Section 2.5. Functions and Surfaces² Brief review for one variable functions and curves:

A (one variable) function is rule that assigns to each member x in asubset D in R1 a unique real number denoted by f (x) . The set D is calledthe domain of f (x) , and the set R of all values f (x) that f takes on, i.e.,

R = ff (x) j x 2 Dg

is called the range of f (x) . The subset of points in R2 de…ned by

G = f(x, f (x)) j x 2 D g

is called the graph of f. The graph of any one variable function is usually acurve in the common sense.

Some times, a curve can be used to de…ne a function provided that ifsatis…es "vertical line test".

A table may also de…ne a function.

² – Determine domains of some commonly used functions:

D (loga (x)) = fx j x > 0gD

¡x1/n

¢= fx j x ¸ 0g (n = even integer)

D (sinx, cos x, ax, xm) = R1 (a > 0, m = positive integer)

D

µf

g

¶= D (f )\D (g)\fg 6= 0g = fx j both f (x) and g (x) are defined, AND g (x) 6= 0g

All these concepts and principles extend to two-variable functions.

² Two-variable functions

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 setD is called the domain of f, and the set R of all values f (x, y) that f takeson, i.e.,

R = ff (x, y) j (x, y) 2 Dg ½ R1

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is called the range of f. The subset of points in R3 de…ned by

G = f(x, y, z) j z = f (x, y) , (x, y) 2 D g = f(x, y, f (x, y)) j (x, y) 2 D gis called the graph of f. The graph of any two-variable function is usually aSURFACE in the common sense.

A surface may be used to de…ne a function as long as it passes the "verticalline test" that each line perpendicular to xy ¡ plane intersects the surface atmost 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,

(b) g (x, y) =

px+ y + 1

x ¡ 1 ,

(c) h (x, y) = x ln (y2 ¡ x) .Solution: (a) f (x, y) is de…ned for all (x, y) . Its graph is called paraboloid.

(b) The function g (x, y) is unde…ned when either x = 1 (denominator =zero) or x+ y + 1 < 0. So

D (g) = f(x, y) j x+ y + 1 ¸ 0, x 6= 1g .

In xy ¡ plane, both x+ y + 1 = 0 and x = 1 are straight lines.

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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)

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(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 parabola

Graph of h (x, y)

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² Graphs of Two-variable Functions

The graph of z = f (x, y) ,

G = f(x, y, f (x, y)) j (x, y) 2 D gmay 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 acurve. With various choices for b, for instance, b = 0, 0.1, 0.2, ..., there is afamily of such curves

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

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

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 acoordinate plane (parallel to yz ¡ plane) x = a, i.e., it is the solution of thesystem

z = f (x, y)

x = a.

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

f(x, y) j T (x, y) = 50g

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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 representsthe path along which the temperature maintains at 50 degree.

When f (x, y) is a polynomial of degree one or two, the function is calleda quadratic function.

² Graphs of some two-variable functions

Example 5.2. (a) The graph of

z = 6¡ 3x ¡ 2y

is the plane3x+ 2y + z ¡ 6 = 0,

passing through P0 (2, 0, 0) (by setting y = z = 0, and then solving for x = 2)with a normal h2, 3, 1i .

One way to graph a plane is to …nd all three intercepts: intersection of theplane and coordinate axis: x ¡ intercept is x = 2 on x ¡ axis, y ¡ interceptis y = 3 on y ¡ axis, and z ¡ intercept is z = 6 on z ¡ axis.

(b) The graph ofz =

p1¡ x2 ¡ y2

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is the upper-half unit sphere.(c) The graph of

z = ¡p1¡ x2 ¡ y2

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

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parabolic cylinder

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 pointP (x0, y0, z0) is on the surface, so is the entire line (x0, y, z0), passing through

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P (x0, y0, z0) and parallel to y¡axis is on the surface. So it is a cylinder withcross section being the parabola. One can also view this surface is generatedby moving a line parallel to y ¡ axis parallel along above parabolic curve.

In general, if one variable, for instance, yis missing, then thegraph 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. Considerhorizontal 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

c = 2x2 + y2 :

½ellipse if c > 0empty if c < 0

on the plane z = c that is parallel to xy ¡ plane. When c > 0,the standardform is

x2³pc/2

´2+ y2

(p

c)2 = 1, horizontal half axis =

pc/2 , vertical Half axis =

pc.

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So as c increases (i.e., moving parallel to xy ¡ plane upward) starting atc = 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 curvez = 2a2 + y2

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

on yz ¡ plane, which is a parabola with vertex y = 0, z = 2a2.Similarly, along zx ¡ plane direction, the cross-section with y = b is a

parabolaz = 2x2 + b2.

In summary, cross-sections are either ellipse or parabola.Example 5.5. z = y2 ¡ x2 (hyperbolic paraboloid)

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

z = y2 ¡ x2,

z = c.

The cross-section, or trace, is the curve

c = y2 ¡ x2 :

½hyperbola (opening along y ¡ axis) if c > 0hyperbola (opening along x ¡ axis) if c < 0

on the plane z = c that is parallel to xy ¡ plane. The standard forms are

y2

(p

c)2 ¡ x2

(p

c)2 = 1 (c > 0), or

y2¡p¡c¢2 ¡ x2¡p¡c

¢2 = ¡1 (c < 0).

So as c increases (moving upward) starting at c = 0, the trace becomesvertical hyperbola with increasing half axis

pc. However, when c decreases

(moving downward) starting at c = 0, the trace becomes horizontal hyperbola

with increasing half axispjcj. y2

(p

c)2 ¡ x2

(p

c)2 = 1

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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 curvez = y2 ¡ a2

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

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

Similarly, along zx ¡ plane direction, the cross-section with y = b is aparabola

z = b2 ¡ x2

opening opposite to z ¡ axis :

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

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In summary, cross-sections are either ellipse or hyperbola. However, thosehyperbola changes from horizontal to vertical as the cross-section parallel toxy ¡ plane moving upward.

