Section 16 7 Surface Integrals Surface Integrals We

Section 16. 7 Surface Integrals

Surface Integrals We now consider integrating functions over a surface S that lies above some plane region D.

Surface Integrals Let’s suppose the surface S is described by z=g(x, y) and we are considering a function f(x, y, z) defined on S The surface integral is given by: Where d. S is the change in the surface AREA!

More useful: i. e. we convert a surface integral into a standard double integral that we can compute!

Example Evaluate Where And S is the portion of the plane 2 x+y+2 z=6 in the first octant.

Surface Integrals • We still need to discuss surface integrals of vector fields…but we need a few notions about surfaces first…. • Recall the vector form of a line integral (which used the tangent vector to the curve): For surface integrals we will make use of the normal vector to the surface!

Normal vector to a surface • If a surface S is given by z=g(x, y), what is the normal vector to the surface at a point (x, y, g(x, y)) on the surface?

Definition: Oriented Surface • Suppose our surface has a tangent plane defined at every point (x, y, z) on the surface • Then at each tangent plane there are TWO unit normal vectors with n 1 = -n 2 • If it is possible to choose a unit normal vector n at every point (x, y, z) so that n varies continuously over S, we say S is an oriented surface

Example Remark: An oriented surface has two distinct sides Positive orientation Negative orientation

Surface Integrals of Vector Fields • If F is a continuous vector field defined on an oriented surface S with unit normal vector n, then the surface integral of F over S is This is often called the flux of F across S

Using our knowledge of the normal vector and the surface area, this can be simplified…

i. e. a more simplified look at this…

An application If is the density of a fluid that is moving through a surface S with velocity given by a vector field, F(x, y, z), then Represents the mass of the fluid flowing across the surface S per unit of time.

Example Let S be the portion of the paraboloid Lying above the xy-plane oriented by an upward normal vector. A fluid with a constant density is flowing through the surface S according to the velocity field F(x, y, z) = <x, y, z>. Find the rate of mass flow through S.