Section 2.5. Functions and Surfaces

Brief review for one variable functions and curves: ² A (one variable) function is rule that assigns to each member x in a subset D in R1 a unique real number denoted by f (x) . The set D is called the domain of f (x) , and the set R of all values f (x) that f takes on, i.e.,

R = f (x) x D f j 2 g is called the range of f (x) . The subset of points in R2 de…ned by

G = (x, f (x)) x D f j 2 g is called the graph of f. The graph of any one variable function is usually a curve in the common sense. Some times, a curve can be used to de…ne a function provided that if satis…es "vertical test". A table may also de…ne a function.

– Determine domains of some commonly used functions: ²

D (loga (x)) = x x > 0 D x1/n = x f xj 0 g(n = even integer) D (sin x, cosfx, aj x, x¸m)g= R1 (a > 0, m = positive integer) ¡ ¢

f D = D (f) D (g) g = 0 = x both f (x) and g (x) are defined, AND g (x) = 0 g \ \f 6 g f j 6 g µ ¶ All these concepts and principles extend to two-variable functions.

Two-variable functions ² De…nition. A function of two variables is rule that assigns to each point (x, y)in a subset D in R2 a unique real number denoted by f (x, y) . The set D is called the domain of f, and the set R of all values f (x, y) that f takes on, i.e., R = f (x, y) (x, y) D R1 f j 2 g ½

1 is called the range of f. The subset of points in R3 de…ned by G = (x, y, z) z = f (x, y) , (x, y) D = (x, y, f (x, y)) (x, y) D f j 2 g f j 2 g is called the graph of f. The graph of any two-variable function is usually a in the common sense. A surface may be used to de…ne a function as long as it passes the "vertical line test" that each line to xy intersects the surface at most once. ¡ A function of two variables may also be de…ned through a 2D table. Example 5.1. Determine domains and discuss ranges for the following functions: (a) f (x, y) = 4x2 + y2, px + y + 1 (b) g (x, y) = , x 1 (c) h (x, y) = x ln (y¡2 x) . Solution: (a) f (x, y) i¡s de…ned for all (x, y) . Its graph is called .

(b) The function g (x, y) is unde…ned when either x = 1 (denominator = zero) or x + y + 1 < 0. So D (g) = (x, y) x + y + 1 0, x = 1 . f j ¸ 6 g In xy plane, both x + y + 1 = 0 and x = 1 are straight lines. ¡ 2 So D (g) consists of all points on one-side of the line x + y + 1 = 0 that containing (0, 0) , including the line x + y + 1 = 0, but excluding those on vertical line x = 1.

Domain of g (x, y)

Graph of g (x, y)

3 (c) h (x, y) = x ln (y2 x) . This function is de…ned as long as ¡ y2 x > 0, or x < y2. ¡

Domain of h : Shaded area, excluding the

Graph of h (x, y)

4 Graphs of Two-variable Functions ² The graph of z = f (x, y) ,

G = (x, y, f (x, y)) (x, y) D f j 2 g may be understood as a surface formed by two families of cross-section curves (or trace) as follows. For any …xed y = b, the one-variable function z = f (x, b) represents a curve. With various choices for b, for instance, b = 0, 0.1, 0.2, ..., there is a family of such curves

z = f (x, 0) , z = f (x, 0.1) , z = f (x, 0.2) , ...

In the same 3D , each curve is the intersection of G and a coordinate plane (parallel to zx-plane) y = b, i.e., it is the solution of the system

z = f (x, y) y = b.

On the other hand, if we …x x = a,the one-variable function z = f (a, y) also represents a curve. With various choices for a, for instance, a = 0, 0.1, 0.2, ..., there is a family of such curves

z = f (0, y) , z = f (0.1, y) , z = f (0.2, y) , ...

In the same 3D coordinate system, each curve is the intersection of G and a coordinate plane (parallel to yz plane) x = a, i.e., it is the solution of the system ¡

z = f (x, y) x = a.

Another way to study two-variable functions, or surfaces are often through one-variable functions, or curves. Consider, for example, z = T (x, y) is the temperature function of Dayton area in a certain time. Then, for each …xed number T0 = 50, for instance, the set

(x, y) T (x, y) = 50 f j g 5 de…nes a one variable function. For each x = a, y is the solution of

T (a, y) = 50.

The graph of this one variable function is a curve called contour. It represents the path along which the temperature maintains at 50 degree. When f (x, y) is a of degree one or two, the function is called a .

Graphs of some two-variable functions ² Example 5.2. (a) The graph of

z = 6 3x 2y ¡ ¡ is the plane 3x + 2y + z 6 = 0, ¡ passing through P0 (2, 0, 0) (by setting y = z = 0, and then solving for x = 2) with a normal 2, 3, 1 . h i

One way to graph a plane is to …nd all three intercepts: intersection of the plane and coordinate axis: x intercept is x = 2 on x axis, y intercept is y = 3 on y axis, and z ¡intercept is z = 6 on z ¡axis. ¡ (b) The gr¡aph of ¡ ¡ z = 1 x2 y2 ¡ ¡ p 6 is the upper-half unit . (c) The graph of z = 1 x2 y2 ¡ ¡ ¡ p

is the other half of the . Example 5.3. Sketch z = x2

7 parabolic

We …rst view this as a one-variable function whose graph is a curve on xz plane. ¡

Now as a two-variable function, since z = x2 is independent of y, if a point P (x0, y0, z0) is on the surface, so is the entire line (x0, y, z0), passing through

8 P (x0, y0, z0) and parallel to y axis is on the surface. So it is a cylinder with being the parabo¡la. One can also view this surface is generated by moving a line parallel to y axis parallel along above parabolic curve. In general, if one varia¡ble, for instance, yis missing, then the graph z = f (x)is a cylinder with generating lines parallel to y axis. ¡ Example 5.4. z = 2x2 + y2 (elliptic paraboloid)

We now try to use trace method to analyze the above graph. Consider horizontal cross-section z = c,i.e., the intersection with a coordinate plane; z = 2x2 + y2, z = c. The cross-section, or trace, is the curve if c > 0 c = 2x2 + y2 : empty if c < 0 ½ on the plane z = c that is parallel to xy plane. When c > 0,the standard form is ¡ x2 y2 2 + 2 = 1, horizontal half axis = c/2 , vertical Half axis = pc. c/2 (pc) p ³p ´ 9 So as c increases (i.e., moving parallel to xy plane upward) starting at c = 0, the trace, which is ellipse, getting larger¡and larger. We next look at cross-sections parallel to yz plane : x = a, or ¡ z = 2x2 + y2, x = a.

The trace is a curve z = 2a2 + y2

a = 0 (solid), a = 1 (dash), a = 2 (dot) on yz plane, which is a parabola with vertex y = 0, z = 2a2. Sim¡ilarly, along zx plane direction, the cross-section with y = b is a parabola ¡ z = 2x2 + b2. In summary, cross-sections are either ellipse or parabola. Example 5.5. z = y2 x2 (hyperbolic paraboloid) ¡

10 We again try to use trace method to analyze this graph. Consider horizontal cross-section z = c,i.e., the intersection with a coordinate plane;

z = y2 x2, ¡ z = c.

The cross-section, or trace, is the curve

(opening along y axis) if c > 0 c = y2 x2 : ¡ ¡ hyperbola (opening along x axis) if c < 0 ½ ¡ on the plane z = c that is parallel to xy plane. The standard forms are ¡ y2 x2 y2 x2 = 1 (c > 0), or = 1 (c < 0). (pc)2 ¡ (pc)2 p c 2 ¡ p c 2 ¡ ¡ ¡ So as c increases (moving upward) s¡tartin¢g at ¡c = ¢0, the trace becomes vertical hyperbola with increasing half axis pc. However, when c decreases (moving downward) starting at c = 0, the trace becomes horizontal hyperbola y2 x2 with increasing half axis c . = 1 j j (pc)2 ¡ (pc)2 p

11 c = 1, 3 (solid), c = 1, 3 (dash) ¡ ¡ We next look at cross-sections parallel to yz plane : x = a, or ¡ z = y2 x2, ¡ x = a.

