Harold’s Calculus 3
Multi-Cordinate System
“Cheat Sheet”
4 March 2015
Rectangular
2-D
𝑓(𝑥) = 𝑦
(𝑥, 𝑦)
(𝑎, 𝑏)
Point
3-D
𝑓(𝑥, 𝑦) = 𝑧
(𝑥, 𝑦, 𝑧)
4-D
𝑓(𝑥, 𝑦, 𝑧) = 𝑤
(𝑥, 𝑦, 𝑧, 𝑤)
•
Polar/Cylindrical
Spherical
Parametric
Vector
Point (a,b) in Rectangular :
𝑥(𝑡) = 𝑎
𝑦(𝑡) = 𝑏
𝑟𝑑
𝑡 = 3 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒, 𝑢𝑠𝑢𝑎𝑙𝑙𝑦 𝑡𝑖𝑚𝑒
< 𝑎, 𝑏 >
⃗ = ⟨𝑥0 , 𝑦0 , 𝑧0 ⟩
𝒓
Matrix
(𝜌 , 𝜃, 𝜙)
𝑥 = 𝜌 𝑠𝑖𝑛 𝜙 𝑐𝑜𝑠 𝜃
𝑦 = 𝜌 sin 𝜙 𝑠𝑖𝑛 𝜃
𝑧 = 𝜌 𝑐𝑜𝑠 𝜙
(𝑟, 𝜃) or 𝑟 ∠ 𝜃
𝑥 = 𝑟 𝑐𝑜𝑠 𝜃
𝑦 = 𝑟 𝑠𝑖𝑛 𝜃
𝑧=𝑧
𝜌2 = 𝑟 2 + 𝑧 2
𝜌2 = 𝑥 2 + 𝑦 2 + 𝑧 2
𝑟2 = 𝑥2 + 𝑦2
𝑟 = ± √𝑥 2 + 𝑦 2
𝑦
𝑡𝑎𝑛 𝜃 = ( )
𝑥
𝑦
𝑡𝑎𝑛 𝜃 = (𝑥 )
𝑦
𝜃 = 𝑡𝑎𝑛−1 ( )
𝑥
𝜙 = cos−1 (
𝑧
√𝑥 2 + 𝑦 2 + 𝑧 2
𝑧
𝜙 = cos−1 ( )
𝜌
[𝑎] [𝑥] = [𝑏]
)
Slope-Intercept Form:
𝑦 = 𝑚𝑥 + 𝑏
Point-Slope Form:
𝑦 − 𝑦0 = 𝑚 (𝑥 − 𝑥0 )
Line
< 𝑥, 𝑦 > = < 𝑥0 , 𝑦0 > + 𝑡 < 𝑎, 𝑏 >
< 𝑥, 𝑦 > = < 𝑥0 + 𝑎𝑡, 𝑦0 + 𝑏𝑡 >
where
< 𝑎, 𝑏 > = < 𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 >
General Form:
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
𝑤ℎ𝑒𝑟𝑒 𝐴 𝑎𝑛𝑑 𝐵 ≠ 0
𝑥(𝑡) = 𝑥0 + 𝑡𝑎
𝑦(𝑡) = 𝑦0 + 𝑡𝑏
𝑓(𝑥) = 𝑓 ′ (𝑎) 𝑥 + 𝑓(0)
where 𝑚 = 𝑓’(𝑎)
3-D:
𝑥 − 𝑥0
𝑦 − 𝑦0
𝑧 − 𝑧0
=
=
𝑎
𝑏
𝑐
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
𝑚=
1
∆𝑦
𝑦2 − 𝑦1 𝑏
=
=
∆𝑥 𝑥2 − 𝑥1 𝑎
⃗ = ⃗⃗⃗⃗
⃗
𝒓
𝒓𝟎 + 𝑡 𝒗
= ⟨𝑥0 , 𝑦0 , 𝑧0 ⟩
+ 𝑡 ⟨𝑎, 𝑏, 𝑐⟩
[𝑎
𝑥
𝑏] [𝑦] = [𝑐]
𝑎
𝑐
𝑒
𝑏 𝑥
] [𝑦] = [𝑓 ]
𝑑
[
Rectangular
Polar/Cylindrical
Spherical
Parametric
Vector
𝑎(𝑥 − 𝑥0 )
+ 𝑏(𝑦 − 𝑦0 )
+ 𝑐(𝑧 − 𝑧0 ) = 0
𝒓 = 𝒓0 + 𝑠𝒗 + 𝑡𝒘
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑
𝑤ℎ𝑒𝑟𝑒 𝑑 = 𝑎𝑥0 + 𝑏𝑦0 + 𝑐𝑧0
(𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 , 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑤ℎ𝑒𝑟𝑒 𝜌 𝑡𝑎𝑘𝑒𝑠 𝑜𝑛 𝑎𝑙𝑙
𝑣𝑎𝑙𝑢𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛
(0 ≤ 𝜌 < ∞)
𝑓(𝑥, 𝑦) = 𝐴𝑥 + 𝐵𝑦 + 𝐶
Plane
𝑥(𝑡) = 𝑟 𝑐𝑜𝑠(𝑡) + ℎ
𝑦(𝑡) = 𝑟 𝑠𝑖𝑛(𝑡) + 𝑘
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [0, 2𝜋)
r = a (constant)
𝜃 = 𝜃 [0, 2𝜋] 𝑜𝑟 [0, 360°]
𝑥2 + 𝑦2 = 𝑟2
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
Circle
General Form:
𝐴𝑥 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹
=0
