TOPICS
Search

Surface Integral


For a scalar function f over a surface parameterized by u and v, the surface integral is given by

Phi=int_Sfda
(1)
=int_Sf(u,v)|T_uxT_v|dudv,
(2)

where T_u and T_v are tangent vectors and axb is the cross product.

For a vector function over a surface, the surface integral is given by

Phi=int_SF·da
(3)
=int_S(F·n^^)da
(4)
=int_Sf_xdydz+f_ydzdx+f_zdxdy,
(5)

where a·b is a dot product and n^^ is a unit normal vector. If z=f(x,y), then da is given explicitly by

 da=+/-(-(partialz)/(partialx)x^^-(partialz)/(partialy)y^^+z^^)dxdy.
(6)

If the surface is surface parameterized using u and v, then

 Phi=int_SF·(T_uxT_v)dudv.
(7)

See also

Integral, Path Integral, Surface Area, Surface Parameterization, Volume Integral

Explore with Wolfram|Alpha

References

Leathem, J. G. Volume and Surface Integrals Used in Physics. Cambridge, England: University Press, 1905.

Referenced on Wolfram|Alpha

Surface Integral

Cite this as:

Weisstein, Eric W. "Surface Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SurfaceIntegral.html

Subject classifications