Surface Area, Part 2

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Math Learning Center Hours

…Have been announced for study day and finals week.

Math majors, minors, and concentrators should have gotten an email about them, and you can find them on the MLC web site (https://www.geneseo.edu/math/mlc, scroll down a bit to see the finals week schedule).

Surface Area and Mathematica

The “hill” I introduced Friday makes a good example of using a tool such as Mathematica to calculate surface area.

Mathematica has all the pieces that you need for calculating surface areas, i.e., (partial) derivatives, cross products, magnitudes, etc., so you can use them to do the work we did by hand last week.

And to really speed the calculation along, Mathematica also has a built-in Area function that will find the surface area of a parametric surface from its parametric equation and bounds on the parameters.

You can download a notebook from Canvas that demonstrates these things for the hill.

Mathematica isn’t always as good as it could be at simplifying things, so there was one place in the notebook where we did a simplification by hand:

Surface Integrals

Analogous to scalar and vector line integrals, except that instead of integrating a function along a curve in its argument space, you’re integrating over a surface in argument space. Note that having a “surface in argument space” implies that the function has at least 3 arguments.

For scalar integrals, the surface provides values for the function’s inputs. Analogous to how a line integral is a sum of function values over infinitesimal lengths of a curve, a scalar surface integral is a sum of function values over infinitesimal areas in a surface. Formulas for scalar surface integrals are thus similar to those for scalar line integrals, except that the length of a tangent vector is replaced with the area of a patch tangent to the surface:

Vector surface integrals are analogous to flux line integrals, and do give the flux of a vector field across a surface. Thus they are basically integrals of the dot product of the vector field with vectors normal to the surface, with those normal vectors again being cross products of the derivatives of a parametric function for the surface:

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