2-dimensional integral substitutions

Description	Transform	Inverse transform	Other properties / Additional notes	Jacobian	Commonly apply to
Polar $\rho$: distance to the origin $\theta$: polar angle	$x=\rho\cos(\theta)$ $y=\rho\sin(\theta)$	$\rho=\sqrt{x^2+y^2}$ $\theta=\mathrm{atan2}(y,x)$	$x^2+y^2=\rho^2$ $\rho\ge0$	$\dfrac{\partial(x,y)}{\partial(\rho,\theta)}=\rho$	Circular integral regions Planar regions described by implicit algebraic equations
Polar *Integrals of symmetrical functions in circular regions		$\mathbf{r}=(x,y)$ $r=|\mathbf{r}|=\sqrt{x^2+y^2}$		$\displaystyle\iint{f(|\mathbf{r}|)\ \mathrm{d}{\mathbf{r}}}=2\pi\int{r f(r)\mathrm{d}{r}}$	Circular regions centered at the origin Infinite 2D plane
Generalized polar $a,b$: scaling on x and y-axis	$x=a\rho\cos(\theta)$ $y=b\rho\sin(\theta)$	$\rho=\sqrt{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}}$ $\theta=\mathrm{atan2}(ay,bx)$	$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=\rho^2$ $\rho\ge0$	$\dfrac{\partial(x,y)}{\partial(\rho,\theta)}=ab\rho$	Elliptical integral regions
Isosceles right triangle formed by axises $u$: distance to origin (Manhattan) $v$: linear interpolation coefficient from x-axis to y-axis	$x=u(1-v)$ $y=uv$	$u=x+y$ $v=\dfrac{y}{x+y}$	Usually, $u\ge 0$ and $0\le v\le 1$.	$\dfrac{\partial(x,y)}{\partial(u,v)}=u$ $\dfrac{\partial(u,v)}{\partial(x,y)}=\dfrac{1}{x+y}$	Region in the first quadrant defined by $x+y\le a$
Triangle defined by origin and two vectors $\vec{a}, \vec{b}$	$\vec{p}=u(1-v)\cdot\vec{a}+uv\cdot\vec{b}$	$\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}\vec{a}&\vec{b}\end{bmatrix}^{-1}\begin{bmatrix}x\\y\end{bmatrix}$ $u=x'+y'$, $v=\frac{y'}{x'+y'}$	$0\le u\le 1$, $0\le v\le 1$	$\dfrac{\partial(x,y)}{\partial(u,v)}=|\vec{a}\times\vec{b}|\cdot u$	Triangular integral regions Green theorem for regions inside polygons
Matrix	$\begin{bmatrix}x\\y\end{bmatrix}=\mathbf{A}\begin{bmatrix}u\\v\end{bmatrix}$	$\begin{bmatrix}u\\v\end{bmatrix}=\mathbf{A}^{-1}\begin{bmatrix}x\\y\end{bmatrix}$		$\dfrac{\partial(x,y)}{\partial(u,v)}=\det(\mathbf{A})$	Parallelogram / Triangle

3-dimensional integral substitutions

Description	Transform	Inverse transform	Other properties / Additional notes	Jacobian	Commonly apply to
Cylindrical $\rho$: distance to the origin $\theta$: polar angle $h$: height	$x=\rho\cos(\theta)$ $y=\rho\sin(\theta)$ $z=h$	$\rho=\sqrt{x^2+y^2}$ $\theta=\mathrm{atan2}(y,x)$ $h=z$	$x^2+y^2=\rho^2$ $\rho\ge0$	$\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,h)}=\rho$	Integral volumes with circular bases
Generalized cylindrical $a,b$: scaling on x and y-axis	$x=a\rho\cos(\theta)$ $y=b\rho\sin(\theta)$ $z=h$	$\rho=\sqrt{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}}$ $\theta=\mathrm{atan2}(ay,bx)$ $h=z$	$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=\rho^2$ $\rho\ge0$	$\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,h)}=ab\rho$	Integral volumes with elliptical bases
Spherical (standard) $\rho$: distance to the origin $\theta$: longitude $\varphi$: angle to the vertical axis	$x=\rho\sin(\varphi)\cos(\theta)$ $y=\rho\sin(\varphi)\sin(\theta)$ $z=\rho\cos(\varphi)$	$\rho=\sqrt{x^2+y^2+z^2}$ $\theta=\mathrm{atan2}(y,x)$ $\varphi=\mathrm{atan2}(\sqrt{x^2+y^2},z)$	$x^2+y^2+z^2=\rho^2$ $x^2+y^2=\rho^2\sin^2(\varphi)$ $\rho\ge 0$, $0\le\varphi\le\pi$	$\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,\varphi)}=\rho^2\sin(\varphi)$	Spherical (ρ) and conical (φ) volumes Volumes defined by implicit algebraic equations
Spherical (latitude) $\rho$: distance to the origin $\theta, \varphi$: longitude and latitude	$x=\rho\cos(\varphi)\cos(\theta)$ $y=\rho\cos(\varphi)\sin(\theta)$ $z=\rho\sin(\varphi)$	$\rho=\sqrt{x^2+y^2+z^2}$ $\theta=\mathrm{atan2}(y,x)$ $\varphi=\mathrm{atan2}(z,\sqrt{x^2+y^2})$	$x^2+y^2+z^2=\rho^2$ $x^2+y^2=\rho^2\cos^2(\varphi)$ $\rho\ge0$, $-\frac{\pi}{2}\le\varphi\le\frac{\pi}{2}$	$\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,\varphi)}=\rho^2\cos(\varphi)$	Spheres
Spherical *Integrals of symmetrical functions in spherical regions		$\mathbf{r}=(x,y,z)$ $r=|\mathbf{r}|=\sqrt{x^2+y^2+z^2}$		$\displaystyle\iiint{f(|\mathbf{r}|)\ \mathrm{d}{\mathbf{r}}}=4\pi\int{r^2f(r)\mathrm{d}{r}}$	Spherical regions centered at the origin Infinite three-dimensional space
Generalized spherical $a,b,c$: scaling on x, y, and z-axis	$x=a\rho\sin(\varphi)\cos(\theta)$ $y=b\rho\sin(\varphi)\sin(\theta)$ $z=c\rho\cos(\varphi)$	$\rho=\sqrt{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}}$ $\theta=\mathrm{atan2}(ay,bx)$ $\varphi=\mathrm{atan2}\left(\sqrt{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}},\dfrac{z}{c}\right)$	$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=\rho^2$ $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=\rho^2\sin^2(\varphi)$ $\rho\ge 0$, $0\le\varphi\le\pi$	$\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,\varphi)}=abc\rho^2\sin(\varphi)$	Ellipsoids
Conical $\rho h$: radius at $z=h$	$x=\rho h \cos(\theta)$ $y=\rho h \sin(\theta)$ $z=h$	$\rho=\dfrac{1}{z}\sqrt{x^2+y^2}$ $\theta=\mathrm{atan2}(y,x)$ $h=z$	$x^2+y^2=\rho^2h^2$ $0\le\rho\le\frac{r}{h_1}$ for integral in a cone with height $h_1$ and base radius $r$	$\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,h)}=\rho h^2$	Cones
