Parametric Surfaces and their Area Part I

Parametric Surfaces Recall: A space curve is described by a vector function of one variable, r(t) As t varies in the interval [𝑎,𝑏], the head of the position vector 𝐫 𝑡 = 𝑥 𝑡 ,𝑦 𝑡 ,𝑧(𝑡) traces the space curve C. Now suppose x, y, z depend on two parameters u, v: As the point (𝑢,𝑣) moves around D, the head of the vector 𝐫(𝑢,𝑣) traces out a surface in 3-D. A vector function of two parameters describes a surface.

Parametric surfaces – Example 1 Identify the parametric surface Let’s try to find a cartesian equation for the surface. We have: 𝑥 2 + 𝑦 2 = (𝑢 cos 𝑣 ) 2 + (𝑢 sin 𝑣 ) 2 = 𝑢 2 ( cos 2 𝑣+ sin 2 𝑣)= 𝑢 2 =𝑧 Thus the cartesian equation of the surface is 𝑧= 𝑥 2 + 𝑦 2 , 0≤𝑧≤9, which we recognize as the part of a paraboloid over the disk 𝑥 2 + 𝑦 2 ≤9.

Parametric surfaces - Gridlines If we fix one parameter, we get curves on the surface called gridlines. Example 2: Consider the paraboloid Let 𝑣= 𝑣 0 = 𝜋 2 , then 𝑥=𝑢 cos 𝜋 2 =0, 𝑦=𝑢 sin 𝜋 2 =𝑢, 𝑧= 𝑢 2 Gridline: segment of the parabola 𝑧= 𝑦 2 in the plane 𝑥=0 (0 ≤ y ≤ 3) Let 𝑣= 𝑣 0 =𝜋, then 𝑥=𝑢 cos 𝜋=−𝑢, 𝑦=𝑢 sin 𝜋, 𝑧= 𝑢 2 Gridline: segment of the parabola 𝑧= 𝑥 2 in the plane 𝑦=0 (−3 ≤ x ≤ 0) Lines of the form v = v0 in the domain D are mapped into gridlines (segments of parabolas) of the surface.

Parametric surfaces – Example 2 continued If we fix u = u0, then The gridlines are the circles on the horizontal planes Lines of the form u = u0 in the domain D are mapped into gridlines (circles) of the surface. The paraboloid with several gridlines:

Parametric surfaces – Example 3 Identify the parametric surface Since 𝑥 2 + 𝑧 2 = cos 2 𝜃+ sin 2 𝜃= 1, the surface is the circular cylinder 𝑥 2 + 𝑧 2 =1 for −2≤𝑦≤2. The gridlines are lines and circles. If we fix 𝜃= 𝜃 0 , we obtain a line parallel to the y-axis: If we fix 𝑦= 𝑦 0 , we obtain the circle 𝑥 2 + 𝑧 2 =1 in the plane 𝑦= 𝑦 0 The cylinder with several gridlines:

Parametric surfaces – Example 4 Parameterize the surface 𝑥 2 + 𝑦 2 + 𝑧 2 =4. The surface is a sphere, thus we can use spherical coordinates. Here ρ equals the radius of the sphere: 𝜌=2 . r is a function of θ and 𝜙 only: Gridlines: 𝜃= 𝜃 0 : meridians (semicircles at constant latitude). 𝜙= 𝜙 0 : circles parallel to the 𝑥𝑦-plane

Parametric surfaces - Example 5 Parameterize the part of the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 =4 that lies between the planes 𝑧=−1 and 𝑧=1. We can use the same r as in the previous example: However, the domain D will be different. Thus, the domain D is given by: