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PARAMETRIC SURFACES Suppose that , , , , Section 16.6 is a vector‐valued function defined on a region D in the ‐plane. So , , and are, component functions of r, are functions of the two variables and with Parametric Surfaces domain D. The set of all points in such that , , , and , varies through D is called a parametric surface S and the highlighted equations above are called the parametric equations of S.

GRID CURVES SURFACES OF REVOLUTION Let S be a parametric surface given by the vector function , . Consider the surface S obtained by rotating the curve , , about the ‐axis, • If we keep constant by putting , where 0. Let θ be the angle of rotation. If then , becomes a vector functions of , , is a point on S, then the single parameter v and defines a curve

C1 lying on S. (See Figure 4 on page 1124.) cos sin • Similarly if keep constant by putting Therefore, we take and θ as parameters and , we get a curve C2 given by , regard the equations above as parametric that lies on S. equations of S. The parameter domain is given We call these curves grid curves. by ,02.

TANGENT PLANES TANGENT PLANES Let S be the parametric surface traced out by a vector (CONCLUDED) Similarly, keeping constant by putting , we function get a grid curve C2 given by , that lies on S, and , , , , its tangent vector at P0 is and let P be a point on the surface with position vector , , , 0 , . Keeping constant by putting , , becomes a vector function of a single parameter If , then the surface S is called smooth (it v and defined a grid curve C1 lying on S. The tangent has no “corners”). For a smooth surface, the tangent vector to C1 at P0 is obtained by taking the partial plane is the plane that contains the tangent vectors derivative of r with respect to : and , and the vector is a normal vector to the tangent plane. , , ,

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SURFACE AREA SURFACE AREA (CONTINUED)

The sides of the patch can be approximated by the Let S be a surface whose parameter domain D is a vectors ∗ ∗ rectangle. We divide it into subrectangles . Let’s Δ and Δ ∗ ∗ choose , to be the lower left corner of . The So, we approximate the area of the patch by the area part of the surface S that corresponds to is of the parallelogram that lies in the tangent plane to S at called a patch and has the point with position . The area of the parallelogram is ∗ ∗ vector , as one of its corners. Let ∗ ∗ ∗ ∗ Δ Δ Δ Δ ∗ ∗ ∗ ∗ ∗ ∗ , and , An approximation to the total surface area of S is be tangent vectors at as given by the equations on ∗ ∗ the two previous slides. Δ Δ

SURFACE ARES OF THE GRAPH SURFACE AREA (CONCLUDED) OF A FUNCTION If a smooth parametric surface S is given by the equation For the special case of a surface S with an equation , , where , lies in D and f has continuous , , , , , ∈ partial derivatives, we take x and y as parameters. The parametric equations are and S is covered just once as , ranges throughout the parameter domain D, then the surface area of S is ,

By doing this, we have 1 and the surface area integral becomes where 1

2