PX275 - D2a - line integrals with fields

normally, in physics, line integral needs to be performed with some field present
eg: integral of a spatially varying force along a path to give the change in energy

for a scalar field

I = \int_{A}^{B} ϕ d \vec{r}

I = \int_{A}^{B} \vec{a} \cdot d \vec{r}

I = \int_{A}^{B} \vec{a} \times d \vec{r}

eg: $\vec{F} = \int q \vec{v} \times B$ , $\vec{F} = I \oint d \vec{r} \times \vec{B}$

\begin{aligned} I & = \int_{O A B} ϕ d \vec{r} \\ = \hat{i} \int_{0}^{1} a x d x + \hat{j} \int_{0}^{1} a d y \\ = \hat{i} {[\frac{a x^{2}}{2}]}_{x = 0}^{1} + \hat{j} [a y]_{y = 1}^{1} \\ = \frac{a}{2} \hat{i} + a \hat{j} \end{aligned}

I = \hat{i} \int_{0}^{1} a x d x = \frac{a}{2} \hat{i}