Generalized conical $a,b,c$ are scaling factors on x, y, z axis	$x=a\rho h \cos(\theta)$ $y=b\rho h \sin(\theta)$ $z=ch$	$\rho=\dfrac{c}{z}\sqrt{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}}$ $\theta=\mathrm{atan2}(ay,bx)$ $h=\dfrac{z}{c}$	$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=\rho^2h^2$ $0\le\rho\le1$, $0\le h\le1$	$\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,h)}=abc\rho h^2$	Elliptic cones with base radius $a,b$ and height $c$
Toric $u$: longitude $v$: latitude $\rho$: radius of small circle $R$: radius of large circle	$x=\cos(u)(R+\rho\cos v)$ $y=\sin(u)(R+\rho\cos v)$ $z=\rho\sin(v)$	$\rho=\sqrt{(\sqrt{x^2+y^2}-R)^2+z^2}$ $u=\mathrm{atan2}(y,x)$ $v=\mathrm{atan2}(z,x^2+y^2-R)$	$x^2+y^2=(R+\rho\cos v)^2$ $0\le u,v\le 2\pi$, $\rho\ge 0$	$\dfrac{\partial(x,y,z)}{\partial(\rho,u,v)}=\rho(R+\rho\cos v)$	Toruses
Tetrahedron formed by coordinate planes and a plane with $\vec{n}=(1,1,1)$ $u\in[0,1]$: xOy to z-axis $v\in[0,1]$: xOz to yOz $w$: x + y + z	$x=u(1-v)w$ $y=uvw$ $z=(1-u)w$	$u=\dfrac{x+y}{x+y+z}$ $v=\dfrac{y}{x+y}$ $w=x+y+z$	$x+y=uw$ Usually, $w\ge0$, $0\le u\le 1$, $0\le v\le 1$.	$\dfrac{\partial(x,y,z)}{\partial(u,v,w)}=uw^2$	Volume in the first octant defined by $x+y+z\le a$
Tetrahedron defined by origin and three vectors $\vec{a}, \vec{b}, \vec{c}$	$\vec{p} = u(1-v)w\cdot\vec{a}+$ $uvw\cdot\vec{b}+(1-u)w\cdot\vec{c}$	Transform $\vec{p}$ using $[\vec{a},\vec{b},\vec{c}]^{-1}$ and apply the formula in the above cell	$0\le u\le 1$ $0\le v\le 1$ $0\le w\le 1$	$\dfrac{\partial(x,y,z)}{\partial(u,v,w)}=\det[\vec{a},\vec{b},\vec{c}]\cdot uw^2$	Gauss divergence theorem for integrals inside polyhedrons

3-dimensional area elements

$\newcommand{\d}{\mathrm{d}}$

Description	Parametric equation	Directional derivatives	Normal vector	Area element
Triangle defined by point $p$ and vectors $\vec{a},\vec{b}$	$p+u(1-v)\vec{a}+uv\vec{b}$ $p+u\vec{a}+uv(\vec{b}-\vec{a})$ $0< u,v< 1$	$P_u'=(1-v)\vec{a}+v\vec{b}$ $P_v'=u(\vec{b}-\vec{a})$	$\mathbf{n}=(\vec{a}\times\vec{b})\cdot u$	$\d{A}=|\vec{a}\times\vec{b}|\cdot u \;\d{u}\d{v}$
Parallelogram defined by point $p$ and vectors $\vec{a},\vec{b}$	$p+u\vec{a}+v\vec{b}$ $0< u,v< 1$	$P_u'=\vec{a},\; P_v'=\vec{b}$	$\mathbf{n}=\vec{a}\times\vec{b}$	$\d{A}=|\vec{a}\times\vec{b}|\;\d{u}\d{v}$
Cylinder with base radius $r$	$x=r\cos(u)$ $y=r\sin(u)$ $z=v$	$P_u'=(-r\cos(u),r\sin(u),0)$ $P_v'=(0,0,1)$	$\mathbf{n}=(r\cos(u),r\sin(u),0)$ $\vec{n}=(\cos(u),\sin(u),0)$	$\d{A}=r\;\d{u}\d{v}$
