Harold’s Calculus 3Multi-Cordinate System
“Cheat Sheet”15 October 2017
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Point
2-Df ( x )= y
( x , y )(a ,b )
3-Df ( x , y )=z
( x , y , z )
4-Df ( x , y , z )=w
( x , y , z ,w )
•
(r , θ) or r∠θx=r cosθy=r sinθ z=z
r2=x2+ y2
r=±√x2+ y2
tanθ=( yx )
θ=tan−1( yx )
(ρ ,θ ,ϕ)
x=ρ sinϕ cosθy= ρsin ϕsin θz=ρ cos ϕ
ρ2=r2+z2
ρ2=x2+ y2+z2
tanθ=( yx )ϕ=cos−1( z
√ x2+ y2+z2 )ϕ=cos−1( zρ )
Point (a,b) in Rectangular :x (t )=ay ( t )=b
t=3rd variable ,usually time,with 1 degree of freedom (df)
r⃗=⟨x0 , y0 , z0 ⟩ [a ] [x ]=[b ]
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 1
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Line
Slope-Intercept Form:y=mx+b
Point-Slope Form:y− y0=m(x−x0)
General Form:Ax+By+C=0where A∧B≠0
Calculus Form:f ( x )=f ' (a ) x+ f (0)where m=f ’(a)
3-D:x−x0a
=y− y0b
=z−z0c
¿ x , y>¿< x0 , y0>+t<a ,b>¿¿ x , y>¿< x0+at , y0+bt>¿
where¿a ,b>¿<x2−x1 , y2− y1>¿
x (t)=x0+tay (t )= y0+tb
m=∆ y∆ x
=y2− y1x2−x1
= ba
r⃗=r⃗0+t v⃗¿ ⟨ x0, y0 , z0 ⟩+t ⟨a ,b , c ⟩
[a b ] [ xy ]= [c ]
[a bc d ][ xy ]=[ef ]
Plane
a (x−x0 )+b ( y− y0 )
+c ( z−z0 )=0
ax+by+cz=dwhere d=ax0+b y 0+c z0
f ( x , y )=Ax+By+C
( variable , constant , constant )where ρ takesonallvalues∈the domain
(0≤ ρ<∞)
r=r0+s v+t w
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
n ∙ (r−r 0 )=0
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 2
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Conics
General Equation for All Conics:
A x2+Bxy+C y2+Dx+Ey+F=0
whereLine : A=B=C=0:̊ A=C∧B=0Ellipse : AC>0¿B2−4 AC<0Parabola : AC=0 or B2−4 AC=0Hyperbola : AC<0¿B2−4 AC>0Note: If A+C=0, square hyperbola
Rotation:If B ≠ 0, then rotate coordinate
system:
cot 2θ= A−CB
x=x ' cosθ− y ' sin θy= y ' cosθ+x ' sinθ
New = (x’, y’), Old = (x, y)rotates through angle θ from x-axis
General Equation for All Conics:
Vertical Axis of Symmetry:
r= p1−ecosθ
Horizontal Axis of Symmetry:
r= p1−esin θ
where p={a (1−e2 )2d
a (e2−1 )for {0≤e<1e=1
e>1
p = semi-latus rectumor the line segment running from the
focus to the curve in a direction parallel to the directrix
Eccentricity:e̊=0
Ellipse0≤e<1Parabolae=1Hyperbolae>1
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 3
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Circle
x2+ y2=r2