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐶 𝑎𝑛𝑑 𝐵 = 0
(ℎ, 𝑘) = 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒
2
(ℎ, 𝑘)
𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜃 = 𝜃 [0, 2𝜋]
𝜙 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 0
Focus and Center:
(ℎ, 𝑘)
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
where:
s and t range over all real
numbers
 v and w are given vectors
defining the plane
 𝒓𝟎 is the vector representing the
position of an arbitrary (but
fixed) point on the plane

2
𝒏 ∙ (𝒓 − 𝒓0 ) = 0
Matrix
Rectangular
2
2
2
Polar/Cylindrical
Spherical
Parametric
Vector
2
𝑥 +𝑦 +𝑧 =𝑟
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 + (𝑧 − 𝑙)2
= 𝑟2
𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜃 = 𝜃 [0, 2𝜋]
𝜙 = 𝜙 [0, 2𝜋]
𝐹𝑜𝑐𝑢𝑠 𝑎𝑛𝑑 𝑐𝑒𝑛𝑡𝑒𝑟:
(ℎ, 𝑘, 𝑙)
Sphere
General Form:
𝐴𝑥 2 + 𝐵𝑦 2 + 𝐶𝑧 2
+ 𝐷𝑥𝑦 + 𝐸𝑦𝑧 + 𝐹𝑥𝑧
+ 𝐺𝑥 + 𝐻𝑦 + 𝐼𝑧 + 𝐽 = 0
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐵 = 𝐶 > 0
Cylindrical to Rectangular:
𝑥 = 𝑟 𝑐𝑜𝑠(𝜃)
𝑦 = 𝑟 𝑠𝑖𝑛(𝜃)
𝑧=𝑧
Spherical to Rectangular:
𝑥 = 𝑟 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜙
𝑦 = 𝑟 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛 𝜙
𝑧 = 𝑟 𝑐𝑜𝑠 𝜃
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
Rectangular to Cylindrical:
𝑟 = √𝑥 2 + 𝑦 2
Spherical to Cylindrical:
𝜌 = 𝑟 𝑠𝑖𝑛(𝜃)
𝜙=𝜙
𝑧 = 𝑟 𝑐𝑜𝑠(𝜃)
Rectangular to Spherical:
𝑟 = √𝑥 2 + 𝑦 2 + 𝑧 2
𝑧
𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑠 ( )
𝑟
𝑦
𝜙 = 𝑎𝑟𝑐𝑡𝑎𝑛 ( )
𝑥
Cylindrical to Spherical:
𝑟 = √𝜌2 + 𝑧 2
𝜌
𝑧
𝜃 = 𝑎𝑟𝑐𝑡𝑎𝑛 ( ) = 𝑎𝑟𝑐𝑐𝑜𝑠 ( )
𝑧
𝑟
𝜙=𝜙
3
Rectangular:
𝑥
𝒓 ≡ [𝑦 ]
𝑧
Cylindrical:
𝑟 𝑐𝑜𝑠(𝜃)
𝒓 ≡ [ 𝑟 𝑠𝑖𝑛(𝜃) ]
𝑧
Spherical:
𝑟 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜙
𝒓 ≡ [ 𝑟 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛 𝜙 ]
𝑟 𝑐𝑜𝑠 𝜃
Matrix
Rectangular
2
Polar/Cylindrical
Spherical
Parametric
Vector
Matrix
2
(𝑥 − ℎ)
(𝑦 − 𝑘)
+
=1
2
𝑎
𝑏2
General Form:
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦
+𝐹 = 0
where 𝐵2 − 4𝐴𝐶 < 0 𝑜𝑟 𝐴𝐶 > 0
𝑒𝑑
1 ± 𝑒 cos 𝜃
𝑜𝑟
𝑒𝑑
𝑟=
1 ± 𝑒 sin 𝜃
𝑟=
Center: (ℎ, 𝑘)
Vertices:
