2-dimensional integral substitutions

DescriptionTransformInverse transformOther properties /
Additional notes
JacobianCommonly apply to
Polar
ρ: distance to the origin
θ: polar angle
x=ρcos(θ)
y=ρsin(θ)
ρ=x2+y2
θ=atan2(y,x)
x2+y2=ρ2
ρ0
(x,y)(ρ,θ)=ρ Circular integral regions
Planar regions described by implicit algebraic equations
Polar
*Integrals of symmetrical functions in circular regions
r=(x,y)
r=|r|=x2+y2
f(|r|) dr=2πrf(r)dr Circular regions centered at the origin
Infinite 2D plane
Generalized polar
a,b: scaling on x and y-axis
x=aρcos(θ)
y=bρsin(θ)
ρ=x2a2+y2b2
θ=atan2(ay,bx)
x2a2+y2b2=ρ2
ρ0
(x,y)(ρ,θ)=abρ Elliptical integral regions
Isosceles right triangle formed by axises
u: distance to origin (Manhattan)
v: linear interpolation coefficient from x-axis to y-axis
x=u(1v)
y=uv
u=x+y
v=yx+y
Usually, u0 and 0v1. (x,y)(u,v)=u
(u,v)(x,y)=1x+y
Region in the first quadrant defined by x+ya
Triangle defined by origin and two vectors a⃗ ,b⃗ p⃗ =u(1v)a⃗ +uvb⃗ [xy]=[a⃗ b⃗ ]1[xy]
u=x+y, v=yx+y
0u1, 0v1(x,y)(u,v)=|a⃗ ×b⃗ |u
Triangular integral regions
Green theorem for regions inside polygons
Matrix [xy]=A[uv][uv]=A1[xy](x,y)(u,v)=det(A)
Parallelogram / Triangle

3-dimensional integral substitutions

DescriptionTransformInverse transformOther properties /
Additional notes
JacobianCommonly apply to
Cylindrical
ρ: distance to the origin
θ: polar angle
h: height
x=ρcos(θ)
y=ρsin(θ)
z=h
ρ=x2+y2
θ=atan2(y,x)
h=z
x2+y2=ρ2
ρ0
(x,y,z)(ρ,θ,h)=ρ Integral volumes with circular bases
Generalized cylindrical
a,b: scaling on x and y-axis
x=aρcos(θ)
y=bρsin(θ)
z=h
ρ=x2a2+y2b2
θ=atan2(ay,bx)
h=z
x2a2+y2b2=ρ2
ρ0
(x,y,z)(ρ,θ,h)=abρ Integral volumes with elliptical bases
Spherical (standard)
ρ: distance to the origin
θ: longitude
φ: angle to the vertical axis
x=ρsin(φ)cos(θ)
y=ρsin(φ)sin(θ)
z=ρcos(φ)
ρ=x2+y2+z2
θ=atan2(y,x)
φ=atan2(x2+y2,z)
x2+y2+z2=ρ2
x2+y2=ρ2sin2(φ)
ρ0, 0φπ
(x,y,z)(ρ,θ,φ)=ρ2sin(φ) Spherical (ρ) and conical (φ) volumes
Volumes defined by implicit algebraic equations
Spherical (latitude)
ρ: distance to the origin
θ,φ: longitude and latitude
x=ρcos(φ)cos(θ)
y=ρcos(φ)sin(θ)
z=ρsin(φ)
ρ=x2+y2+z2
θ=atan2(y,x)
φ=atan2(z,x2+y2)
x2+y2+z2=ρ2
x2+y2=ρ2cos2(φ)
ρ0, π2φπ2
(x,y,z)(ρ,θ,φ)=ρ2cos(φ) Spheres
Spherical
*Integrals of symmetrical functions in spherical regions
r=(x,y,z)
r=|r|=x2+y2+z2
f(|r|) dr=4πr2f(r)dr Spherical regions centered at the origin
Infinite three-dimensional space
Generalized spherical
a,b,c: scaling on x, y, and z-axis
x=aρsin(φ)cos(θ)
y=bρsin(φ)sin(θ)
z=cρcos(φ)
ρ=x2a2+y2b2+z2c2
θ=atan2(ay,bx)
φ=atan2(x2a2+y2b2,zc)
x2a2+y2b2+z2c2=ρ2
x2a2+y2b2=ρ2sin2(φ)
ρ0, 0φπ
(x,y,z)(ρ,θ,φ)=abcρ2sin(φ) Ellipsoids
Conical
ρh: radius at z=h
x=ρhcos(θ)
y=ρhsin(θ)
z=h
ρ=1zx2+y2
θ=atan2(y,x)
h=z
x2+y2=ρ2h2
0ρrh1 for integral in a cone with height h1 and base radius r
(x,y,z)(ρ,θ,h)=ρh2 Cones
Generalized conical
a,b,c are scaling factors on x, y, z axis
x=aρhcos(θ)
y=bρhsin(θ)
z=ch
ρ=czx2a2+y2b2
θ=atan2(ay,bx)
h=zc
x2a2+y2b2=ρ2h2
0ρ1, 0h1
(x,y,z)(ρ,θ,h)=abcρh2 Elliptic cones with base radius a,b and height c
Toric
u: longitude
v: latitude
ρ: radius of small circle
R: radius of large circle
x=cos(u)(R+ρcosv)
y=sin(u)(R+ρcosv)
z=ρsin(v)
ρ=(x2+y2R)2+z2
u=atan2(y,x)
v=atan2(z,x2+y2R)
x2+y2=(R+ρcosv)2
0u,v2π, ρ0
(x,y,z)(ρ,u,v)=ρ(R+ρcosv) Toruses
Tetrahedron formed by coordinate planes and a plane with n⃗ =(1,1,1)
u[0,1]: xOy to z-axis
v[0,1]: xOz to yOz
w: x + y + z
x=u(1v)w
y=uvw
z=(1u)w
u=x+yx+y+z
v=yx+y
w=x+y+z
x+y=uw

Usually, w0, 0u1, 0v1.
(x,y,z)(u,v,w)=uw2 Volume in the first octant defined by x+y+za
Tetrahedron defined by origin and three vectors a⃗ ,b⃗ ,c⃗ p⃗ =u(1v)wa⃗ +
uvwb⃗ +(1u)wc⃗ 
Transform p⃗  using [a⃗ ,b⃗ ,c⃗ ]1 and apply the formula in the above cell 0u1
0v1
0w1
(x,y,z)(u,v,w)=det[a⃗ ,b⃗ ,c⃗ ]uw2 Gauss divergence theorem for integrals inside polyhedrons

3-dimensional area elements

DescriptionParametric equationDirectional derivativesNormal vectorArea element
Triangle defined by point p and vectors a⃗ ,b⃗ p+u(1v)a⃗ +uvb⃗ 
p+ua⃗ +uv(b⃗ a⃗ )
0<u,v<1
Pu=(1v)a⃗ +vb⃗ 
Pv=u(b⃗ a⃗ )
n=(a⃗ ×b⃗ )udA=|a⃗ ×b⃗ |ududv
Parallelogram defined by point p and vectors a⃗ ,b⃗ p+ua⃗ +vb⃗ 
0<u,v<1
Pu=a⃗ ,Pv=b⃗ n=a⃗ ×b⃗ dA=|a⃗ ×b⃗ |dudv
Cylinder with base radius rx=rcos(u)
y=rsin(u)
z=v
Pu=(rcos(u),rsin(u),0)
Pv=(0,0,1)
n=(rcos(u),rsin(u),0)
n⃗ =(cos(u),sin(u),0)
dA=rdudv
Standard sphere with radius rx=rsin(φ)cos(θ)
y=rsin(φ)sin(θ)
z=rcos(φ)
Pθ=(rsinφsinθ,rsinφcosθ,0)
Pφ=(rcosφcosθ,rcosφsinθ,rsinφ)
n⃗ =1r(x,y,z)
n=r2sin(φ)n⃗ 
dA=r2sin(φ)dφdθ
Sphere with radius r parametrized on latitude
x=rcos(φ)cos(θ)
y=rcos(φ)sin(θ)
z=rsin(φ)
Pθ=r(cosφsinθ,cosφcosθ,0)
Pφ=r(sinφcosθ,sinφsinθ,cosφ)
n⃗ =1r(x,y,z)
n=r2cos(φ)n⃗ 
dA=r2cos(φ)dφdθ
Cone with k equals to the tangent of one half of the opening angle x=khcos(θ)
y=khsin(θ)
z=h
Pθ=(khsin(θ),khcos(θ),0)
Ph=(kcos(θ),ksin(θ),1)
n=kh(cos(θ),sin(θ),k)dA=k1+k2hdθdh
Cone with base radius r and height hx=rvcos(θ)
y=rvsin(θ)
z=hv
Pθ=(rvsin(θ),rvcos(θ),0)
Pv=(rcos(θ),rsin(θ),h)
n=rv(hcos(θ),hsin(θ),r)dA=rr2+h2vdθdv
Torus with major radius R and minor radius rx=cos(u)(R+rcosv)
y=sin(u)(R+rcosv)
z=rsin(v)
Pu=(R+rcosv)(sinu,cosu,0)
Pv=r(cosusinv,sinusinv,cosv)
n=r(R+rcosv)
(cosucosv,sinucosv,sinv)
dA=r(R+rcos(v))dudv
Elliptic cylinder with scaling factor a,b on x, y axis x=acos(u)
y=bsin(u)
z=v
Pu=(asin(u),bcos(u),0)
Pv=(0,0,1)
n=(bcos(u),asin(u),0)dA=a2sin2(u)+b2cos2(u)dudv
Ellipsoid with radius a,b,c on x, y, z axis x=asin(φ)cos(θ)
y=bsin(φ)sin(θ)
z=ccos(φ)
Pθ=(asinφsinθ,bsinφcosθ,0)
Pφ=(acosφcosθ,bcosφsinθ,csinφ)
n=sinφ(bcsinφcosθ,
acsinφsinθ,abcosφ)
dA=sin(φ)[a2b2cos2(φ)+
c2sin2(φ)(a2sin2(θ)+b2cos2(θ))]1/2dφdθ
Elliptic cone with radius a,b on cross section with z=1x=ahcos(θ)
y=bhsin(θ)
z=h
Pθ=(ahsin(θ),bhcos(θ),0)
Ph=(acos(θ),bsin(θ),1)
n=h(bcos(θ),asin(θ),ab)dA=h[a2b2+
a2sin2(θ)+b2cos2(h)]1/2dθdh