The trace is a curve z = y2 a2 ¡ on yz plane, which is a parabola with vertex y = 0, z = a2. ¡ ¡

12 a = 0 (solid), a = 1 (dash), a = 2 (dot) Similarly, along zx plane direction, the cross-section with y = b is a parabola ¡ z = b2 x2 ¡ opening opposite to z axis : ¡

b = 0 (solid), b = 1 (dash), b = 2 (dot)

13 In summary, cross-sections are either ellipse or hyperbola. However, those hyperbola changes from horizontal to vertical as the cross-section parallel to xy plane moving upward. ¡ Example 5.6.

y2 z2 x2 + + = 1 9 4

It is easy to see cross-sections from all three directions are . Example 5.7. of One Sheet

y2 z2 x2 + = 1 4 ¡ 4

14 Let us look at traces in all three directions. Along xy plane z = c ¡ y2 z2 x2 + = 1 4 ¡ 4 z = c, the trace y2 c2 x2 + = 1 + 4 4 is a ellipse on xy plane with the standard form ¡ x2 y2 2 + 2 = 1. c2 c2 1 + 2 1 + Ãr 4 ! Ã r 4 ! Along yz plane, the trace is ¡ y2 z2 x2 + = 1 4 ¡ 4 x = a, or hyperbola y2 z2 = 1 a2 4 ¡ 4 ¡ 15 on yz plane. As a moves across a = 1, i.e., as (1 a2) changes signs, the dir¡ection of opening of the hyperbo§la changes from¡horizontal (or y axis, when 1 a2 > 0) to vertical (or z axis if 1 a2 < 0). Similarly, th¡e traces on zx ¡plane, ¡ ¡ ¡ y2 z2 x2 + = 1 4 ¡ 4 y = b, is hyperbola z2 b2 x2 = 1 ¡ 4 ¡ 4 b2 on xz plane whose direction changes when 1 changes signs. ¡ ¡ 4 Example 5.8. Hyperboloid of Two Sheetµs ¶

y2 z2 x2 + = 1. 4 ¡ 4 ¡

16 The traces along three directions are, respectively,

y2 c2 c2 x2 + = 1 (z = c) ellipse if 1 > 0 4 4 ¡ 4 ¡ z2 b2 x2 = 1 (y = b) hyperbola (opening along z axis) ¡ 4 ¡ ¡ 4 ¡ y2 z2 = 1 a2 (x = a) hyperbola (opening along z axis) 4 ¡ 4 ¡ ¡ ¡ Note that here there is not directional change. Classi…cation of Quadratic Surfaces ² Consider in general quadratic equations of three variables

Ax2 + By2 + Cz2 + Dx + Ey + Fz + G + Hxy + Iyz + Jzx = 0.

By a rotation, it can be reduced to

Ax2 + By2 + Cz2 + Dx + Ey + Fz + G = 0.

We then complete squares, if possible. There are several cases analogous to 2Dsituations. (1) If ABC = 0,it reduces to 6 A (x h)2 + B (y k)2 + C (z l)2 = R. ¡ ¡ ¡ The signs of A, B, C, R determine shapes of surfaces. We suppose that R = 0. 6 (a) A, B, C have the same sign (either all positive or all three are nega- tive). In this case, we have ellipsoid with the standard form

(x h)2 (y k)2 (z l)2 ¡ + ¡ + ¡ = 1 a2 b2 c2

C (h, k, l) = Center of ellipsoid a = half axis in x axis direction ¡ b = half axis in y axis direction ¡ c = half axis in z axis direction. ¡

