chapter 1...4 chapter 1. the cartesian spaces 1.1 the cartesian space r2 key ideas. • cartesian...

Chapter 1

The Cartesian spaces

Portrait of Rene Descartes by Frans Halspublic domain image acquired from:

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4 CHAPTER 1. THE CARTESIAN SPACES

1.1 The Cartesian space R2

Key Ideas.

• Cartesian plane R2 and Cartesian coordinates (x, y).

• Implicit curves. Classic examples:

– Line ax+ by = 1

– Ellipse⇣x

a

⌘2+⇣y

b

⌘2= 1

– Hyperbola⇣x

a

⌘2�⇣y

b

⌘2= ±1

– Parabola y = ax2 and x = ay

2.

Shift center to (x⇤, y⇤)

• Regions given by inequalities. Convert between

– Pictures/images

– Implicit inequalities

– Explicit iterated inequalities

Exercises.

1. Sketch the regions of the R2 plane described by the following:

(a) x = 2;

(b) x < 2y;

(c) x2 + y

2> 4;

(d) 4x2 + (y + 1)2 1;

(e) 1 x2 + 3y2 3;

(f) �3 x sin(2y).

2. Sketch the regions of the xy-plane described by the following. You need to be able to do thiswithout any help of “technology”.

(a) x2 + 4y2 = 9

(b) x2 + 4y2 > 9

(c) x2 � y

2 = 9;

(d) x2 � y

2< 9;

(e) x2 � 4y2 = 9;

(f) x2 � 4y2 < 9;

(g) x2 � 4y2 = �9.

(h) x2 � 4y2 < �9.

3. Express the following regions in the format described below. (You just need to present onesolution, though there are multiple correct solutions.)

1.1. THE CARTESIAN SPACE R2 5

(a) Example: Triangle with vertices (1, 0), (0, 2) and (1, 2).

Solution 1: We have 0 x 1. For each fixed value of x we require 2� 2x y 2.

Solution 2: We have 0 y 2. For each fixed value of y we require 1� 12y x 1.

(b) The region inside the unit circle that is also in the third quadrant;

(c) The square with vertices (1, 0), (0, 1), (�1, 0) and (0,�1);

(d) The region bounded by the parabola y = 9� x2 and the line 8x+ y = 0;

(e) The disk (i.e interior of the circle) of radius 3 centered at (1, 1);

(f) The region contained in both the disk of radius 2 centered at (0, 0) and the disk of radius2 centered at (2, 0).

6 CHAPTER 1. THE CARTESIAN SPACES

1.2 The Cartesian space R3

Key Ideas.

• Cartesian space R3 and Cartesian coordinates (x, y, z).

• Implicit surfaces in R3:

– Line ax+ by + cz = a

– Sphere x2 + y

2 + z2 = 1

– Cylinder x2 + y

2 = 1

• 3D regions in R3

– Ball x2 + y2 + z

2 1

– Cylindrical solid x2 + y

2 1

• Convert between

– Image/picture

– Implicit inequality

– Explicit iterated inequalities

Exercises.

1. What do the following describe in R3? Sketch a picture and express in words.

(a) x = 2;

(b) y = 2z;

(c) �1 x 1 and �1 y 1 and �1 z 1;

(d) (x� 1)2 + (y + 2)2 + z2 = 1;

(e) x2 + 4y2 + z

2 16;

(f) x2 + y

2 = 1;

(g) 1 y2 + 4(z � 1)2 4;

(h) x+ 1 = z2.

2. Express the following volumes in the format described below. (You just need to present onesolution, though there are multiple correct solutions.)

(a) Example: The interior of the sphere centered at the origin and passing through (1, 1, 1);

Solution 1: pancakes The equation for the sphere is

x2 + y

2 + z2 = 3,

and the radius isp3. Thus we have �

p3 z

p3. For each fixed value of z we

have a “pancake” slice given by

x2 + y

2 3� z2.

1.2. THE CARTESIAN SPACE R3 7

Solution 2: french fries For each location (x, y), which is described by x2 + y

2 3,in the equatorial disk there is a “French fry” described by

�p3� x2 � y2 z

p3� x2 � y2.

