MAT3100: Multivariable Calculus - Surfaces, Lecture notes of Mathematics

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MAT3100: MULTIVARIABLE CALCULUS

MWALE David

Contents

• 1 FUNCTIONS OF SEVERAL VARIABLES
• 2 Further Differential Calculus
• 3 Surfaces
  • 3.1 Three Dimensional Coordinate Systems
  • 3.2 Representation of a surface
    • 3.2.1 Planes
    • 3.2.4 Equation of a Sphere
    • 3.2.7 Quadratic Surfaces
• Bibliography

Chapter 2

Further Differential Calculus

Chapter 3

Surfaces

The aim of this chapter is to present the study of surfaces. However, in these notes we will look at some basic concepts regarding surfaces.

3.1 Three Dimensional Coordinate Systems

To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair of real numbers ( a; b ) ; where a is the x -coordinate and b is the y -coordinate. For this reason, a plane is called two-dimensional. To locate a point in space, three numbers are required. We represent any point in space by an ordered triple of real numbers ( a; b; c ) : In order to represent points in space, we first choose a fixed point O (the origin) and three directed lines through O that are perpendicular to each other, called the coordinate axes and labeled the x -axis, y -axis, and z -axis. Usually we think of the x - and y -axes as being horizontal and the z -axis as being vertical, and we draw the orientation of the axes in figure 3.1. The three coordinate axes determine the three coordinate planes. Because many people have some difficulty visualizing diagrams of three-

2. B = (3 ; − 2 ; − 6)

Solution As in figure 3.

Figure 3.2: Points in Space

3.2 Representation of a surface

You are already familiar with the representations of conic sections (MAT2100). Here, we will study surfaces and their representations.

3.2.1 Planes

In two-dimensional analytic geometry, the graph of an equation involving x and y is a curve in R^2. In three-dimensional analytic geometry, an equation in x; y , and z represents a surface in R^3 : Thus, a plane is a flat, two-dimensional surface that extends infinitely far.

Example 3.2.2. What surface in R^3 are represented by the following equations?

1. z = 3
2. y = 5

Solution

1. The equation z = 3 represents the set { ( x; y; z ) |z = 3 } , which is the set of all points in R^3 whose z -coordinate is 3 : This is the horizontal plane that is parallel to the xy -plane and three units above it as in the figure 3.3.
2. The equation y = 5 represents the set of all points in R^3 whose y -coordinate is 5 : This is the vertical plane that is parallel to the xz -plane and five units to the right of it as in figure 3.3.

Figure 3.3: Planes in Space

Example 3.2.3. Describe and sketch the surface in R^3 represented by the equation y = x: Solution The equation represents the set of all points in R^3 whose x - and y -coordinates are equal, that is, { ( x; x; z ) |x ∈ R ; z ∈ R }: This is a vertical plane that intersects the xy -plane in the lines y = x and z = 0 : See figure 3.

4 x − 6 y + 2 z + 6 = 0 becomes

( x + 2)^2 + ( y − 3)^2 + ( z + 1)^2 = 8 :

Consequently, ( − 2 ; 3 ; − 1) is the center and the radius is √ 8 = 2 √ 2 : See figure 3.5 for the sketch of x^2 + y^2 + z^2 + 4 x − 6 y + 2 z + 6 = 0 :

Figure 3.5: Graph of a Sphere

Example 3.2.6. What region is represented by the following inequalities?

1 ≤ x^2 + y^2 + z^2 ≤ 4 and z ≤ 0 :

Solution The inequalities

1 ≤ x^2 + y^2 + z^2 ≤ 4 and z ≤ 0

represents the set of all points ( x; y; z ) which are of distance at least 1 and at

most 2 from the origin. But we are also given that z ≤ 0 ; so the points lie on or below the xy -plane. Thus the given inequalities represent the region that lies between (or on) the spheres x^2 + y^2 + z^2 = 1 and x^2 + y^2 + z^2 = 4 ; and beneath (or on) the xy -plane as shown in figure 3.

Figure 3.6: Graph of Spheres

3.2.7 Quadratic Surfaces

We shall take the general equation of a quadric surface in space to be a second degree equation Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0 : By translation and rotation this equation can be reduced to one of the standard forms Ax^2 + By^2 + Cz^2 + J = 0 or Ax^2 + By^2 + Iz + J = 0 : Seven basic types of these surfaces are given below:

x^2 + y^2 = r^2 ; x^2 + z^2 = r^2 and y^2 + z^2 = r^2 ; the type of an equation in which one of the variables is missing, such an equation represents a cylinder. The missing variable is the axis of the cylinder. Since we are considering this equation x^2 + y^2 = r^2 in space, the equation does not represent a circle but a cylinder. See figure 3.7.

Figure 3.8: Graph of Ellipsoids

3. The equation x^2 a^2 +^

y^2 b^2 −^

z^2 c^2 = 1 is a Hyperboloid of one sheet. The axis of the hyperboloid is the axis of the variable with a negative coefficient. Figure 3.9 illustrates the graph of x^2 + 2 y^2 − 4 z^2 = 4 :

Figure 3.9: Graph of Hyperboloid

4. The equation x^2 a^2 −^

y^2 b^2 −^

z^2 c^2 = 1 represents a Hyperboloid of two sheets. The axis corresponds to the axis of the variable with positive coefficient. Figure ?? illustrates the graph of x^2 − 2 y^2 − 4 z^2 = 4 : It is easy to get the graph in figure 3.10, if we intersect the graphs of x^2 − 2 y^2 = 4 and x^2 − 4 z^2 = 4 :

Figure 3.11: Graph of an Elliptic Cone

6. The equation z = x a^22 + y b^22 represents an Elliptic paraboloid. Figure 3.12 shows the graph of 4 x^2 + 9 y^2 = 36 z:

Figure 3.12: Graph of an Elliptic Paraboloid

7. The equation z = y 2 b^2 −^

x^2 a^2 is a Hyperboloid paraboloid.

Figure 3.13 shows the graph of − 4 x^2 + 9 y^2 = 36 z:

Figure 3.13: Graph of a Hyperboloid Paraboloid

Example 3.2.8. Classify the surface 4 x^2 − 3 y^2 + 12 z^2 + 12 = 0 and determine

Figure 3.14: Graph of a HyperboloidOf2Example A level curve of f is the set of points ( x; y ) in the domain of f for which f ( x; y ) = c for some constant c in the range of f: The level curve is the projection of the contour curve on the xy -plane (with same c ).

Example 3.2.10. Consider the function f ( x; y ) = 100 − x^2 − y^2 : Clearly, its domain is R^2 and the range is ( −∞; 100] : For f ( x; y ) = 0 ; the level curve is { ( x; y ) : x^2 + y^2 = 100 }: For f ( x; y ) = 51 ; the level curve is { ( x; y ) : x^2 + y^2 = 49 }: