2-dimensional integral substitutions

Description	Transform	Inverse transform	Other properties / Additional notes	Jacobian	Commonly 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 ρ≥0	∂(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=2π∫rf(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 ρ≥0	∂(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(1−v) y=uv	u=x+y v=yx+y	Usually, u≥0 and 0≤v≤1.	∂(x,y)∂(u,v)=u ∂(u,v)∂(x,y)=1x+y	Region in the first quadrant defined by x+y≤a
Triangle defined by origin and two vectors a⃗ ,b⃗	p⃗ =u(1−v)⋅a⃗ +uv⋅b⃗	[x′y′]=[a⃗ b⃗ ]−1[xy] u=x′+y′, v=y′x′+y′	0≤u≤1, 0≤v≤1	∂(x,y)∂(u,v)=|a⃗ ×b⃗ |⋅u	Triangular integral regions Green theorem for regions inside polygons
Matrix	[xy]=A[uv]	[uv]=A−1[xy]		∂(x,y)∂(u,v)=det(A)	Parallelogram / Triangle

3-dimensional integral substitutions

Description	Transform	Inverse transform	Other properties / Additional notes	Jacobian	Commonly 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 ρ≥0	∂(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 ρ≥0	∂(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) ρ≥0, 0≤φ≤π	∂(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) ρ≥0, −π2≤φ≤π2	∂(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=4π∫r2f(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) ρ≥0, 0≤φ≤π	∂(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 0≤ρ≤rh1 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 0≤ρ≤1, 0≤h≤1	∂(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+y2−−−−−−√−R)2+z2−−−−−−−−−−−−−−−−−−√ u=atan2(y,x) v=atan2(z,x2+y2−R)	x2+y2=(R+ρcosv)2 0≤u,v≤2π, ρ≥0	∂(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(1−v)w y=uvw z=(1−u)w	u=x+yx+y+z v=yx+y w=x+y+z	x+y=uw Usually, w≥0, 0≤u≤1, 0≤v≤1.	∂(x,y,z)∂(u,v,w)=uw2	Volume in the first octant defined by x+y+z≤a
Tetrahedron defined by origin and three vectors a⃗ ,b⃗ ,c⃗	p⃗ =u(1−v)w⋅a⃗ + uvw⋅b⃗ +(1−u)w⋅c⃗	Transform p⃗  using [a⃗ ,b⃗ ,c⃗ ]−1 and apply the formula in the above cell	0≤u≤1 0≤v≤1 0≤w≤1	∂(x,y,z)∂(u,v,w)=det[a⃗ ,b⃗ ,c⃗ ]⋅uw2	Gauss divergence theorem for integrals inside polyhedrons

3-dimensional area elements

Description	Parametric equation	Directional derivatives	Normal vector	Area element
Triangle defined by point p and vectors a⃗ ,b⃗	p+u(1−v)a⃗ +uvb⃗  p+ua⃗ +uv(b⃗ −a⃗ ) 0<u,v<1	P′u=(1−v)a⃗ +vb⃗  P′v=u(b⃗ −a⃗ )	n=(a⃗ ×b⃗ )⋅u	dA=|a⃗ ×b⃗ |⋅ududv
Parallelogram defined by point p and vectors a⃗ ,b⃗	p+ua⃗ +vb⃗  0<u,v<1	P′u=a⃗ ,P′v=b⃗	n=a⃗ ×b⃗	dA=|a⃗ ×b⃗ |dudv
Cylinder with base radius r	x=rcos(u) y=rsin(u) z=v	P′u=(−rcos(u),rsin(u),0) P′v=(0,0,1)	n=(rcos(u),rsin(u),0) n⃗ =(cos(u),sin(u),0)	dA=rdudv
Standard sphere with radius r	x=rsin(φ)cos(θ) y=rsin(φ)sin(θ) z=rcos(φ)	P′θ=(−rsinφsinθ,rsinφcosθ,0) P′φ=(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)	P′θ=r(−cosφsinθ,cosφcosθ,0) P′φ=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	P′θ=(−khsin(θ),khcos(θ),0) P′h=(kcos(θ),ksin(θ),1)	n=kh(cos(θ),sin(θ),−k)	dA=k1+k2−−−−−√⋅hdθdh
Cone with base radius r and height h	x=rvcos(θ) y=rvsin(θ) z=hv	P′θ=(−rvsin(θ),rvcos(θ),0) P′v=(rcos(θ),rsin(θ),h)	n=rv(hcos(θ),hsin(θ),−r)	dA=rr2+h2−−−−−−√⋅vdθdv
Torus with major radius R and minor radius r	x=cos(u)(R+rcosv) y=sin(u)(R+rcosv) z=rsin(v)	P′u=(R+rcosv)(−sinu,cosu,0) P′v=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	P′u=(−asin(u),bcos(u),0) P′v=(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)	P′θ=(−asinφsinθ,bsinφcosθ,0) P′φ=(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=1	x=ahcos(θ) y=bhsin(θ) z=h	P′θ=(−ahsin(θ),bhcos(θ),0) P′h=(acos(θ),bsin(θ),1)	n=h(bcos(θ),asin(θ),−ab)	dA=h⋅[a2b2+ a2sin2(θ)+b2cos2(h)]1/2dθdh