(ℎ ± 𝑎, 𝑘)
(ℎ, 𝑘 ± 𝑏)
(ℎ, 𝑘) = 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑒𝑙𝑙𝑖𝑝𝑠𝑒
𝐸𝑐𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑡𝑦: 0 < 𝑒 < 1
𝑟(𝜃) =
Foci: (ℎ ± 𝑐, 𝑘)
Ellipse
𝑥(𝑡) = 𝑎 𝑐𝑜𝑠(𝑡) + ℎ
𝑦(𝑡) = 𝑏 𝑠𝑖𝑛(𝑡) + 𝑘
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [0, 2𝜋]
---------------------------------------------------Rotated Ellipse:
𝑥(𝑡) = 𝑎 𝑐𝑜𝑠 𝑡 𝑐𝑜𝑠 𝜃 − 𝑏 𝑠𝑖𝑛 𝑡 𝑠𝑖𝑛 𝜃 + ℎ
𝑦(𝑡) = 𝑎 𝑐𝑜𝑠 𝑡 𝑠𝑖𝑛 𝜃 + 𝑏 𝑠𝑖𝑛 𝑡 𝑐𝑜𝑠 𝜃 + 𝑘
𝑎𝑏
√(𝑏 𝑐𝑜𝑠 𝜃)2 + (𝑎 𝑠𝑖𝑛 𝜃)2
relative to center
𝜃 = the angle between the x-axis and the
major axis of the ellipse
Focus length, c, from center:
𝑐 = √𝑎2 − 𝑏 2
Eccentricity:
𝑐
√𝑎2 − 𝑏 2
𝑒= =
𝑎
𝑎
If B ≠ 0, then rotate coordinate
system:
𝐴−𝐶
𝑐𝑜𝑡 2𝜃 =
𝐵
𝑥 = 𝑥′ 𝑐𝑜𝑠 𝜃 − 𝑦′ 𝑠𝑖𝑛 𝜃
𝑦 = 𝑦′ 𝑐𝑜𝑠 𝜃 + 𝑥′ 𝑠𝑖𝑛 𝜃
New = (x’, y’), Old = (x, y)
rotates through angle 𝜃 from xaxis
Ellipsoid
(𝑥 − ℎ)2 (𝑦 − 𝑘)2 (𝑧 − 𝑙)2
+
+
𝑎2
𝑏2
𝑐2
=1
2
2
2
2
2
𝑟 𝑐𝑜𝑠 𝜃 𝑟 𝑠𝑖𝑛 𝜃 𝑧
+
+ 2=1
𝑎2
𝑏2
𝑐
𝑟 2 𝑐𝑜𝑠 2 𝜃 𝑠𝑖𝑛2 𝜙 𝑟 2 𝑠𝑖𝑛2 𝜃 𝑠𝑖𝑛2 𝜙
+
𝑎2
𝑏2
𝑟 2 𝑐𝑜𝑠 2 𝜙
+
=1
𝑐2
𝑥(𝑡, 𝑢) = 𝑎 𝑐𝑜𝑠(𝑡) 𝑐𝑜𝑠(𝑢) + ℎ
𝑦(𝑡, 𝑢) = 𝑏 𝑐𝑜𝑠(𝑡) 𝑠𝑖𝑛(𝑢) + 𝑘
𝑧(𝑡, 𝑢) = 𝑐 𝑠𝑖𝑛(𝑡) + 𝑙
𝜋 𝜋
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [− , ]
2 2
[𝑢𝑚𝑖𝑛 , 𝑢𝑚𝑎𝑥 ] = [−𝜋, 𝜋]
(ℎ, 𝑘, 𝑙) = 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑒𝑙𝑙𝑖𝑝𝑠𝑜𝑖𝑑
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
4
Centered at vector v
Rectangular
2
Polar/Cylindrical
Spherical
Parametric
2
(𝑥 − ℎ)
(𝑦 − 𝑘)
−
=1
2
𝑎
𝑏2
General Form:
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹
=0
where 𝐵2 − 4𝐴𝐶 > 0 𝑜𝑟 𝐴𝐶 < 0
If 𝐴 + 𝐶 = 0, square hyperbola
𝑒𝑑
1 ± 𝑒 cos 𝜃
𝑜𝑟
𝑒𝑑
𝑟=
1 ± 𝑒 sin 𝜃
Center: (ℎ, 𝑘)
Vertices: (ℎ ± 𝑎, 𝑘)
Foci: (ℎ ± 𝑐, 𝑘)
𝐸𝑐𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑡𝑦: 𝑒 > 1
Focus length, c, from center:
𝑐 = √𝑎2 + 𝑏 2
Hyperbola
Eccentricity:
𝑐
√𝑎2 + 𝑏 2
𝑒= =
= sec 𝜃
𝑎
𝑎
Left-Right Opening Hyperbola:
𝑥(𝑡) = 𝑎 𝑠𝑒𝑐(𝑡) + ℎ
𝑦(𝑡) = 𝑏 𝑡𝑎𝑛(𝑡) + 𝑘
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐]
(h, k) = vertex of hyperbola
𝑟=
Alternate Form:
𝑥(𝑡) = ±𝑎 𝑐𝑜𝑠ℎ(𝑡) + ℎ
𝑦(𝑡) = 𝑏 𝑠𝑖𝑛ℎ(𝑡) + 𝑘
𝑎(𝑒 2 − 1)
𝑟=
