Gauss’, Green’s and Stokes’ Theorems

If \Omega is a domain in R^3 with boundary \Gamma with outward unit normal n=(n_1,n_2,n_3), and u:\Omega\rightarrow R^3 and v,w:\Omega\rightarrow R, then we obtain applying the Divergence Theorem to the product vw,

  • \int_\Omega \frac{\partial v}{\partial x_i}w\,dx=\int_\Gamma vw\,n_{i}\,ds-\int_\Omega v\frac{\partial w}{\partial x_i}\,dx,\quad i=1,2,3.

Further, similarly,

  • \int_\Omega\nabla v\cdot\nabla w\, dx =\int_\Gamma v\partial_nw\, ds - \int_\Omega v\Delta w\, dx

and

  • \int_\Omega v\Delta w\, dx -\int_\Omega \Delta v w\, dx =\int_\Gamma v\partial_nw\, ds -\int_\Gamma\partial_nvw\, ds.

These formulas are referred to as Green’s Formulas and express 3d analogs to integration by parts in 1d.

If S is a surface in R^3 bounded by a closed curve \Gamman is a unit normal to S, \Gamma is oriented in a clockwise direction following the positive direction of the normal n, and u:R^3\rightarrow R^3 is differentiable, then

  • \int_S(\nabla\times u)\cdot n\, ds=\int_\Gamma u\cdot ds,

which is Stokes’ Theorem.

Read More

Proof that 1+1=2 in Principia Mathematica

 

Previous: Integration in Several Dimensions

 

 

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