Please help to calculate the following integral.
Calculate $$\int_\gamma \frac{x\,dx + y\,dy+z\,dz}{x^2+y^2+z^2}$$ where $\gamma$ is the way of class $\mathcal C^1$ which unites point on the sphere $x^2+y^2+z^2=a^2$ with a point on the sphere $x^2+y^2+z^2=b^2$ with $a,b>0$.
Note that if $F(x,y,z)=\frac{1}{2}\log(x^2+y^2+z^2)$, then $$ \frac{x}{x^2+y^2+z^2}=\frac{1}{2}\frac{\partial}{\partial x} \log(x^2+y^2+z^2)=\frac{\partial}{\partial x}F, $$ and hence $$ \frac{1}{x^2+y^2+z^2}(x,y,z)=\nabla \left(\frac{1}{2}\frac{\partial}{\partial x} \log(x^2+y^2+z^2)\right)=\nabla F, $$ since $$ \nabla =\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right) \quad\text{and}\quad \nabla F=\left(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z}\right). $$ and hence if $\gamma(t)=\big(x(t),y(t),z(t)\big) : [0,T]\to \mathbb R^3$, is $C^1$, then \begin{align} \int_\gamma \frac{x\,dx+y\,dy+z\,dz}{x^2+y^2+z^2} &= \int_0^T \nabla F\big(\gamma(t)\big)\cdot \gamma'(t)\,dt=F\big(\gamma(T)\big)-F\big(\gamma(0)\big) \\ &=\frac{1}{2}\log\left(\frac{\|\gamma(T)\|^2}{\|\gamma(0)\|^2}\right)=\log\left(\frac{\|b\|}{\|a\|}\right). \end{align}