1 + 𝑒 𝑐𝑜𝑠 𝜃
Up-Down Opening Hyperbola:
𝑥(𝑡) = 𝑎 𝑡𝑎𝑛(𝑡) + ℎ
𝑦(𝑡) = 𝑏 𝑠𝑒𝑐(𝑡) + 𝑘
1
1
−𝑐𝑜𝑠 −1 (− ) < 𝜃 < 𝑐𝑜𝑠 −1 (− )
𝑒
𝑒
3-D
(𝑥 − ℎ)2 (𝑦 − 𝑘)2 (𝑧 − 𝑙)2
+
−
𝑎2
𝑏2
𝑐2
=1
Alternate Form:
𝑥(𝑡) = 𝑎 𝑠𝑖𝑛ℎ(𝑡) + ℎ
𝑦(𝑡) = ±𝑏 𝑐𝑜𝑠ℎ(𝑡) + 𝑘
(𝑥 − ℎ)2
(𝑦 − 𝑘)2 (𝑧 − 𝑙)2
−
−
+
𝑎2
𝑏2
𝑐2
=1
General Form:
𝑥(𝑡) = 𝐴𝑡 2 + 𝐵𝑡 + 𝐶
𝑦(𝑡) = 𝐷𝑡 2 + 𝐸𝑡 + 𝐹
where A and D have different signs
If B ≠ 0, then rotate coordinate
system:
𝐴−𝐶
𝑐𝑜𝑡 2𝜃 =
𝐵
𝑥 = 𝑥′ 𝑐𝑜𝑠 𝜃 − 𝑦′ 𝑠𝑖𝑛 𝜃
𝑦 = 𝑦′ 𝑐𝑜𝑠 𝜃 + 𝑥′ 𝑠𝑖𝑛 𝜃
New = (x’, y’), Old = (x, y)
rotates through angle 𝜃 from xaxis
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
5
Vector
Matrix
Rectangular
Polar/Cylindrical
Vertical Axis of Symmetry:
𝑥 2 = 4 𝑝𝑦
(𝑥 − ℎ)2 = 4𝑝(𝑦 − 𝑘)
Vertex: (ℎ, 𝑘)
Focus: (ℎ, 𝑘 + 𝑝)
Directrix: 𝑦 = 𝑘 − 𝑝
𝑒𝑑
1 ± 𝑒 cos 𝜃
𝑜𝑟
𝑒𝑑
𝑟=
1 ± 𝑒 sin 𝜃
Limit
𝐸𝑐𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑡𝑦: 𝑒 = 1
Where 𝑑 = 2𝑝
General Form:
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹
=0
where 𝐵2 − 4𝐴𝐶 = 0
or 𝐴𝐶 = 0
Projectile Motion:
𝑥(𝑡) = 𝑥0 + 𝑣𝑥 𝑡
𝑦(𝑡) = 𝑦0 + 𝑣𝑦 𝑡 − 16𝑡 2 feet
𝑦(𝑡) = 𝑦0 + 𝑣𝑦 𝑡 − 4.9𝑡 2 meters
𝑣𝑥 = 𝑣 𝑐𝑜𝑠 𝜃
𝑣𝑦 = 𝑣 𝑠𝑖𝑛 𝜃
General Form:
𝑥 = 𝐴𝑡 2 + 𝐵𝑡 + 𝐶
𝑦 = 𝐿𝑡 2 + 𝑀𝑡 + 𝑁
where A and L have the same sign
New = (x’, y’), Old = (x, y)
rotates through angle 𝜃 from xaxis
(𝑥 − ℎ)2 (𝑦 − 𝑘)2 (𝑧 − 𝑙)2
+
=
𝑎2
𝑏2
𝑐2
lim 𝑓(𝑥) = 𝐿
𝑥→𝑐
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
ℎ→0
ℎ
𝑓(𝑥) − 𝑓(𝑐)
′ (𝑐)
𝑓
= lim
𝑥→𝑐
𝑥−𝑐
𝑑𝑦
′ (𝑥)
𝑓
=
= 𝑦′ = 𝐷𝑥
𝑑𝑥
𝑓 ′ (𝑥) = lim
1st
Derivative
Vector
Horizontal axis of symmetry:
𝑥(𝑡) = 𝑝𝑡 2 + ℎ
𝑦(𝑡) = 2𝑝𝑡 + 𝑘 (opens right) or
𝑦(𝑡) = −2𝑝𝑡 + 𝑘 (opens left)
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐]
If B ≠ 0, then rotate coordinate
system:
𝐴−𝐶
𝑐𝑜𝑡 2𝜃 =
𝐵
𝑥 = 𝑥′ 𝑐𝑜𝑠 𝜃 − 𝑦′ 𝑠𝑖𝑛 𝜃
𝑦 = 𝑦′ 𝑐𝑜𝑠 𝜃 + 𝑥′ 𝑠𝑖𝑛 𝜃
Nose Cone
Parametric
Vertical axis of symmetry:
𝑥(𝑡) = 2𝑝𝑡 + ℎ
𝑦(𝑡) = 𝑝𝑡 2 + 𝑘 (opens upwards) or
𝑦(𝑡) = −𝑝𝑡 2 + 𝑘 (opens downwards)
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐]
(h, k) = vertex of parabola
𝑟=
Horizontal Axis of Symmetry:
𝑦 2 = 4 𝑝𝑥
(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ)
