       r={2 Cos[t],2 Sin[t],0}
       dr=D[r,t]
       Fdotdr=F.dr/.{x->2 Cos[t],y->2 Sin[t],z->0}
       Integrate[Fdotdr,{t,0,2 Pi}]