(b) The intersection of balls of radius 2 centered at (0, 0, 0) and (0, 0, 2);

(c) The portion of the ball of radius 2 centered at the origin and located above the planez = 1.

(d) The volume inside the infinitely long cylinder of unit radius centered along the x-axis.

8 CHAPTER 1. THE CARTESIAN SPACES

1.3 Polar coordinates

Key Ideas.

• Polar coordinates:x = r cos ✓

y = r sin ✓

Here r � 0 and we can choose

�⇡ ✓ ⇡ or 0 ✓ ⇡.

Remember to use radians for all angles!

• Key skills:

– convert between r✓ values and xy values

– describe curves and regions using polar coordinates

• Polar coordinate examples: Draw both r✓ plane and xy plane.

– Use for any 2D region with a center point.

– Circle r = constant

– Sector 1 r 2 and ⇡/4 ✓ 3⇡/4

– Line x = 2 becomes r = 2/ cos ✓ for �⇡/2 < ✓ < ⇡/2

• Describe curves/regions in formatx =???

y =???

The point is that there is correspondence between r✓ plane and xy plane. Really emphasizethis!

Exercises.

1. Find:

(a) the polar coordinates of the following Cartesian points:

i. (�2, 0);

ii. (�3, 3);

iii. (1,�3);

iv. (�2,�1).

(b) the Cartesian coordinates of the point whose polar coordinates are (r, ✓) = (2, ⇡3 ).

2. The following equations describe a curve or a region of the Cartesian plane by means of polarcoordinates. Identify (and draw) these regions without any help of “technology”.

(a) r 1, and any ✓;

1.3. POLAR COORDINATES 9

(b) r = ⇡4 , �

⇡4 ✓ ⇡

4 ;

(c) 1 r 4, ⇡4 ✓ 3⇡

4 ;

(d) r = e2✓, �1 < ✓ < 1;

(e) r = e� ✓

2 , �1 < ✓ < 1.

3. Express the following geometric objects using polar coordinates. Follow the template providedbelow.

(a) Example: Washer centered at (3, 1), of inner radius 2 and outer radius 4;

Solution: We havex = 3 + r cos(✓)

y = 1 + r sin(✓)with

2 r 4

0 ✓ 2⇡

Note: Since we are describing a region with area, it makes sense that we have twofree variables r and ✓, indicating two independent directions of motion inside thiswasher.

(b) Disk of radius 2 centered at the origin;

(c) The first quadrant of the Cartesian plane;

(d) The line x = 1;

(e) The interior of the triangle with the vertices at (0, 0), (1, 1) and (1,�1).

10 CHAPTER 1. THE CARTESIAN SPACES

1.4 Cylindrical coordinates

Key Ideas.

• Cylindrical coordinates:x = r cos ✓

y = r sin ✓

z = z

Here r � 0 and 0 ✓ 2⇡ (or �⇡ ✓ ⇡).

• It is often helpful to draw the “cutaway” view in the rz plane.

• Famous examples:

– Cylinder r = 1

– Parabolic bowl z = r2

– Hyperboloid of one sheet r2 � z2 = 1

– Hyperboloid of two sheets z2 � r

2 = 1

– Sphere r2 + z

2 = 1

– Cone z = r

Exercises

1. Find:

(a) the cylindrical coordinates of the Cartesian point (x, y, z) = (0,�1, 0);

(b) the cylindrical coordinates of the Cartesian point (x, y, z) = (0, 1,�1);

(c) the Cartesian coordinates of the point whose cylindrical coordinates are (r, ✓, z) =(2, 2⇡

3 ,⇡4 ).

2. Express the following geometric objects using cylindrical coordinates. Follow the templateprovided below.

(a) Example: The surface of the infinite upright cylinder of unit radius centered along thez-axis;

Solution: We have r = 1, while ✓ with 0 ✓ 2⇡ and z with �1 < z < 1. Thus theCartesian coordinate expressions are

x = cos ✓,

y = sin ✓,

z = z.

