Parametric Surfaces We can use parametric equations to describe a curve. Because a curve is one dimensional, we only need one parameter. If we want to describe a surface (two dimensions) using parametric equations, we will need two parameters. r(u,v) where the domain refers to values in the uv-plane. Ex. Identify the surface with vector equation r(u,v) = 2cos ui + vj + 2sin uk. Ex. Identify the surface with vector equation r(u,v) = 5ui + (2u + v)j + v2k. Ex. Find the rectangular equation of the surface with vector equation r(u,v) = 2ucos vi + u2 j + 2usin vk. Ex. Find a parametric representation of the sphere x2 + y2 + z2 = a2. Ex. Find a parametric representation of the elliptic paraboloid z = 2x2 + y2. For surfaces created by rotating the function y =f (x) about the x-axis, the parametric equations would be x = uy =f (u) cos vz =f (u) sin v These can be adapted for rotation around the y- or z-axis. Ex. Find the parametric equation of the surface generated by rotating z = ln y about the y-axis. Let r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k
Each of these are tangent to the surface Ex. Find the equation of the plane tangent to the surface r(u,v) = u2i + v2j + (u + 2v)k at the point (1,1,1). Thm. If a surface S is defined by the vector function r(u,v), defined on the region D in the uv-plane, then the surface area of S is Ex. Find the surface area of the sphere of radius a. Ex. Find the area of the surface defined by z =f (x,y). Ex. Find the area of the part of the paraboloid z = x2 + y2 that lies under the plane z = 9. Surface Integrals Line integrals added the values of a function at every point on a curve. Surface integrals add the value of a 3-D function at every point on a surface. Let S be a surface with equation z = g(x,y), and let R be the projection of S onto the xy-plane. Ex. Evaluate , where S is the surface z = x + y2, Ex. Evaluate , where S is the first octant
portion of 2x + y + 2z = 6. If the surface can not be written as z = g(x,y), then we need to parameterize it like last time.
Thm. If S can be represented parametrically by , then where D is the domain in the uv-plane. Ex. Evaluate, where S is given by Ex. Evaluate , where S is the unit sphere. Ex. Evaluate , where S is the surface of the region
bounded by x2 + y2 = 1, z = 0, and z = 1 + x. Surface Integrals of Vector Fields
Let S be an oriented surface with unit normal vector n.The surface integral of F over S is also called the flux integral of F over S. If F is a force causing energy to flow through our surface, the flux integral gives the rate of flow through S. For surface defined by z = g(x,y), the unit normal vector is Ex. Evaluate , where F = yi + xj + zk and S is
the boundary of the solid enclosed by z = 1 x2 y2 and z = 0. For a surface that is defined parametrically, the unit normal vector is Ex. Find the flux of the vector field F = zi + yj + xk across the unit sphere.