(x−h)2+( y−k )2=r2
General Form:A x2+Bxy+C y2+Dx+Ey+F=0
where A=C∧B=0
Focus and Center:(h , k )
(h , k )
Centered at Origin:r = a (constant)
θ=θ [0 ,2 π ]∨[0 ,360 ° ]
Centered at (r0 , ϕ ):r2+r0
2−2 rr 0cos (θ−ϕ)=R2
Hint: Law of Cosinesor
r=r0 cos (θ−ϕ )+√a2−r 02 sin2(θ−ϕ)
ρ=constantθ=θ[0 ,2π ]
ϕ=constant=0
x (t)=r cos(t )+hy (t )=r sin(t)+k
[ tmin , tmax ]=¿
(h , k )=center of ¿̊
Sphere
x2+ y2+z2=r2
(x−h)2+( y−k )2+(z−l)2=r2
Focus∧center :(h , k , l)
General Form:A x2+B y2+C z2
+D xy+E yz+Fxz+G x+Hy+ Iz+J=0where A=B=C > 0
Cylindrical to Rectangular:x=r cos (θ)y=rsin (θ)
z=z
Spherical to Rectangular:x=r sin θ cos ϕy=rsin θ sin ϕz=r cosθ
Rectangular to Cylindrical:
r=√x2+ y2
Spherical to Cylindrical:ρ=r sin (θ)
ϕ=ϕz=r cos (θ)
ρ=constantθ=θ[0 ,2π ]ϕ=ϕ [0 ,2 π ]
Rectangular to Spherical:
r=√x2+ y2+z2
θ=arccos( zr )ϕ=arctan( yx )
Cylindrical to Spherical:
r=√ ρ2+z2
θ=arctan( ρz )=arccos( zr )ϕ=ϕ
Rectangular:
r ≡[ xyz ]Cylindrical:
r ≡ [rcos (θ)r sin (θ)z ]
Spherical:
r ≡[r sin θ cos ϕr sinθ sin ϕr cosθ ]
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 4
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Ellipse
(x−h)2
a2+
( y−k)2
b2=1
General Form:A x2+Bxy+C y2+Dx+Ey
+F=0where B2−4 AC<0∨AC>0
Center: (h , k )Vertices: (h±a , k )∧ (h , k±b )
Foci: (h±c ,k )
Focus length, c, from center:
c=√a2−b2
Eccentricity:
e= ca=√a2−b2
a
If B ≠ 0, then rotate coordinate system:
cot 2θ= A−CB
x=x ' cosθ− y ' sinθy= y ' cosθ+x ' sin θ
New = (x’, y’), Old = (x, y)rotates through angle θ from x-
axis
Vertical Axis of Symmetry:
r= a(1−e2)1±ecosθ
Horizontal Axis of Symmetry:
r= a (1−e2)1±e sin θ
Eccentricity : 0<e<1
r (θ )= ab
√(bcosθ)2+(a sin θ)2
relative to center (h,k)
Interesting Note:The sum of the distances from each
focus to a point on the curve is constant.
|d1+d2|=k
x (t)=acos( t)+hy (t )=b sin(t ¿)+k ¿[ tmin , tmax ]=[0 ,2 π ]
(h , k )=center of ellipse
Rotated Ellipse:x (t )=acos t cosθ−b sin t sin θ+hy ( t )=acos t sinθ+b sin t cosθ+k
θ = the angle between the x-axis and the major axis of the ellipse
Ellipsoid(x−h)2
a2+
( y−k)2
b2+
(z−l)2
c2=1 r2cos2θ
a2+ r2 sin2θ
b2+ z2
c2=1
r2cos2θ sin2ϕa2
+r2sin 2θ sin2ϕb2
+r2 cos2ϕc2
=1
x (t , u)=acos(t)cos (u)+hy (t ,u)=bcos ( t )sin (u¿)+k ¿
z (t , u )=c sin(t )+l
[ tmin , tmax ]=[−π2
, π2 ]
