for and , compute .

We generalize the idea of line integrals to higher dimensions.

### Generalizing to parametric surfaces

We’ve learned that given an explicit function that graphs a surface in , we can compute its surface area with where We will now generalize this idea to parametric surfaces. To do this, we need to be able to compute when our surface is drawn by a parametric function. Let’s remind ourselves how we compute for the surface . Consider the surface area of a “patch” of the surface, determined by and below:

hence

Now suppose we have a parametric surface: This case is essentially the same as before, though now we define our patch by looking at tangent vectors

### Flux: The flow across a surface

There are many specialized applications where one is interested in the rate that a
“fluid” passes through a “surface” per unit time. We call this rate **flux** or the **flow
across a surface**. To compute the flux, we see how aligned field vectors are with
vectors normal to the surface.

- When the field vectors are going the same direction as the vectors normal to the surface, the flux is positive.
- When the field vectors are going the opposite direction as the vectors normal to the surface, the flux is negative.
- When the field vectors are orthogonal to the vectors normal to the surface, the flux is zero.

**orientation**means choosing one of these vectors to be “positive” and the other to be “negative.”

In essence, to compute the flow across a surface, we demand that the surface has **two
sides**. While it might seem reasonable to assume that *every* surface have two sides,
in fact this is false, there are surfaces that **cannot be oriented**. Consider For your
viewing pleasure, we’ve included a graph:

*without*crossing the edge! Möbius strips are really cool, and this author invites you, the young mathematician, to explore their mysteries on your own.

If you have a closed surface, the normal vector pointing outward indicates the “positive” direction, and the normal vector pointing inward indicates the “negative” direction. Moreover, given any parameterization of an orientable surface, there is a natural orientation based on the parameterization.

**orientation given by the parameterization**is given by the direction of

Now we have all the “parts” of a surface integral, it is time to explain what they are.

### Surface integrals

To compute the flow across a surface, also known as flux, we’ll use a *surface integral*.
While line integrals allow us to integrate a vector field along a curve that is
parameterized by : A *surface integral* allows us to integrate a vector field across a
surface that is parameterized by Consider a patch of a surface along with a unit
vector normal to the surface :

A **surface integral** is an integral of the form:

Now let’s work some examples.

for and . Using the orientation given by the parameterization of , compute:

for and . Using the outward pointing orientation, compute:

Let’s now examine our normal vector. If we check at the angles or the angles something strange happens, we find the vector . This is an unavoidable consequence of the so-called “Hairy ball theorem.” Regardless, since there are only a finite number of isolated “bad points” our computation is unaffected. Checking the orientation at the angles , we find the normal vector meaning that these normal vectors are oriented inwardoutward . Hence we will now set Now write: So, write with me,

*The Expanse*, the

*Canterbury*(a space ship hulling water) is destroyed. Several “lucky” crew members escape destruction in a small shuttle meters away, only later to be pummeled by debris.

Setting

where , , and the surface integral will compute the momentum (in kilograms-per-second) that will impact the shuttle. Find this value.

For some interesting extra reading check out: