# Scalar field and 2 types of line integrals

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?

• Would Mathematics be a better home for this question? – Qmechanic Nov 19 '18 at 9:58
• Where have you ever seen the first one? – probably_someone Nov 19 '18 at 10:22

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 $$\hat a = \hat b$$