[umin ,umax ]=[−π , π ]
(h , k , l )=center of ellipsoid
( x−v )T A−1 ( x−v )=1Centered at vector v
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 5
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Parabola
Vertical Axis of Symmetry:x2=4 py
( x−h )2=4 p ( y−k )Vertex: (h , k )
Focus: (h , k+ p )Directrix: y=k−p
Horizontal Axis of Symmetry:y2=4 px
( y−k )2=4 p(x−h)Vertex: (h , k )
Focus: (h+ p , k )Directrix: x=h−p
General Form:A x2+Bxy+C y2+Dx+Ey+F=0
where B2−4 AC=0or AC=0
If B ≠ 0, then rotate coordinate system:
cot 2θ= A−CB
x=x ' cosθ− y ' sinθy= y ' cosθ+x ' sin θ
New = (x’, y’), Old = (x, y)rotates through angle θ from x-axis
Vertical Axis of Symmetry:
r= ed1±ecosθ
Horizontal Axis of Symmetry:
r= ed1±e sin θ
Eccentricity : e=1where d=2 p
Vertical axis of symmetry:x (t )=2 pt+h
y ( t )=p t 2+k (opens upwards) or
y ( t )=−p t2+k (opens downwards)
[ tmin , tmax ]=[−c , c ](h, k) = vertex of parabola
Horizontal axis of symmetry:x (t)=p t 2+h
y (t )=2 pt+k (opens right) ory (t )=−2 pt+k (opens left)
[ tmin , tmax ]=[−c , c ]
Projectile Motion:x (t )=x0+vx t
y (t )= y0+v y t−16 t2 feet
y ( t )= y0+v y t−4.9 t2 meters
vx=vcosθv y=v sin θ
General Form:x=A t 2+Bt+Cy=L t 2+Mt+N
where A and L have the same sign
Nose Cone(x−h)2
a2+
( y−k)2
b2=
(z−l)2
c2
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 6
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Hyperbola
( x−h )2
a2− ( y−k )2
b2=1
General Form:A x2+Bxy+C y2+Dx+Ey+F=0
where B2−4 AC>0∨AC<0
If A+C=0, square hyperbola
Center: (h , k )Vertices: (h±a ,k )
Foci: (h±c ,k )
Focus length, c, from center:
c=√a2+b2
Eccentricity:
e= ca=√a2+b2
a=sec θ
3-D( x−h )2
a2+
( y−k )2
b2−
(z−l )2
c2=1
−( x−h )2
a2− ( y−k )2
b2+ (z−l )2
c2=1
If B ≠ 0, then rotate coordinate system:
cot 2θ= A−CB
x=x ' cosθ− y ' sinθy= y ' cosθ+x ' sin θ
New = (x’, y’), Old = (x, y)rotates through angle θ from x-
axis
Vertical Axis of Symmetry:
r= a(e2−1)1±ecosθ
Horizontal Axis of Symmetry:
r= a (e2−1)1±e sin θ
Eccentricity : e>1
where e= ca=√a2+b2
a=secθ>1
relative to center (h,k)
−cos−1(−1e )<θ<cos−1(−1e )
p = semi-latus rectum or the line segment running from the
focus to the curve in the directions
θ=± π2
Interesting Note:The difference between the distances
from each focus to a point on the curve is constant.
