In this video the professor walks us through calculating the electric potential different. I am having a problem understanding why $d\vec{r}$ is in $\hat{r}$ direction . Although he says that $d\vec{r} = dr\hat{r}$ where $dr$ can be negative. But I having problem linking this to single variable calculus where our integration variable is always positive. I think that the limits of integral must be in same direction as the integration variable ,though I don't understand why. Can someone please explain this.
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Does A doubt in the derivation for determining the electric potential difference between concentric spherical shells answer your question? – peek-a-boo Jun 04 '21 at 14:19
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@peek-a-boo Yes it does to some extend. To clarify if we chose a particular co-ordinate system for $d\vec{s}$ and then wrote the limits in the same system, would we still get the right answer. – saket kumar Jun 04 '21 at 14:38
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yes. Always write the line element as per usual (in any coordinate system you fancy), and do not try to use physical arguments to take the sign of the line element and the vector field into account. ALWAYS let the limits of integration take the signs into account. Following this one piece of advice in Griffiths' textbook has saved me countless sign errors. – peek-a-boo Jun 04 '21 at 14:40
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@peek-a-boo Here is where I had the problem- In the video he says that integration variable can be negative, but that is not true in single variable calculus. Is that correct? – saket kumar Jun 04 '21 at 14:43
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If $a<b$ and we write the integral $\int_a^bf(r),dr$ one thinks of $dr$ as a positive quantity, but if you write $\int_b^af(r),dr$, then by definition, this is equal to $-\int_a^bf(r),dr$, so in the case of $\int_b^af(r),dr$, we interpret $dr$ as a negative quantity. – peek-a-boo Jun 04 '21 at 14:47
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@peek-a-boo https://math.stackexchange.com/questions/2220045/changing-signs-of-integration-limits, here in a comment by levap , they say that $dr>0$ in our case also – saket kumar Jun 04 '21 at 14:54
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Let us continue this discussion in chat. – peek-a-boo Jun 04 '21 at 14:56