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Consider the line integral,

$\int _ c$f(x,y)$\vec dr$ , where $f(x,y)$ is a scalar field, and it is evaluvated on a curve $c $. After integration we get a vector let it be $\vec I$ .

$\int _ c$f(x,y)$\ dr$, hear differential element is a scalar ( magnitude of $\vec dr$ ), After integration we get a scalar, Let it be K , As usually I think

|$\vec I$| = K ,

But most of the cases it is not true, Can u please clear my concept?

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  • $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Nov 19 '18 at 9:58
  • $\begingroup$ Where have you ever seen the first one? $\endgroup$ – probably_someone Nov 19 '18 at 10:22
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In simple terms just consider two small steps $\Delta \vec a = \Delta a \,\hat a$ and $\Delta \vec b = \Delta b \,\hat b$.

The vector line integral will be something like $F_{\rm a} \,\Delta a \,\hat a + F_{\rm b} \,\Delta b \,\hat b$ whereas the scalar line integral will be something like $F_{\rm a} \,\Delta a + F_{\rm b} \,\Delta b $ and so in general $|F_{\rm a} \,\Delta a \,\hat a + F_{\rm b} \,\Delta b \,\hat b| \ne F_{\rm a} \,\Delta a + F_{\rm b} \,\Delta b $ unless $\hat a = \hat b$

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