Standard sphere with radius $r$	$x=r\sin(\varphi)\cos(\theta)$ $y=r\sin(\varphi)\sin(\theta)$ $z=r\cos(\varphi)$	$P_\theta'=(-r\sin\varphi\sin\theta,r\sin\varphi\cos\theta,0)$ $P_\varphi'=(r\cos\varphi\cos\theta,r\cos\varphi\sin\theta,-r\sin\varphi)$	$\vec{n}=\dfrac{1}{r}(x,y,z)$ $\mathbf{n}=r^2\sin(\varphi)\cdot\vec{n}$	$\d A=r^2\sin(\varphi) \;\d\varphi\d\theta$
Sphere with radius $r$ parametrized on latitude	$x=r\cos(\varphi)\cos(\theta)$ $y=r\cos(\varphi)\sin(\theta)$ $z=r\sin(\varphi)$	$P_\theta'=r(-\cos\varphi\sin\theta,\cos\varphi\cos\theta,0)$ $P_\varphi'=r(-\sin\varphi\cos\theta,-\sin\varphi\sin\theta,\cos\varphi)$	$\vec{n}=\dfrac{1}{r}(x,y,z)$ $\mathbf{n}=r^2\cos(\varphi)\cdot\vec{n}$	$\d A=r^2\cos(\varphi) \;\d\varphi\d\theta$
Cone with $k$ equals to the tangent of one half of the opening angle	$x=k h\cos(\theta)$ $y=k h\sin(\theta)$ $z=h$	$P_\theta'=(-k h\sin(\theta),k h\cos(\theta),0)$ $P_h'=(k\cos(\theta),k\sin(\theta),1)$	$\mathbf{n}=kh(\cos(\theta),\sin(\theta),-k)$	$\d A=k\sqrt{1+k^2}\cdot h \;\d\theta\d h$
Cone with base radius $r$ and height $h$	$x=r v\cos(\theta)$ $y=r v\sin(\theta)$ $z=h v$	$P_\theta'=(-rv\sin(\theta),rv\cos(\theta),0)$ $P_v'=(r\cos(\theta),r\sin(\theta),h)$	$\mathbf{n}=rv(h\cos(\theta),h\sin(\theta),-r)$	$\d A=r\sqrt{r^2+h^2}\cdot v \;\d\theta\d v$
Torus with major radius $R$ and minor radius $r$	$x=\cos(u)(R+r\cos v)$ $y=\sin(u)(R+r\cos v)$ $z=r\sin(v)$	$P_u'=(R+r\cos v)(-\sin u,\cos u,0)$ $P_v'=r(-\cos u\sin v,-\sin u\sin v,\cos v)$	$\mathbf{n}=r(R+r\cos v)\cdot$ $(\cos u\cos v,\sin u\cos v,\sin v)$	$\d A=r(R+r\cos(v)) \;\d{u}\d{v}$
Elliptic cylinder with scaling factor $a,b$ on x, y axis	$x=a\cos(u)$ $y=b\sin(u)$ $z=v$	$P_u'=(-a\sin(u),b\cos(u),0)$ $P_v'=(0,0,1)$	$\mathbf{n}=(b\cos(u),a\sin(u),0)$	$\d{A}=\sqrt{a^2\sin^2(u)+b^2\cos^2(u)} \;\d{u}\d{v}$
Ellipsoid with radius $a,b,c$ on x, y, z axis	$x=a\sin(\varphi)\cos(\theta)$ $y=b\sin(\varphi)\sin(\theta)$ $z=c\cos(\varphi)$	$P_\theta'=(-a\sin\varphi\sin\theta,b\sin\varphi\cos\theta,0)$ $P_\varphi'=(a\cos\varphi\cos\theta,b\cos\varphi\sin\theta,-c\sin\varphi)$	$\mathbf{n}=\sin\varphi(bc\sin\varphi\cos\theta,$ $ac\sin\varphi\sin\theta,ab\cos\varphi)$	$\d A=\sin(\varphi){\large[}a^2b^2\cos^2(\varphi)+$ $c^2\sin^2(\varphi)(a^2\sin^2(\theta)+b^2\cos^2(\theta)){\large]}^{1/2} \;\d\varphi\d\theta$
Elliptic cone with radius $a,b$ on cross section with $z=1$	$x=ah\cos(\theta)$ $y=bh\sin(\theta)$ $z=h$	$P_\theta'=(-ah\sin(\theta),bh\cos(\theta),0)$ $P_h'=(a\cos(\theta),b\sin(\theta),1)$	$\mathbf{n}=h(b\cos(\theta),a\sin(\theta),-ab)$	$\d A=h\cdot {\large[}a^2b^2+$ $a^2\sin^2(\theta)+b^2\cos^2(h){\large]}^{1/2} \;\d\theta\d h$