|d1−d2|=k
Left-Right Opening Hyperbola:x (t)=asec(t )+hy (t )=b tan(t)+k
[ tmin , tmax ]=[−c , c ](h, k) = vertex of hyperbola
Alternate Form:x (t)=±acosh ( t)+hy (t )=b sinh(t )+k
Up-Down Opening Hyperbola:x (t)=a tan( t)+hy (t )=bsec (t)+k
Alternate Form:x (t)=a sinh(t)+hy (t )=±b cosh (t)+k
General Form:x (t)=A t 2+Bt +Cy (t )=D t2+Et+F
where A and D have different signs
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 7
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Limit limx→c
f (x )=L
1st
Derivative
f ' ( x )= limh→0
f ( x+h )−f (x)h
f ' (c )=limx→c
f ( x )−f (c )x−c
f ' ( x )=dydx
= y '=D x
dydx
=
dydθdxdθ
=
drdθsin θ+rcos θ
drdθcosθ−r sin θ
Hint: Use Product Rule fory=rsinθx=r cosθ
dydx
=
dydtdxdt
, provided dxdt
≠0
ddt
( r⃗ )=r⃗ '
Unit tangent vector
T⃗ ( t )= r⃗ ' (t)‖r⃗ ' (t)‖
where r⃗ ' (t )≠ 0⃗
2nd
Derivativef ' ' ( x )= d
dx ( dydx )=d2 yd x2
= y ' ' d2 yd x2
= ddx ( dydx )=
ddθ ( dydx )
dxdθ
d2 yd x2
= ddx ( dydx )=
ddt ( dydx )dxdt
Unit normal vector
N⃗ (t )= T⃗ ' (t)‖T⃗ ' (t)‖
whereT⃗ ' (t )≠ 0⃗
Integral F ( x )=∫a
b
f ( x )dx=F (b )−F (a)
Riemann Sum:
S=∑i−1
n
f ( y i)(x i−x i−1)
Left Sum:
S=( 1n ) [f (a )+ f (a+ 1n )+¿ f (a+ 2n )+…+ f (b−1n)]
Middle Sum:
S=( 1n ) [ f (a+ 12n )+ f (a+ 32n )+…+ f (b− 1
2n)]
Right Sum:
S=( 1n ) [f (a+ 1n )+f (a+ 2n )+…+ f (b)]
∫a
b
r⃗ (t )dt=⟨∫ab
f (t )dt ,∫a
b
g (t )dt ,∫a
b
h (t )dt ⟩
Double Integral ∫
a
b
∫c( y)
d ( y)
f ( x , y )dx dySame asrectangular , but
f ( x , y )⟶ f (ρ cosϕ , ρsin ϕ)
Triple Integral ∫
a
b
∫c(z )
d ( z)
∫e( y, z )
g( y , z)
f ( x , y , z )dx dy dz Same asrectangular , butf ( x , y , z )⟶ f (ρ cos ϕ, ρ sin ϕ, z )
Same asrectangular ,butf ( x , y , z )⟶ f ¿
ρ sin θ sinϕ ,ρ cos ϕ¿NA NA NA
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 8
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Inverse Functions
If f ( x )= y , then f−1( y )=x
Inverse Function Theorem:
f−1 (b )= 1f ' (a )
where b=f ' (a )
if y=sinθif y=cosθif y=tan θ
if y=cscθif y=sec θif y=cot θ
thenθ=sin−1 ythen θ=cos−1 ythenθ= tan−1 y
thenθ=csc−1 ythenθ=sec−1 ythenθ=cot−1 y
θ=arcsin yθ=arccos yθ=arctan y
θ=arccsc yθ=arcsec yθ=arccot y
NA NA NA
Arc Length
L=∫a
b
√1+[ f ' ( x )]2dx
Proof:
∆ s=√ (x−x0 )2+ ( y− y0 )2
∆ s=√(∆x )2+¿¿ds=√dx2+d y2
ds=√dx2+d y2( d x2d x2 )ds=√dx2+( dydx )
2
d x2
ds=√dx2(1+( dydx )2)
ds=√1+( dydx )2
dx
L=∫ ds
Polar:
L=∫√r2+( drdθ )2
dθ
Where r=f (θ)
Circle:L=s=r θ
Proof:L=( fraction of circumference ) ∙