Since we are describing a surface, it makes sense that we have two free variables✓ and z, indicating two independent directions of motion along the surface of thecylinder.

1.4. CYLINDRICAL COORDINATES 11

(b) The interior of the infinite upright cylinder of unit radius centered along the z-axis;

(c) The surface of infinite cylinder of radius 1 centered around the y-axis;

(d) Unit sphere centered at the origin;

(e) Unit ball centered at the origin;

(f) Upper unit hemi-sphere centered at the origin;

(g) Polar cap of the sphere of radius 2 centered at the origin, located to the “north” of the60�-parallel;

(h) The surface of a circular cone of your choice going around the z-axis with the tip at theorigin.

12 CHAPTER 1. THE CARTESIAN SPACES

1.5 Spherical coordinates

Key Ideas.

• Spherical coordinatesx = r cos ✓ sin�

y = r sin ✓ sin�

z = r cos�

were r � 0, �⇡ ✓ ⇡, 0 � ⇡.

Remember to use radians for all angles.

• Warning: We use “North Pole” spherical coordinates. Other conventions exist, especially inphysics land (where ✓ and � are sometimes swapped).

Exercises.

1. Find:

(a) the spherical coordinates of the Cartesian point (0,�1, 0);

(b) the spherical coordinates of the Cartesian point (0, 1,�1);

(c) the Cartesian coordinates of the point whose spherical coordinates (r, ✓,�) are (2, 2⇡3 ,

⇡4 ).

2. The following equations describe regions in spherical coordinates. What regions are those?Explain in words and try to draw. Avoid using “technology”.

(a) r � 4;

(b) 1 r 4, 0 � ⇡2 ;

(c) r 1, 0 � ⇡4 ;

(d) 1 r 4, ⇡4 � 3⇡

4 .

3. Express the following geometric objects in spherical coordinates. Follow the template providedbelow.

(a) Example: Unit sphere centered at the origin.

Solution: For the unit sphere we have r = 1. We want ✓ to vary over the range0 ✓ 2⇡ and � to vary over the range 0 � ⇡. The result is that

x = cos(✓) sin(�)

y = sin(✓) sin(�)

z = cos(�)

with0 ✓ 2⇡

0 � ⇡

Since we are describing a surface, it makes sense that we have two free variables✓ and �, indicating two independent directions of motion along the surface of thesphere: ✓ going from west to east and � going from north to south.

(b) Upper unit hemi-sphere centered at the origin;

1.5. SPHERICAL COORDINATES 13

(c) Unit ball centered at the origin;

(d) The “north” 60�-parallel of the sphere of radius 2 centered at the origin;

(e) Polar cap of the sphere of radius 2 centered at the origin, located to the “north” of the60�-parallel;

(f) The “north-south” meridian passing through the point (1, 1,�p2) of the sphere of radius

2 centered at the origin;

(g) The “east-west” parallel passing through the point (1, 1,�p2) of the sphere of radius 2

centered at the origin;

(h) The surface of an infinite circular cone of your choice going around the z-axis whose tipis at the origin;

(i) The volume inside of an infinite circular cone of your choice going around the z-axiswhose tip is at the origin.

14 CHAPTER 1. THE CARTESIAN SPACES

1.6 Paths R ! Rn

Key Ideas.

• These functions describe a path in R2 or R3

• Polar, cylindrical, spherical coordinates are helpful when constructing paths

• Important examples:

– Traversing the unit circle P (t) = (cos(t), sin(t))

– Straight line connecting (a, b) and (p, q)

P (t) = (a+ (p� a)t, b+ (q � b)t)

Exercises.

1. Sketch a plot of the following paths.

(a) The path given byx(t) = 2 cos(t)

y(t) = 3 sin(t)with 0 t ⇡

(b) The path given byx(t) = �2t

y(t) = 5twith � 1 t 1

(c) The path given byx(t) = e

t cos(3t)

y(t) = et sin(3t)

z(t) = 3etwith � ⇡ t ⇡

2. Construct a formula for the following paths:

(a) A path that travels along the standard parabola y = x2 from the point (�1, 1) to the

point (1, 1).