Vertex: (ℎ, 𝑘)
Focus: (ℎ + 𝑝, 𝑘)
Directrix: 𝑥 = ℎ − 𝑝
Parabola
Spherical
𝑑
(𝒓
⃗)= 𝒓
⃗ ′
𝑑𝑡
𝑑𝑦 𝑑𝑟
𝑑𝑦 𝑑𝜃 𝑑𝜃 𝑠𝑖𝑛 𝜃 + 𝑟 𝑐𝑜𝑠 𝜃
=
=
𝑑𝑟
𝑑𝑥 𝑑𝑥
𝑐𝑜𝑠 𝜃 − 𝑟 𝑠𝑖𝑛 𝜃
𝑑𝜃
𝑑𝜃
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
𝑑𝑦
𝑑𝑦 𝑑𝑡
=
,
𝑑𝑥 𝑑𝑥
𝑑𝑡
Hint: Use Product Rule for
𝑦 = 𝑟 𝑠𝑖𝑛 𝜃
𝑥 = 𝑟 𝑐𝑜𝑠 𝜃
6
𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑
𝑑𝑥
≠0
𝑑𝑡
Unit tangent vector
⃗𝑇(𝑡)
⃗ ′(𝑡)
𝒓
⃗ ′ (𝑡)
=
𝑤ℎ𝑒𝑟𝑒 𝒓
‖𝒓
⃗ ′(𝑡)‖
≠ ⃗0
Matrix
Rectangular
2nd
Derivative
𝑓
′′ (𝑥)
Polar/Cylindrical
Spherical
𝑑 𝑑𝑦
( )
𝑑2 𝑦
𝑑 𝑑𝑦
𝑑𝜃 𝑑𝑥
=
(
)
=
𝑑𝑥
𝑑𝑥 2 𝑑𝑥 𝑑𝑥
𝑑𝜃
𝑑 𝑑𝑦
𝑑2 𝑦
=
( )=
= 𝑦′′
𝑑𝑥 𝑑𝑥
𝑑𝑥 2
Parametric
Vector
Matrix
𝑑 𝑑𝑦
( )
𝑑2 𝑦
𝑑 𝑑𝑦
𝑑𝑡 𝑑𝑥
=
(
)
=
𝑑𝑥
𝑑𝑥 2 𝑑𝑥 𝑑𝑥
𝑑𝑡
Unit normal vector
⃗𝑁(𝑡)
⃗𝑻′(𝑡)
=
𝑤ℎ𝑒𝑟𝑒 ⃗𝑻′ (𝑡)
⃗ ′(𝑡)‖
‖𝑻
⃗
≠0
Riemann Sum:
𝑛
𝑆=∑
𝑓(𝑦𝑖 )(𝑥𝑖 − 𝑥𝑖−1 )
𝑖−1
Left Sum:
1
𝑓(𝑎) + 𝑓 (𝑎 + ) +
1
𝑛
𝑆 = ( )[
]
2
1
𝑛
𝑓 (𝑎 + ) + ⋯ + 𝑓(𝑏 − )
𝑛
𝑛
𝑏
Integral
Middle Sum:
1
1
3
𝑆 = ( ) [𝑓 (𝑎 + ) + 𝑓 (𝑎 + ) + ⋯
𝑛
2𝑛
2𝑛
1
+ 𝑓(𝑏 − )]
2𝑛
𝐹(𝑥) = ∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
𝑎
𝑏
𝑏
𝑏
𝑏
⃗ (𝑡) 𝑑𝑡 = ⟨∫ 𝑓(𝑡)𝑑𝑡 , ∫ 𝑔(𝑡)𝑑𝑡 , ∫ ℎ(𝑡)𝑑𝑡⟩
∫𝒓
𝑎
𝑎
𝑎
𝑎
Right Sum:
1
1
2
𝑆 = ( ) [𝑓 (𝑎 + ) + 𝑓 (𝑎 + ) + ⋯
𝑛
𝑛
𝑛
+ 𝑓(𝑏)]
𝑏 𝑑(𝑦)
Double
Integral
Triple
Integral
𝑆𝑎𝑚𝑒 𝑎𝑠 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟, 𝑏𝑢𝑡
𝑓(𝑥, 𝑦) ⟶ 𝑓(𝜌 cos 𝜙 , 𝜌 sin 𝜙)
∫ ∫ 𝑓(𝑥, 𝑦) 𝑑𝑥 𝑑𝑦
𝑎 𝑐(𝑦)
𝑏 𝑑(𝑧) 𝑔(𝑦,𝑧)
∫ ∫
∫ 𝑓(𝑥, 𝑦, 𝑧) 𝑑𝑥 𝑑𝑦 𝑑𝑧
𝑆𝑎𝑚𝑒 𝑎𝑠 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟, 𝑏𝑢𝑡
𝑓(𝑥, 𝑦, 𝑧) ⟶ 𝑓(𝜌 cos 𝜙 , 𝜌 sin 𝜙 , 𝑧)
𝑎 𝑐(𝑧) 𝑒(𝑦,𝑧)
𝐼𝑓 𝑓(𝑥) = 𝑦, 𝑡ℎ𝑒𝑛 𝑓 −1 (𝑦) = 𝑥
Inverse
Functions
Inverse Function Theorem:
1
𝑓 −1 (𝑏) = ′
𝑓 (𝑎)
where 𝑏 = 𝑓 ′ (𝑎)
𝑖𝑓 𝑦 = 𝑠𝑖𝑛 𝜃
𝑖𝑓 𝑦 = 𝑐𝑜𝑠 𝜃
𝑖𝑓 𝑦 = 𝑡𝑎𝑛 𝜃
𝑖𝑓 𝑦 = 𝑐𝑠𝑐 𝜃
𝑖𝑓 𝑦 = 𝑠𝑒𝑐 𝜃
𝑖𝑓 𝑦 = 𝑐𝑜𝑡 𝜃
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
𝑡ℎ𝑒𝑛 𝜃 = 𝑠𝑖𝑛−1 𝑦
then 𝜃 = 𝑐𝑜𝑠 −1 𝑦
𝑡ℎ𝑒𝑛 𝜃 = 𝑡𝑎𝑛−1 𝑦
−1
𝑡ℎ𝑒𝑛 𝜃 = 𝑐𝑠𝑐 𝑦
𝑡ℎ𝑒𝑛 𝜃 = 𝑠𝑒𝑐 −1 𝑦
𝑡ℎ𝑒𝑛 𝜃 = 𝑐𝑜𝑡 −1 𝑦
𝑆𝑎𝑚𝑒 𝑎𝑠 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟, 𝑏𝑢𝑡
𝑓(𝑥, 𝑦, 𝑧) ⟶ 𝑓(𝜌 cos 𝜃 sin 𝜙,
𝜌 sin 𝜃 sin 𝜙 , 𝜌 cos 𝜙)
NA
NA
NA
NA
NA
NA
𝜃 = arcsin 𝑦
𝜃 = arccos 𝑦
𝜃 = 𝑎𝑟𝑐𝑡𝑎𝑛 𝑦
𝜃 = 𝑎𝑟𝑐𝑐𝑠𝑐 𝑦
𝜃 = 𝑎𝑟𝑐𝑠𝑒𝑐 𝑦
𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑡 𝑦
7
Rectangular
Polar/Cylindrical
Spherical
Parametric
Vector
Matrix
𝑏
Rectangular 2D:
𝐿 = ∫ √1 + [𝑓 ′ (𝑥)]2 𝑑𝑥
𝑑𝑥 2
𝑑𝑦 2
𝐿 = ∫ √( ) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
Polar:
𝑎
Proof :
∆𝑠 = √(𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2
∆𝑠 = √(∆𝑥)2 + (∆𝑦)2
𝑑𝑠 = √𝑑𝑥 2 + 𝑑𝑦 2
𝑑𝑠 = √𝑑𝑥 2 + 𝑑𝑦 2 (
Arc Length
𝑑𝑦
𝑑𝑠 = √𝑑𝑥 2 + ( ) 𝑑𝑥 2
𝑑𝑥
√𝑑𝑥 2
𝑑𝑦 2
(1 + ( ) )
𝑑𝑥
𝑑𝑠 = √1 + (
𝐿 = ∫ √𝑟 2 + (
𝑑𝑟 2
) 𝑑𝜃
𝑑𝜃
𝛼
Curvature
Circle:
𝐿 = 𝑠 = 𝑟𝜃
𝛼
𝑑𝑥 2
𝑑𝑦 2
𝑑𝑧 2
) + ( ) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑡2
𝜃
𝐿 = ( ) 𝜋 (2𝑟) = 𝑟𝜃
2𝜋
Spherical:
𝐿=
𝑑𝑢
𝑡2
𝑑𝜌 2
𝑑𝜃 2
𝑑𝜑 2
2
2
2
√
∫ ( ) + 𝜌 𝑠𝑖𝑛 𝜑 ( ) + 𝜌 ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑡1
𝜅(𝜃) =
3
𝑦 ′2 ) ⁄2
(𝑧 ′′ 𝑦 ′
⃗
𝑑𝑇
𝜅= | |
𝑑𝑠
𝑦 ′′ 𝑧 ′ )2
−
+
′′
′
2
−𝑧 𝑥 ) +
(𝑦 ′′ 𝑥 − 𝑥 ′′ 𝑦 ′ )2
|𝑟 2 + 2𝑟 ′2 − 𝑟𝑟 ′′ |
√(𝑥 ′′ 𝑧 ′
3⁄
2
(𝑟 2 + 𝑟 ′2 )
NA
𝜅=
(𝑥 ′2
+
𝑦 ′2
+
𝜅=
3
𝑧 ′2 ) ⁄2
where f(t) = (x(t), y(t), z(t))
Square: P = 4s
Rectangle: P = 2l + 2w
Triangle : P = a + b + c
s(t) =
NA
𝑡
⃗ ′(𝑢)‖
∫0 ‖𝒓
𝑡1
for r(𝜃)
Perimeter
𝑎
Cylindrical:
𝑑𝑟 2
𝑑𝜃 2
𝑑𝑧 2
𝐿 = ∫ √( ) + 𝑟 2 ( ) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑦 2
) 𝑑𝑥
𝑑𝑥
𝑏
⃗ ′(𝑡)‖ 𝑑𝑡
𝐿 = ∫ ‖𝒓
Proof:
𝐿 = (𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒) ∙
𝜋 ∙ (𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟)
|𝑦 ′′ |
(1 +
𝛽
𝐿 = ∫ √(
𝐿 = ∫ 𝑑𝑠
𝜅=
Rectangular 3D:
C = πd = 2πr
Where 𝑟 = 𝑓(𝜃)
𝑑𝑥 2
)
𝑑𝑥 2
2
𝑑𝑠 =
𝛽
𝜅=
⃗ ′(𝑡)‖
‖𝑇
‖𝑟′(𝑡)‖
(See Wikipedia : Curvature)
‖𝑟 ′ (𝑡) × 𝑟′′(𝑡)‖
‖𝑟′(𝑡)‖3
Circle: C = πd = 2πr
Ellipse: 𝐶 ≈ 𝜋(𝑎 + 𝑏)
Circle: C = 2πr
𝜋
2
Ellipse: 𝐶 = 4𝑎 ∫0 √1 −
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
𝑐 2
( ) 𝑠𝑖𝑛2 𝜃 𝑑𝜃
𝑎
8
NA
NA
NA
Rectangular
Polar/Cylindrical
Square: A = s²
Rectangle: A = lw
Rhombus: A = ½ ab
Parallelogram: A = bh
(𝑏 +𝑏 )
Trapezoid: 𝐴 = 1 2 ℎ
Kite: 𝐴 =
Area
Triangle: 𝐴 =
√𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐),
𝑎+𝑏+𝑐
𝑤ℎ𝑒𝑟𝑒 𝑠 = 2
Parametric
𝛼
where 𝑓(𝑡) = 𝑥 and 𝑔(𝑡) = 𝑦
or
x(t) = f(t) and y(t) = g(t)
