In multivariable calculus the line integrals was parameterized and denoted:
\begin{gather*}
\int_l \mathbf{A} \cdot \, d\mathbf{r} = \int_\gamma \mathbf{A}(\mathbf{r}(t)) \cdot \, \frac{d\mathbf{r}(t)}{dt}dt \tag{1}\\
\text{where} \quad \mathbf{r}(t)=x(t)\mathbf{\hat i}+y(t)\mathbf{\hat j}+z(t)\mathbf{\hat k}
\end{gather*}
However in electromagnetism the line integrals are confusing.
Exemple: Suppose a charge is moving from point $b(\rho_b,\phi_b,z_b)$ to point $a(\rho_a,\phi_a,z_a)$ along the direction of $\rho$ (cylindrical coordinates):
\begin{gather*}
-\int_b^a \frac{Q\phi_L}{2\pi\epsilon_0 \rho}\mathbf{\hat \rho} \cdot (\mathbf{\hat \rho}\, d\rho+\mathbf{\hat \phi} \, \rho d\phi + \hat z \, dz)=\\
=-\int_b^a \frac{Q\phi_L}{2\pi\epsilon_0 \rho}\mathbf{\hat \rho} \cdot \mathbf{\hat \rho} \, d\rho = -\int_b^a \frac{Q\phi_L}{2\pi\epsilon_0 \rho} \, d\rho
\end{gather*}
This notation confuses me.
In math the procedure was; parametrize the curve, take the derivative of it, dot it with the parameterized field, as in (1). How can I use this approach in the exemple?
How can I derive the notation in the
second integral from the notation in (1)?
Can I calculate the exemple using $\mathbf{r}(t)$ in cartesian coordinates?
Best Answer
That is exactly what you did here. You'r parametrization is - $$\vec{r} = (t,\phi_a,z_a),\;t\in[\rho_a,\rho_ b]$$ so - $$\vec{r}'=(1,0,0)$$ since $\phi_a=\phi_b, z_a=z_b$.