Example 5.6. Ellipsoid

x2 +y2

9+

z2

4= 1

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

x2 +y2

4¡ z2

4= 1

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Let us look at traces in all three directions. Along xy ¡ plane z = c

x2 +y2

4¡ z2

4= 1

z = c,

the trace

x2 +y2

4= 1 +

c2

4is a ellipse on xy ¡ plane with the standard form

x2Ãr1 +

c2

4

!2 +y2Ã

2

r1 +

c2

4

!2 = 1.

Along yz ¡ plane, the trace is

x2 +y2

4¡ z2

4= 1

x = a,

or hyperbolay2

4¡ z2

4= 1¡ a2

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on yz ¡ plane. As a moves across a = §1, i.e., as (1¡ a2) changes signs,the direction of opening of the hyperbola changes from horizontal (or y ¡axis, when 1¡ a2 > 0) to vertical (or z ¡ axis if 1¡ a2 < 0). Similarly, thetraces on zx ¡ plane,

x2 +y2

4¡ z2

4= 1

y = b,

is hyperbola

x2 ¡ z2

4= 1¡ b2

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on xz ¡ plane whose direction changes whenµ1¡ b2

4

¶changes signs.

Example 5.8. Hyperboloid of Two Sheets

x2 +y2

4¡ z2

4= ¡1.

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The traces along three directions are, respectively,

x2 +y2

4=

c2

4¡ 1 (z = c) ellipse if

c2

4¡ 1 > 0

x2 ¡ z2

4= ¡1¡ b2

4(y = b) hyperbola (opening along z ¡ axis)

y2

4¡ z2

4= ¡1¡ a2 (x = a) hyperbola (opening along z ¡ axis)

Note that here there is not directional change.

² Classi…cation of Quadratic Surfaces

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 to2Dsituations.

(1) If ABC 6= 0,it reduces to

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 thatR 6= 0.

(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

a2+(y ¡ k)2

b2+(z ¡ l)2

c2= 1

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.

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For simpli…cation, we take h = k = l = 0 :

x2

a2+

y2

b2+

z2

c2= 1.

We use traces to see the graph. Set z = l be a constant. The cross sectionin the direction parallel to xy ¡ plane is

x2

a2+

y2

b2+

z2

c2= 1

z = l

or

x2

a2+

y2

b2= 1¡ l2

c2

z = l.

If jlj · c,this is an ellipse. If jlj > c, then

1¡ l2

c2< 0

so there is no solution for the system and the curve is empty. Similarly, inother directions, all cross-sections are ellipses or the empty set.

(b) A, B,C don’t have the same signs. Assuming that AB > 0. Theequation

A (x ¡ h)2 +B (y ¡ k)2 + C (z ¡ l)2 = R.

reduces to either

(x ¡ h)2

a2+(y ¡ k)2

b2¡(z ¡ l)2

c2= 1 (Hyperboloid of One Sheet, z¡axis is axis of symmetry)

or

(x ¡ h)2

a2+(y ¡ k)2

b2¡ (z ¡ l)2

c2= ¡1 (Hyperboloid of Two Sheets)

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 cone. For instance,

2x2 + 3y2 ¡ 4z2 = 0

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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 6= 0.The equation

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

reduce toA (x ¡ h)2 +B (y ¡ k)2 = F (z ¡ l) .

This is an elliptic paraboloid if AB > 0 and a hyperbolic paraboloid ifAB < 0.

We summarize by the Table 2 in page 682:(1) Ellipsoid:

x2

a2+

y2

b2+

z2

c2= 1

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

z

c=

x2

a2+

y2

b2(opening up if c > 0, down if c < 0)

y

b=

x2

a2+

z2

c2(opening towards positiv if y¡direction if b > 0, opposite if b < 0)

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x

a=

y2

b2+

z2

c2(opening towards positiv if x¡direction if a > 0, opposite if a < 0)

(3) Hyperbolic Paraboloid (Saddle)

z

c=

x2

a2¡ y2

b2

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y

b=

z2

c2¡ x2

a2

x

a=

y2

b2¡ z2

c2

(4) Conez2

c2=

x2

a2+

y2

b2

y2

b2=

x2

a2+

z2

c2

x2

a2=

y2

b2+

z2

c2

(5) Hyperboloid of One Sheet

x2

a2+

y2

b2¡ z2

c2= 1

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x2

a2¡ y2

b2+

z2

c2= 1

¡x2

a2+

y2

b2+

z2

c2= 1

(6) Hyperboloid of Two Sheets

x2

a2+

y2

b2¡ z2

c2= ¡1

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x2

a2¡ y2

b2+

z2

c2= ¡1

¡x2

a2+

y2

b2+

z2

c2= ¡1

Homework:

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1. Find and sketch the domain of the function.

(a) f (x, y) =

py ¡ 4x2x2 ¡ 1

(b) g (x, y) =p4¡ x2 ¡ y2 + ln (x2 + y2 ¡ 1)

2. Identify and sketch the trace x = k, y = k, and z = k, and then usethese 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: square both sides)

(e) x = ¡py2 + 2z2 (hint: square both sides)

(f) x2 + 4y2 + 2z2 = ¡4

4. Give a concrete example.

(a) An elliptical paraboloid openning to the negative x ¡ axis withx ¡ 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) forwhich the distance from P to the x ¡ axis is twice the distance fromP to the yz ¡ plane. identify the surface.

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