Let .   Let be a piecewise smooth closed curve in the plane enclosing a region , traversed anti-clockwise.   

Evaluating the line integral 

gives....

Let be a solid body in three-dimensional space, with boundary given by a closed, (piecewise smooth) surface .

Let be the vector field given by 

for some constants .

What is the flux through , oriented with an outward-pointing normal?

The Opera House:

The magnitude of the surface integral of the vector field though the surface :

is:

The Opera House:

The magnitude of the surface integral of the vector field though the surface :

is:

The Opera House:

The magnitude of the surface integral of the vector field though the surface :

is:

The surface is  :

 ,  ,  ,    

and its spherical parametrisation can be :

Any smooth surface has two unit normals at each point, if one is then the other is ; they point in opposite directions.

Choose all of the true statements about how this choice of orientation relates to the calculation of surface integrals.