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Problem 45 Hard Difficulty

If $C$ is a smooth curve given by a vector function $\mathbf{r}(t)$ $a \leqslant t \leqslant b,$ and $\mathbf{v}$ is a constant vector, show that
$$\int_{C} \mathbf{v} \cdot d \mathbf{r}=\mathbf{v} \cdot[\mathbf{r}(b)-\mathbf{r}(a)]$$

Answer

$\int_{C} \mathbf{v} \cdot d \mathbf{r}=\mathbf{v} \cdot[\mathbf{r}(b)-\mathbf{r}(a)]$

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Video Transcript

Okay, So in this video, we are supposed to show a verify that the integral over, um, Kurt of a constant vector So constant vector field, which is to be so he's gonna be a constant vector field don t r is equal to p dot r b minus are all right. So since bars are position vector, we can write it. Except he I have plus my t j at policy of TK have all right now. So our position vector another condensed way of writing it. It's just accept IQ of a wife becomes a 50 that they were gonna take the derivative of that with respect. That's just going to be expressed. 50 called the wide prime off to become a Z. And now, since RV is our constant about vector field, it's, uh I'm gonna right, So since it's because I want to read in this case, that's K one I have plus key to J. A happ must be three key. So he is basically we want to write it in the more convinced ways just k one complicate to come. All right, so now we're gonna write everything in terms of bars of the integral over a long curb. CEO Vidal tr just the integral from a to B of the dot are crying Lefty Bt So now we're gonna again Our limits points are probably so no, uh, we already know that the is K one k creepy three. And our prime is just x private t call Ally, prime of decomposition privacy ever DT. It's an hour. If we multiply the terms out, it's gonna be K one. Okay, One, uh, ex prime prosecute to lie prime plus King three z, Prime DT and the Integral Post from A to B. Since the case are constants, we can pull them to the outside. So we're gonna have, for example, Kale on the Integral from A to B of ex prime tea. Well, again, from the fundamental theory of capitalist we haven't integral and a derivative. So they cancel out and we're just left with, except T as it goes from a to B. So this is where the first year same thing for the second term gonna pull that K two out and then the integral of the derivative. It's just gonna be the function itself. And it goes from a to B Same thing, quite a search rates that now if we take the differences were going to get kay. Okay, one times. Except be lying. It's exit they k two times lie of B minus wildly plus K three times you'll be fine. Zero. Now, another way. We can write this. There's a three pull out that they want to keep three, and then we don't product it with X of a minus X api. Why have b minus sorry except beeline. Sex of a life B minus viably covers because basically, when we do the dot product, we're just gonna multiply this first start by this rooster that's don't product. And then this second term, with this second term with their second and the that this searcher with this critter that's gonna be our 30. So this and this are equivalent their equal to each other. But then recall that Ah, the vector field K one K two k three. Well, that's just be. And then exit B minus x of a lie of B minus Lively's new B minus zero level. This is just a difference of two vectors. We're the vectors. Are R V minus R R is or the difference of these two. It's exit B minus X away from the life being liners while they follow zero B minus. So we finally were able to verify this. So this is what we wanted to verify.

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