π ∙ (diameter)
L=( θ2π ) π (2 r)=r θ
C = πd = 2πr
Rectangular 2D:
L=∫α
β
√( dxdt )2+( dydt )2
dt
Rectangular 3D:
L=∫α
β
√( dxdt )2+( dydt )2
+(dzdt )2
dt
Cylindrical:
L=∫t1
t2 √( drdt )2+r2( dθdt )2+( dzdt )2
dt
Spherical:L=¿
∫t1
t2 √( dρdt )2+ ρ2sin2φ ( dθdt )2
+ρ2( dφdt )2
dt
L=∫a
b
‖r⃗ ' (t)‖dt
s(t)¿∫0
t
‖r⃗ ' (u)‖du
NA
Curvatureκ=
|y ' '|
(1+ y ' 2)32
κ (θ)=|r2+2 r ' 2−r r ' '|
(r2+r '2)32
for r(θ)
NA
κ=√ ( z ' ' y '− y ' ' z' )2+¿ (x ' ' z '−z ' ' x ' )2+¿( y' ' x−x ' ' y ' )2
(x ' 2+ y' 2+z '2)32
where f(t) = (x(t), y(t), z(t))
κ=|d T⃗ds |κ=
‖T⃗ ' (t)‖‖r⃗ ' (t )‖
κ=‖r⃗ ' ( t )× r⃗ ' ' (t)‖
‖r⃗ ' (t)‖3
(See Wikipedia : Curvature)
PerimeterSquare: P = 4s
Rectangle: P = 2l + 2wTriangle : P = a + b + c
Circle: C = πd = 2πrEllipse: C≈ π (a+b )
Ellipse: C=4 a∫0
π2
√1−( ca )2
sin 2θd θ
Circle: C = 2πr NA NA NA
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 9
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Area
Square: A = s²Rectangle: A = lw
Rhombus: A = ½ abParallelogram: A = bh
Trapezoid: A=(b1+b2 )2
h
Kite: A=d1d22
Triangle: A = ½ bhTriangle: A = ½ ab sin(C)
Triangle:
A=√s ( s−a ) (s−b )(s−c) ,where s=a+b+c2
Equilateral Triangle: A=¼√3 s2
Frustum: A=13 ( b1+b22 )h
Circle: A = πr²Circular Sector: A = ½ r²θ
Ellipse: A = πab
A=∫α
β 12[ f (θ)]2dθ
where r=f (θ)
Proof:Area of a sector:
A=∫ s dr=¿∫r ∆θdr=12r2∆θ ¿
where arc length s=r ∆θ NA
A=∫α
β
g ( t ) f ' ( t ) dt
where f (t )=x and g (t )= yor
x(t) = f(t) and y(t) = g(t)
Simplified:
A=∫α
β
y (t ) dx (t )dt
dt
Proof:
∫a
b
f ( x )dx
y = f(x) = g(t)
dx=df ( t )dt
dt=f ’ (t )dt
A=∬D
|∂r∂u × ∂r∂ v|dudv NA
Lateral Surface
Area
Cylinder: S = 2 rhπCone: S = rlπ
S=2π∫a
b
f ( x ) √1+[ f ' ( x )]2dx
For rotation about the x-axis:S=∫ 2πy ds
For rotation about the y-axis:S=∫ 2πx ds
ds=√r2+( drdθ )2
dθ
r=f (θ ) , α ≤ θ≤β
Sphere: S = 4πr²
For rotation about the x-axis:S=∫ 2πy ds
For rotation about the y-axis:S=∫2πx ds
ds=√( dxdt )2
+( dydt )2
dt
if x=f (t ) , y=g (t ) , α ≤ t ≤ β
Total Surface
Area
Cube: S = 6s²Rectangular Box: S = 2lw + 2wh +
2hlRegular Tetrahedron: S = 2bh
Cylinder: S = 2πr (r + h)Cone: S = πr² + πrl = πr (r + l)
Sphere: S = 4πr²
Ellipsoid: S ≈
4 π ( apbp+a pc p+b p cp
3 )1p
Where p ≈1.6075 ,|E|≤1.061%(Knud Thomsen’s Formula)
Ellipsoid: S =
where
Surface of Revolution
For revolution about the x-axis:
A=2π∫a
b
f (x)√1+( dydx )2
dx
For revolution about the y-axis:
A=2π∫a
b
x√1+( dxdy )2
dy
For revolution about the x-axis:
