Random Calculus Results

These are some useful results that are not easily found on the web.

Integral over a sphere

With unit vectors defined in spherical Cartesian coordinates

$$\begin{array}{rl}{n}_1& =\sin \theta \cos \varphi \\ n_2& =\sin \theta \sin \varphi \\ n_3& =\cos \theta \end{array}$$

The integral of a tensor product of unit vectors over the surface of a unit sphere

$$\frac{1}{4\pi }\int d\Omega N_{A_{2l+1}}=0$$ $$\frac{1}{4\pi }\int d\Omega N_{A_{2l+1}}=\frac{1}{(2l+1)(2l+1)!!}\sum _{j_2,j_4,\cdots ,j_{2k}}{\delta }_{a1,a{j}_2}{\delta }_{a{j}_3,a{j}_4}\cdots {\delta }_{a{j}_{2l-1},a{j}_{2l}}$$

where $N_{A_l}=n_{a_1}{n}_{a_2}\cdots n_{a_l}$, $j_2$ is summed from $2$ to $2l$, $j_3$ is the smallest integer not equal to $1$ or $j_2$, $j_4$ is summed over all integers $2$ to $2l$ not equal to $j_2$ or $j_3$, $j_5$ is the smallest integer not equal to 1 or $j_2$ or $j_3$ or $j_4$ …

Ref: Kip S. Thorne, “Multipole Exapansions of gravitational radiation”, Rev. of Mod. Phys., Vol. 52, No.2, Apr. 1980.

Leibniz integral rule

Let $f(x,t)$ be a function such that both $f(x,t)$ and its partial derivative $f_x(x,t)$ are continuous in $t$ and $x$ in some region of the $xt$-plane, including $a(x)\le t\le b(x)$, $x_0\le x\le x_1$. Also suppose that the functions $a(x)$ and $b(x)$ are both continuous and both have continuous derivatives for $x_0\le x\le x_1$, then

$$\frac{d}{dx}\left({\int }_{a(x)}^{b(x)}f(x,t)dt\right)=f(x,b(x))\cdot \frac{d}{dx}b(x)-f(x,a(x))\cdot \frac{d}{dx}a(x)+{\int }_{a(x)}^{b(x)}\frac{\partial }{\partial x}f(x,t)dt$$

Three-dimensional, time-dependent case

Integral rule for a two dimensional surface moving in three dimensional space

$$\frac{d}{dt}\int {\int }_{\Sigma (t)}\vec{F}(\vec{r},t)\cdot d\vec{A}=\int {\int }_{\Sigma (t)}\left(\frac{\partial }{\partial t}\vec{F}(\vec{r},t)+\left[\nabla \cdot \vec{F}(\vec{r},t)\right]\vec{v}\right)\cdot d\vec{A}-{\oint }_{\partial \Sigma (t)}\left[\vec{v}\times \vec{F}(\vec{r},t\right]\cdot d\vec{s}$$

where $\vec{F}$ is a vector field, $\Sigma$ is a surface bounded by the closed curve $\partial \Sigma$, $d\vec{A}$ is a vector element of the surface $\Sigma$, $d\vec{s}$ is a vector element of the curve $\partial \Sigma$, $\vec{v}$ is the velocity of the surface.