17 For simpli…cation, we take h = k = l = 0 : x2 y2 z2 + + = 1. a2 b2 c2 We use traces to see the graph. Set z = l be a constant. The cross section in the direction parallel to xy plane is ¡ x2 y2 z2 + + = 1 a2 b2 c2 z = l or x2 y2 l2 + = 1 a2 b2 ¡ c2 z = l. If l c,this is an ellipse. If l > c, then j j · j j l2 1 < 0 ¡ c2 so there is no solution for the system and the curve is empty. Similarly, in other directions, all cross-sections are ellipses or the empty set. (b) A, B, C don’t have the same signs. Assuming that AB > 0. The equation A (x h)2 + B (y k)2 + C (z l)2 = R. ¡ ¡ ¡ reduces to either (x h)2 (y k)2 (z l)2 ¡ + ¡ ¡ = 1 (Hyperboloid of One Sheet, z axis is axis of ) a2 b2 ¡ c2 ¡ or (x h)2 (y k)2 (z l)2 ¡ + ¡ ¡ = 1 (Hyperboloid of Two Sheets) a2 b2 ¡ c2 ¡ If R = 0,then we have A (x h)2 + B (y k)2 + C (z l)2 = 0, ¡ ¡ ¡ and depending on the signs of A, B, C,its graph is a . For instance, 2x2 + 3y2 4z2 = 0 ¡ 18 is a cone centered at (0, 0, 0) .Its axis is parallel to z axis. (3) Assume that only one of three numbers A, B, C¡is zero. For simplicity, assuming C = 0, but AB = 0.The equation 6 Ax2 + By2 + Cz2 + Dx + Ey + Fz + G = 0 reduce to A (x h)2 + B (y k)2 = F (z l) . ¡ ¡ ¡ This is an elliptic paraboloid if AB > 0 and a hyperbolic paraboloid if AB < 0. We summarize by the Table 2 in page 682: (1) Ellipsoid: x2 y2 z2 + + = 1 a2 b2 c2

19 An ellipsoid becomes a sphere if a = b = c. (2) Elliptic Paraboloid

z x2 y2 = + (opening up if c > 0, down if c < 0) c a2 b2

y x2 z2 = + (opening towards positiv if y direction if b > 0, opposite if b < 0) b a2 c2 ¡

20 x y2 z2 = + (opening towards positiv if x direction if a > 0, opposite if a < 0) a b2 c2 ¡ (3) Hyperbolic Paraboloid (Saddle) z x2 y2 = c a2 ¡ b2

21 y z2 x2 = b c2 ¡ a2 x y2 z2 = a b2 ¡ c2 (4) Cone z2 x2 y2 = + c2 a2 b2

y2 x2 z2 = + b2 a2 c2 x2 y2 z2 = + a2 b2 c2 (5) Hyperboloid of One Sheet

x2 y2 z2 + = 1 a2 b2 ¡ c2

22 x2 y2 z2 + = 1 a2 ¡ b2 c2 x2 y2 z2 + + = 1 ¡a2 b2 c2 (6) Hyperboloid of Two Sheets

x2 y2 z2 + = 1 a2 b2 ¡ c2 ¡

23 x2 y2 z2 + = 1 a2 ¡ b2 c2 ¡

x2 y2 z2 + + = 1 ¡a2 b2 c2 ¡ Homework:

24 1. Find and sketch the domain of the function.

y 4x2 (a) f (x, y) = ¡ x2 1 p ¡ (b) g (x, y) = 4 x2 y2 + ln (x2 + y2 1) ¡ ¡ ¡ 2. Identify and skpetch the trace x = k, y = k, and z = k, and then use these traces to sketch the graph of y = x2 + 4z2

3. Identify (i.e., spell the name, openning and axis of symmetry, if any) and sketch the graph.

(a) 4x2 + y2 4z2 = 4 ¡ (b) 2y2 + z2 + 4x = 0 (c) 4x2 y2 4z2 = 4 ¡ ¡ (d) y = p16 x2 z2 (hint: both sides) ¡ ¡ (e) x = y2 + 2z2 (hint: square both sides) ¡ (f) x2 + 4y2 + 2z2 = 4 p ¡ 4. Give a concrete example.

(a) An elliptical paraboloid openning to the negative x axis with x axis as its axis of symmetry. ¡ ¡ (b) One branch of a hyperboloid with two sheets whose axis of sym- metry is y axis. The branch is open to the negative y axis. ¡ ¡ (c) The upper-half of a hyperboloid with one sheet whose axis of sym- metry is x axis. ¡ 5. Find an equation for the surface consisting of all points P (x, y, z) for which the distance from P to the x axis is twice the distance from P to the yz plane. identify the sur¡face. ¡

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