Math 212-Lecture 5 12-7: Cylinders and quadric surfaces Surface and trace The graph of F (x; y; z) = 0 where F is a functions of three variables −−! (the input can also be regarded as the position vector OP = hx; y; zi) is a surface in 3D space. The trace of surface S in the plane P is simply the intersection of S and P , which usually is a curve. Example:p Find the trace of the sphere centered at (1; 0; −1) with radius 5 in the xy plane. Cylinders and rulings To define a cylinder, we need two things: 1. A curve C in a plane. 2. A line L that is not parallel to the plane. The cylinder is the set of all lines that are parallel to L and intersect with C. The lines are called rulings. Intuitively, we just shift the curve along the direction of L and get a cylinder. Surfaces with missing variables are cylinders. Examples: Circular cylinder x2 +y2 = 1. We can obtain it by shifting the circle C : x2 + y2 = 1; z = 0 along z axis since z variable is missing in the equation. Elliptic cylinder 4y2 + 9z2 = 36. x variable is missing. It is hence the cylinder by shifting y2=9 + z2=4 = 1 along x-axis. Parabolic cylinder: z = 4 − x2: Revolution To generate a surface, another way is to use revolution. We also need two things: 1. A curve C in a plane. 2. A line in the same plane, called L. We revolve the curve around L and then get a surface. L is called the axis of the surface of revolution. p 2 2 The curve f(x; y) = 0 rotated aboutpx-axis then generates f(x; y + z ) = 0. If we rotate about y-axis, we have f( x2 + z2; y) = 0. Comment: If we are going to revolve about x axis. Then, we fix x variable in the equation and replace the other one (y or z) with py2 + z2 Example: Sketch the graph of z = x2 + y2. 1 This is the surface of revolution by revolving z = x2; y = 0 about z axis. Example: Describe the surface x2 + py2 + z2 = 4. Solution. Here, we see it can be obtained by revolving about about x axis, since we can fix x variable in the process. The trace of the surface in z = 0 (xy plane) is given by x2 + jyj = 4, which is the union of two portions of parabolas. Hence, the surface is the surface of revolution by revolving the two portions of parabola x2 + y = 2 4; y ≥ 0 and x − y = 4; y < 0 about x axis. Example: x = py2 + z2: This is a cone by revolving x = jyj; z = 0 about x-axis. Quadratic surfaces The surfaces with equations of the form Ax2 + By2 + Cz2 + Dx + Ey + F z + H = 0: x2 y2 z2 Ellipsoid: a2 + b2 + c2 = 1. x2 y2 2 Consider 4 + 4 +z = 1. The general shape is like a football. Imagine stretching a sphere. To get a rough picture, we can sketch the traces in the z = z0 planes. We can shift the center as well. For example, if the center is shifted to (x−1)2 (y−1)2 2 (1; 1; 2), then we have 4 + 4 + (z − 2) = 1. Elliptic paraboloid: x2=a2 + y2=b2 = z=c. In the plane z = z0 > 0, we have an ellipse. As z grows larger, the ellipse is larger. In the x = x0 plane, we have a parabola. The special paraboloid z = x2 + y2 can be obtained by revolving z = x2 about z-axis. Hyperbolic paraboloid: y2=b2 − x2=a2 = z=c. In each z = z0, we have a hyperbola. Elliptic cone: x2=a2 + y2=b2 = z2=c2 In each y = kx plane, the trace is just two lines. In the z = z0 plane, the trace is an ellipse. This is a cone. The special cone x2 + y2 = z2 can be obtained by revolving z = ±x about z axis. Hyperboloid of one sheet: x2=a2 + y2=b2 − z2=c2 = 1. x2 y2 z2 a2 + b2 = 1+ c2 . In any z = z0 plane, the trace is an ellipse. The whole surface has only one piece. The surface y2=4 + z2 − x2 = 1 is certainly also a hyperboloid of one sheet. The difference is that the axis is x-axis. 2 Hyperboloid of two sheets: z2=c2 − x2=a2 − y2=b2 = 1. x2=a2 + y2=b2 = z2=c2 − 1. If jzj < c, there is no point. Only in z = z0 > c or z = z0 < −c, the surface has a trace which is an ellipse. There are two pieces. 13.2:functions of several variables Basic concepts A function of two or three variables: f(x; y) or f(x; y; z). Comment: The input could be regarded as the position vector r = hx; yi or r = hx; y; zi. Domain: The region where the function is defined on. Example: p f(x; y) = 4 − x2 − y2: The domain is f(x; y): x2 + y2 ≤ 4g. (This is NOT f(x; y): −2 ≤ x ≤ 2; −2 ≤ y ≤ 2g). An important concept to describe a function of multiple variables is the level set or contour. For a given constant k, the level set is the set of all points (x; y) such that f(x; y) = k or all points (x; y; z) such that f(x; y; z) = k. 1 Example: Describe the level curves of f(x; y) = 1−x2−y2 . 2 2 1 Solution. f(x; y) = k and thus x + y = 1 − k . It's a family of circles in xy plane. For k > 1, the radius is less than 1; for k < 0, the radius is bigger than 1. 3.