Surface Integral - Surface Integrals of Scalar Fields

Surface Integrals of Scalar Fields

To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by

$\int_{S} f \,dS = \iint_{T} f(\mathbf{x}(s, t)) \left|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right| ds\, dt$

where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(s, t), and is known as the surface element.

For example, if we want to find the surface area of some general scalar function, say, we have

$A = \int_S \,dS = \iint_T \left\|{\partial \mathbf{r} \over \partial x}\times {\partial \mathbf{r} \over \partial y}\right\| dx\, dy$

where . So that, and . So,

$\begin{align} A &{} = \iint_T \left\|\left(1, 0, {\partial f \over \partial x}\right)\times \left(0, 1, {\partial f \over \partial y}\right)\right\| dx\, dy \\ &{} = \iint_T \left\|\left(-{\partial f \over \partial x}, -{\partial f \over \partial y}, 1\right)\right\| dx\, dy \\ &{} = \iint_T \sqrt{\left({\partial f \over \partial x}\right)^2+\left({\partial f \over \partial y}\right)^2+1}\, \, dx\, dy \end{align}$

which is the familiar formula we get for the surface area of a general functional shape. One can recognize the vector in the second line above as the normal vector to the surface.

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

Read more about this topic: Surface Integral

Famous quotes containing the words surface and/or fields:

“How easily it falls, how easily I let drift
On the surface of morning feathers of self-reproach:
How easily I disperse the scolding of snow.”
—Philip Larkin (1922–1986)

“On fields all drenched with blood he made his record in war, abstained from lawless violence when left on the plantation, and received his freedom in peace with moderation. But he holds in this Republic the position of an alien race among a people impatient of a rival. And in the eyes of some it seems that no valor redeems him, no social advancement nor individual development wipes off the ban which clings to him.”
—Frances Ellen Watkins Harper (1825–1911)