(b) A path that spirals three times around (in a clockwise manner) while traveling from(4, 0) to (1, 0).

(c) A path that travels along a semicircle (centered at the origin) from (1, 0) to (�1, 0).

(d) A path that traverses a circle of radius 7 that is parallel to the xy plane, but is at aheight of 12.

(e) A helicoidal path on the cylinder of radius 5 that winds around the cylinder three times.

(f) A path on the sphere of radius 10 that traverses the 45� north parallel.

(g) A path on the unit sphere (meaning radius 1) from the north pole to the south pole thatspirals twice around the sphere.

1.7. TRANSFORMATIONS RN ! RN 15

1.7 Transformations Rn ! Rn

Key Ideas.

• Transformations are used to construct “coordinate grids” in R2 or R3.

• Visualize using domain/codomain pictures.

• Important examples:

– Diagonal coordinates x = u+ v, y = u� v

– Polar coordinates

– Modified polar coordinates x = Au cos(v), y = Bu sin(v)

– Cylindrical coordinates; modified cylindrical coordinates

– Spherical coordinates; modified spherical coordinates

Exercises.

1. Draw the domain/codomain sketch of each coordinate transformation.

(a) The polar coordinate transformation T (r, ✓) = (r cos ✓, r sin ✓).

(b) The cylindrical coordinate transformation T (r, ✓, z) = (r cos ✓, r sin ✓, z).

(c) The spherical coordinate transformation T (r, ✓,�) = (r cos ✓ sin�, r sin ✓ sin�, r cos�).

2. Use modified polar coordinates to parametrize each region of R2. Make a domain/codomainpicture that indicates the range of u and v.

(a) 4x2 + 9y2 36

(b) 1 x2 + 4y2 16

(c) x2 + 9y2 64 and y � 0

3. Use modified coordinates to parametrize each region of R3. What range of u, v, w are required?Draw pictures are you are able.

(a) x2 + 4y2 4 and 0 z 4

(b) 4x2 + y2 + 9z2 1

4. For each transformation create a domain/codomain sketch to visualize the function. Thendescribe transformation in words.

(a) T (u, v) = (1 + 2u, 3� v)

(b) T (u, v) = (u+ v, u� v)

(c) T (u, v) = (2u� 3v, 3u)

(d) T (u, v) = (u2 � v2, 2uv), restricted to u � 0.

(e) T (u, v, w) = (u, v, u2 + v2 + w).

(Hint: think about where w =constant slices go.)

16 CHAPTER 1. THE CARTESIAN SPACES

1.8 Parametrizations R2 ! R3

Key Ideas.

• Parametrizations are used to put “coordinate grids” on surfaces inside R3

• Visualize using domain/codomain pictures.

• Use polar, cylindrical, spherical coordinates as appropriate.

• Important examples:

– Cylinders

– Parabolic bowls

– Spheres and ellipsoids

– Cones

Exercises.

1. Make a domain/codomain sketch the following parametrizations. What is the appropriaterange of u and v?

(a) The surface given byF (u, v) = (u, v, cos(u))

(b) The sphere given by

F (u, v) = (3 cos(u) sin(v), 3 sin(u) sin(v), 3 cos(v))

(c) The ellipse given by

F (u, v) = (cos(u) sin(v), sin(u) sin(v), 2 cos(v))

(d) The bowl given byF (u, v) = (u cos(v), u sin(v), u2)

(e) The dome given by

F (u, v) = (u cos(v), u sin(v),p1� u2)

(f) The helicoid given byF (u, v) = (u cos(v), u sin(v), v)

2. Find a parametrizing function of each object. Then make a domain/codomain sketch of yourfunction. Indicate the range of u and v.

(a) The surface of the ellipsoid x2 + y

2 + 4z2 = 4

(b) The elliptical cylinder x2 + 9y2 = 36

(c) The portion of the bowl z = 4x2 + y2 below the plane z = 4

(d) The torus (surface of a bagel) with large radius 4 and small radius 1.