𝛼
where 𝑟 = 𝑓(𝜃)
Proof:
Area of a sector:
Simplified :
1
𝐴 = ∫ 𝑠 𝑑𝑟 = ∫ 𝑟 ∆𝜃 𝑑𝑟 = 𝑟 2 ∆𝜃
2
where arc length 𝑠 = 𝑟 ∆𝜃
𝛽
𝐴 = ∫ 𝑦(𝑡)
NA
𝛼
𝐴 = ∬|
𝐷
𝜕𝒓 𝜕𝒓
× | 𝑑𝑢 𝑑𝑣
𝜕𝑢 𝜕𝑣
NA
Proof:
∫ 𝑓(𝑥) 𝑑𝑥
𝑎
1 𝑏 +𝑏
y = f(x) = g(t)
For rotation about the x-axis:
𝑑𝑓(𝑡)
𝑑𝑡 = 𝑓’(𝑡) 𝑑𝑡
𝑑𝑡
For rotation about the x-axis:
𝑆 = ∫ 2𝜋𝑦 𝑑𝑠
𝑆 = ∫ 2𝜋𝑦 𝑑𝑠
For rotation about the y-axis:
For rotation about the y-axis:
𝑑𝑥 =
Cylinder: S = 2πrh
Cone: S = πrl
Sphere: S = 4πr²
𝑆 = ∫ 2𝜋𝑥 𝑑𝑠
𝑏
𝑆 = ∫ 2𝜋𝑥 𝑑𝑠
𝑆 = 2𝜋 ∫ 𝑓(𝑥) √1 + [𝑓 ′ (𝑥)]2 𝑑𝑥
𝑑𝑟 2
2
√
𝑑𝑠 = 𝑟 + ( ) 𝑑𝜃
𝑑𝜃
𝑟 = 𝑓(𝜃),
𝛼≤ 𝜃≤𝛽
𝑎
Cube: S = 6s²
Rectangular Box: S = 2lw + 2wh +
2hl
Regular Tetrahedron: S = 2bh
Cylinder: S = 2πr (r + h)
Cone: S = πr² + πrl = πr (r + l)
Sphere: S = 4πr²
For revolution about the x-axis:
𝑏
Surface of
Revolution
𝑑𝑥(𝑡)
𝑑𝑡
𝑑𝑡
𝑏
Frustum: 𝐴 = ( 1 2 ) ℎ
3
2
Circle: A = πr²
Circular Sector: A = ½ r²𝜃
Ellipse: A = πab
Total
Surface
Area
Matrix
𝐴 = ∫ 𝑔(𝑡) 𝑓 ′ (𝑡) 𝑑𝑡
1
𝐴 = ∫ [𝑓(𝜃)]2 𝑑𝜃
2
Equilateral Triangle: 𝐴 = ¼√3𝑠 2
Lateral
Surface
Area
Vector
𝛽
𝛽
2
𝑑1 𝑑2
2
Triangle: A = ½ bh
Triangle: A = ½ ab sin(C)
Spherical
𝑑𝑦 2
𝐴 = 2𝜋 ∫ 𝑓(𝑥) √1 + ( ) 𝑑𝑥
𝑑𝑥
𝑎
For revolution about the y-axis:
𝑏
2
𝑑𝑥
𝐴 = 2𝜋 ∫ 𝑥 √1 + ( ) 𝑑𝑦
𝑑𝑦
𝑎
𝑑𝑥 2
𝑑𝑦 2
) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑖𝑓 𝑥 = 𝑓(𝑡), 𝑦 = 𝑔(𝑡), 𝛼 ≤ 𝑡 ≤ 𝛽
𝑑𝑠 = √(
Ellipsoid: S ≈
1⁄
𝑎 𝑏 + 𝑎𝑝 𝑐 𝑝 + 𝑏 𝑝 𝑐 𝑝 𝑝
4𝜋 (
)
3
Where p ≈ 1.6075, |𝐸| ≤ 1.061%
(Knud Thomsen’s Formula)
𝑝 𝑝
Ellipsoid: S =
where
For revolution about the x-axis:
For revolution about the x-axis:
𝛽
𝑏
𝑑𝑟 2
2
√
𝐴 = 2𝜋 ∫ 𝑟 𝑐𝑜𝑠 𝜃 𝑟 + ( ) 𝑑𝜃
𝑑𝜃
𝐴𝑥 = 2𝜋 ∫ 𝑦(𝑡) √(
𝛼
Sphere: S = 4πr²
For revolution about the y-axis:
𝛽
For revolution about the y-axis:
𝑏
2
𝑑𝑥 2
𝑑𝑦 2
√
𝐴𝑦 = 2𝜋 ∫ 𝑥(𝑡) ( ) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑟
𝐴 = 2𝜋 ∫ 𝑟 𝑠𝑖𝑛 𝜃 √𝑟 2 + ( ) 𝑑𝜃
𝑑𝜃
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
𝑎
𝑑𝑥 2
𝑑𝑦 2
) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝛼
𝑎
9
NA
NA
Rectangular
Volume
Cube: V = s³
Rectangular Prism: V = lwh
Cylinder: V = πr²h
Triangular Prism: V= Bh
Tetrahedron: V= ⅓ bh