A=2π∫α
β
r cosθ√r2+( drdθ )2
d θ
For revolution about the y-axis:
A=2π∫α
β
r sin θ√r2+( drdθ )2
dθ
Sphere: S = 4πr²
For revolution about the x-axis:
Ax=2π∫a
b
y ( t ) √( dxdt )2
+( dydt )2
dt
For revolution about the y-axis:
A y=2π∫a
b
x (t ) √( dxdt )2
+( dydt )2
dt
NA NA
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 10
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Volume
Cube: V = s³Rectangular Prism: V = lwh
Cylinder: V = πr²hTriangular Prism: V= Bh
Tetrahedron: V= ⅓ bhPyramid: V = ⅓ Bh
Sphere: V= 43π r3
Ellipsoid: V = 43πabc
Cone: V = ⅓ bh = ⅓ πr²h
∫∫∫ f (x , y , z )dx dydz
∫∫∫ f (r cosθ , rsin θ , z ) rdz dr dθ
∫∫∫ f (ρ sinφ cosθ ,ρsinφ sin θ ,ρcos φ )
…ρ2 sinφdρ dφdθ
Ellipsoid:
V= 43π √det ( A−1 )
Volume of Revolution
Disc Method - Rotation about the x-axis:
V=∫a
b
π [ f ( x ) ]2dx
Washer Method - Rotation about the x-axis:
V=∫a
b
π ¿¿¿
Cylinder Method - Rotation about the y-axis:
V=∫a
b
2 πx f (x ) dx
¿∫a
b
(circumference) (hight )dx
Disc Method:Cylindrical Shell Method:
Moments of Inertia
I=∑i=1
N
mi r i2=∫
0
a
mr2dr NA NA I=∭V
ρ (r )d (r )2dV (r) (see Wikipedia)
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 11
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Center of Mass
R= 1M ∑
i=1
N
mi ri
where M=∑i=1
N
mi
1-D for Discrete:
xcm=m1 x1+m2 x2m1+m2
2-D for Discrete:
M y=∑i=1
N
mi xi
M x=∑i=1
N
mi y i
x=M y
M, y=
M x
M
3-D for Discrete:
xcm=x= 1M ∑
i=1
N
mi x i
ycm= y= 1M ∑
i=1
N
mi yi
zcm=z= 1M ∑
i=1
N
mi zi
3-D for Continuous:
x= 1M∫
0
M
xdm
y= 1M∫
0
M
y dm
z= 1M∫
0
M
zdm
where M=∫0
M
dm
and dm=ρdz dy dx
R= 1M∫r dm
R= 1M∭
Vρ (r ) r dV
Where r is distance from the axis of
rotation, not origin.
Gradient ∇ƒ= ∂ f∂ x
i+ ∂ f∂ y
j+ ∂ f∂ z
k ∇ƒ (ρ ,ϕ , z )= ∂ f∂ ρ
eρ+1ρ∂ f∂ϕ
eϕ+∂ f∂ z
ez
∇ƒ (r , θ ,ϕ )=¿
∂ f∂ r
er+1r∂ f∂θ
eθ+1
r sinθ∂ f∂ϕ
eϕ
(∇ƒ (x ) ) • v=Dv f (x)
∇ƒ=∂ f i
∂ x je i e j
where ƒ=(ƒ1 , ƒ2 , ƒ3)
Line Integral ∫
Cf ds=∫
a
b
f (r ( t ) )|r ' (t)|dt NA NA ∫CF (r )• dr=∫
a
b
F (r ( t ) ) •r ' (t )dt
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 12
Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix
Surface Integral
∫Sf dS=¿
∬Tf ( x (s ,t ) )|∂x∂ s × ∂ x
∂ t |ds dtwhere
x (s , t )=¿(x (s ,t ) , y (s , t ) , z ( s , t ) )
and
( ∂ x∂ s × ∂x∂ t )=¿
( ∂( y , z)∂(s ,t ), ∂(z , x )∂(s , t)
, ∂(x , y )∂(s , t) )
NA NA
∫Sv • d S=¿
∫S
( v • n )d S=¿
∬Tv (x ( s , t ) )•( ∂ x∂ s × ∂x
∂ t )ds dt
Copyright © 2011-2017 by Harold Toomey, WyzAnt Tutor 13
