Surface integral

Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value).

If a region R is not flat, then it is called a surface as shown in the illustration.

Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.

Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere.

Then, the surface integral is given by where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of r(s, t), and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere, where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced).

The surface integral can also be expressed in the equivalent form where g is the determinant of the first fundamental form of the surface mapping r(s, t).

[1][2] For example, if we want to find the surface area of the graph of some scalar function, say z = f(x, y), we have where r = (x, y, z) = (x, y, f(x, y)).

Because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space.

This can be seen as integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface.

Suppose now that it is desired to integrate only the normal component of the vector field over the surface, the result being a scalar, usually called the flux passing through the surface.

This illustration implies that if the vector field is tangent to S at each point, then the flux is zero because the fluid just flows in parallel to S, and neither in nor out.

Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above.

In other words, we have to integrate v with respect to the vector surface element

We find the formula The cross product on the right-hand side of this expression is a (not necessarily unital) surface normal determined by the parametrisation.

This formula defines the integral on the left (note the dot and the vector notation for the surface element).

We may also interpret this as a special case of integrating 2-forms, where we identify the vector field with a 1-form, and then integrate its Hodge dual over the surface.

is the induced volume form on the surface, obtained by interior multiplication of the Riemannian metric of the ambient space with the outward normal of the surface.

Let be a differential 2-form defined on a surface S, and let be an orientation preserving parametrization of S with

denotes the determinant of the Jacobian of the transition function from

Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, magnetic flux, and its generalization, Stokes' theorem.

For example, if we move the locations of the North Pole and the South Pole on a sphere, the latitude and longitude change for all the points on the sphere.

A natural question is then whether the definition of the surface integral depends on the chosen parametrization.

For integrals of vector fields, things are more complicated because the surface normal is involved.

It follows that given a surface, we do not need to stick to any unique parametrization, but, when integrating vector fields, we do need to decide in advance in which direction the normal will point and then choose any parametrization consistent with that direction.

This is indeed how things work, but when integrating vector fields, one needs to again be careful how to choose the normal-pointing vector for each piece of the surface, so that when the pieces are put back together, the results are consistent.

For the cylinder, this means that if we decide that for the side region the normal will point out of the body, then for the top and bottom circular parts, the normal must point out of the body too.

This means that at some junction between two pieces we will have normal vectors pointing in opposite directions.