Introduction to a Surface Integral of a Scalar-Valued Function
Math Insight
Three are many parallels between parameterized surfaces1 and parameterized
curves2. Both are parameterized by vector-valued functions, with the difference
being the dimension of their inputs (one-dimensional t for parameterized curves
and two-dimensional (u,v) for parameterized surfaces). Both have two possible
orientations (oriented by choice of tangent vector3 for curves and by choice
of normal vector4 for surfaces). Both have integrals giving their measure (arc
length or area5) in terms of derivatives of the parameterization that describe how
the domain is stretched and shrunk as it is mapped onto the image.
In the same way, surface integrals are similar to line integrals6. Just as there are
two types of integrals over curves (line integrals of scalar functions7 and of vector
fields8) there are two types of surface integrals: surface integrals of scalar
functions (discussed on this page) and surface integrals of vector fields9. The
surface integral of a scalar function is a simple generalization of a double
integral10. Like the line integral of vector fields11, the surface integrals of vector
fields12 will play a big role in the fundamental theorems of vector calculus.
Let S be a surface parameterized by Φ(u,v) for (u,v) in some region D. Imagine
you wanted to calculate the mass of the surface given its density at each
point x described by the scalar-valued function f(x). The mass will be the integral
of the density over the surface (just like the mass of a wire was the integral of a
density13 over the curve).
If the surface happened to be lying in the xy-plane rather than floating in threedimensional space, the situation would be identical to that of changing variables
in double integrals14 (other than the region we call D here is called D∗ in that
context changing variables). We need only slight modifications for the case of a
surface floating in three-dimensional space.
To calculate the mass, we integrate f(Φ(u,v)) as (u,v) range over the domain D.
In order to multiply the density f by area on the surface not by area in D, we must
account for any stretching or shrinking by Φ as it maps D onto the surface, as
illustrated in the below image.
Source URL: http://mathinsight.org/surface_integral_scalar_function_introduction
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 1 of 3
A parametrized helicoid with surface area elements. The
function Φ(u,v)=(ucosv,usinv,v) parametrizes a helicoid when (u,v)∈D, where D is the
rectangle [0,1]×[0,2π]. The region D is shown as the green rectangle floating above the helicoid,
and it is divided into small rectangles. This rectangle is mapped into the small region of the
helicoid that is outlined in red and surrounds the red point Φ(u,v). The area of the red region on
the helicoid depends on how Φ stretches or shrinks the small green rectangle as it maps it on the
surface.
We already calculated the stretching or shrinking by Φ when we
calculated surface area15. The area of the region on the surface that is the image
of a small Δu×Δv rectangle in D is
If we multiply this by the density f(Φ(u,v)), we will obtain the mass of that small
section. We could approximate the mass of the whole surface by a Riemann sum
of such terms. Then, by letting Δu and Δv go to zero, we would discover that the
mass of the whole surface is the integral
This integral is a two-dimensional analog of the line integral of a scalar-valued
function16 except that ∥Φ′(t)∥ is replaced by
You can read some examples17 of calculating surface integrals of scalar
functions.
Source URL: http://mathinsight.org/surface_integral_scalar_function_introduction
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 2 of 3
Notes and Links:
1. http://mathinsight.org/parametrized_surface_introduction
2. http://mathinsight.org/parametrized_curve_introduction
3. http://mathinsight.org/parametrized_curve_orient
4. http://mathinsight.org/parametrized_surface_orient
5. http://mathinsight.org/parametrized_surface_area
6. http://mathinsight.org/line_integral_scalar_function_introduction
7. http://mathinsight.org/line_integral_scalar_function_introduction
8. http://mathinsight.org/line_integral_vector_field_introduction
9. http://mathinsight.org/surface_integral_vector_field_introduction
10. http://mathinsight.org/double_integral_introduction
11. http://mathinsight.org/line_integral_vector_field_introduction
12. http://mathinsight.org/surface_integral_vector_field_introduction
13. http://mathinsight.org/line_integral_scalar_function_introduction
14. http://mathinsight.org/double_integral_change_variables_introduction
15. http://mathinsight.org/parametrized_surface_area
16. http://mathinsight.org/line_integral_scalar_function_introduction
17. http://mathinsight.org/surface_integral_scalar_examples
Source URL: http://mathinsight.org/surface_integral_scalar_function_introduction
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 3 of 3