Pyramid: V = ⅓ Bh
4
Sphere: 𝑉 = 3 𝜋𝑟 3
4
Polar/Cylindrical
∫∫∫
𝑓(𝑟 𝑐𝑜𝑠 𝜃 , 𝑟 𝑠𝑖𝑛 𝜃, 𝑧)𝑟
𝑑𝑧 𝑑𝑟 𝑑𝜃
Ellipsoid: V = 3 πabc
Cone: V = ⅓ bh = ⅓ πr²h
Spherical
Parametric
Vector
𝜌 𝑠𝑖𝑛 𝜑 𝑐𝑜𝑠 𝜃 ,
∫ ∫ ∫ 𝑓 (𝜌 𝑠𝑖𝑛 𝜑 sin 𝜃 ,)
𝜌 𝑐𝑜𝑠 𝜑
… 𝜌2 𝑠𝑖𝑛 𝜑 𝑑𝜌 𝑑𝜑 𝑑𝜃
Matrix
Ellipsoid:
4
𝑉 = 𝜋√𝑑𝑒𝑡(𝐴−1 )
3
∫ ∫ ∫ 𝑓(𝑥, 𝑦, 𝑧) 𝑑𝑥 𝑑𝑦 𝑑𝑧
Disc Method - Rotation about the
x-axis:
𝑏
𝑉 = ∫𝜋
[𝑓(𝑥)]2
Cylindrical Shell Method:
Disc Method:
𝑑𝑥
𝑎
Washer Method - Rotation about
the x-axis:
𝑏
Volume of
Revolution
𝑉 = ∫ 𝜋 { [𝑓(𝑥)]2 − [𝑔(𝑥)]2 } 𝑑𝑥
𝑎
Cylinder Method - Rotation about
the y-axis:
𝑏
𝑉 = ∫ 2𝜋𝑥 𝑓(𝑥) 𝑑𝑥
𝑎
𝑏
= ∫(𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒) (ℎ𝑖𝑔ℎ𝑡) 𝑑𝑥
𝑎
𝑁
Moments
of Inertia
𝐼=
∑ 𝑚𝑖 𝑟𝑖2
𝑖=1
= ∫ 𝑚 𝑟 2 𝑑𝑟
NA
0
𝑁
1
𝑹=
∑ 𝑚𝑖 𝑟𝑖
𝑀
Center of
Mass
𝐼
𝑎
𝑖=1
where 𝑀 =
𝑁
𝑀𝑦 = ∑ 𝑚𝑖 𝑥𝑖
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
𝑖=1
𝑁
𝑀𝑥 = ∑ 𝑚𝑖 𝑦𝑖
𝑁
𝑥𝑐𝑚
1
= 𝑥̅ =
∑ 𝑚𝑖 𝑥𝑖
𝑀
𝑖=1
𝑁
𝑦𝑐𝑚 = 𝑦̅ =
𝑀𝑦
𝑀𝑥
, 𝑦̅ =
𝑀
𝑀
1
∑ 𝑚𝑖 𝑦𝑖
𝑀
𝑖=1
𝑁
𝑖=1
𝑥̅ =
= ∭ 𝜌(𝒓) 𝑑(𝒓)2 𝑑𝑉(𝒓)
3-D for Discrete:
2-D for Discrete:
∑𝑁
𝑖=1 𝑚𝑖
1-D for Discrete:
𝑚1 𝑥1 + 𝑚2 𝑥2
𝑥𝑐𝑚 =
𝑚1 + 𝑚2
NA
𝑧𝑐𝑚 = 𝑧̅ =
1
∑ 𝑚𝑖 𝑧𝑖
𝑀
𝑖=1
10
3-D for Continuous:
1 𝑀
𝑥̅ =
∫ 𝑥 𝑑𝑚
𝑀 0
1 𝑀
𝑦̅ =
∫ 𝑦 𝑑𝑚
𝑀 0
1 𝑀
𝑧̅ =
∫ 𝑧 𝑑𝑚
𝑀 0
𝑀
where 𝑀 = ∫0 𝑑𝑚
and 𝑑𝑚 = 𝜌 𝑑𝑧 𝑑𝑦 𝑑𝑥
𝑉
𝑹=
𝑹=
1
∫ 𝒓 𝑑𝑚
𝑀
1
∭ 𝜌(𝒓) 𝒓 𝑑𝑉
𝑀
𝑉
Where 𝒓 is distance
from the axis of
rotation, not origin.
(see Wikipedia)
Rectangular
Polar/Cylindrical
Spherical
Parametric
Vector
(∇ ƒ(x)) • 𝒗 = 𝐷𝒗 𝑓(𝑥)
𝜕𝑓
𝜕𝑓
𝜕𝑓
∇ƒ=
𝒊+
𝒋+
𝒌
𝜕𝑥
𝜕𝑦
𝜕𝑧
Gradient
Line
Integral
𝜕𝑓
1 𝜕𝑓
𝒆ρ +
𝒆
𝜕ρ
ρ 𝜕ϕ ϕ
𝜕𝑓
+
𝒆
𝜕𝑧 z
∇ ƒ(r, θ, ϕ) =
𝜕𝑓
1 𝜕𝑓
𝟏 𝜕𝑓
𝒆r +
𝒆θ +
𝒆
𝜕r
𝑟 𝜕θ
𝑟 sin θ 𝜕ϕ ϕ
𝜕𝑓𝑖
𝒆𝒆
𝜕𝑥𝑗 i j
where ƒ = (ƒ𝟏 , ƒ𝟐 , ƒ𝟑 )
∇ƒ=
∫ 𝑭(𝑟) • 𝑑𝑟
𝑏
′
∫ 𝑓 𝑑𝑠 = ∫ 𝑓(𝒓(𝑡))|𝒓 (𝑡)| 𝑑𝑡
𝐶
∇ ƒ(ρ, ϕ, z) =
NA
NA
𝐶
𝑏
= ∫ 𝑭(𝒓(𝑡)) • 𝒓′ (𝑡) 𝑑𝑡
𝑎
𝑎
∫ 𝑓 𝑑𝑆 =
𝑆
∬ 𝑓(𝒙(𝑠, 𝑡)) |
𝑇
Surface
Integral
𝜕𝒙
𝜕𝒙
×
| 𝑑𝑠 𝑑𝑡
𝜕s
𝜕t
where
𝒙(𝑠, 𝑡) =
(𝑥(𝑠, 𝑡), 𝑦(𝑠, 𝑡), 𝑧(𝑠, 𝑡))
and
𝜕𝒙
𝜕𝒙
(
×
)=
𝜕s
𝜕t
∫ 𝒗 • 𝑑𝑺 =
𝑆
∫ (𝒗 • 𝒏) 𝑑𝑺 =
NA
NA
∬ 𝒗(𝒙(𝑠, 𝑡))
𝑇
𝜕𝒙
𝜕𝒙
•(
×
) 𝑑𝑠 𝑑𝑡
𝜕s
𝜕t
𝜕(𝑦, 𝑧) 𝜕(𝑧, 𝑥) 𝜕(𝑥, 𝑦)
(
,
,
)
𝜕(𝑠, 𝑡) 𝜕(𝑠, 𝑡) 𝜕(𝑠, 𝑡)
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
𝑺
11
Matrix