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Problem 49 Hard Difficulty

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

Answer

$$
\begin{array}{c}{\int_{C} \mathbf{v} \cdot d \mathbf{r}=v_{1} \int_{a}^{b} x^{\prime}(t) d t+v_{2} \int_{a}^{b} y^{\prime}(t) d t+v_{3} \int_{a}^{b} z^{\prime}(t) d t} \\ {\int_{C} \mathbf{v} \cdot d \mathbf{r}=v_{1}[x(b)-x(a)]+v_{2}[y(b)-y(a)]+v_{3}[z(b)-z(a)]}\end{array}
$$

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Video Transcript

Okay, let's try Toe derives the given formula. So in this case, we're not giving any information about what other Arab tears. But we can still follow standard procedure. Let's say we just have our tea is it could be to our street. I mentioned this. That's off key for simplicity. All right into dimension. But you can add a sir component. We should change the derivation by much. And how do we rise? Eleven side Elevens? I tease from A to B as a visa. Constant vetters. Always we write be horseless, eh? Be one too. And modesty are well, the R should be no ex pry off t. Why Proud t So what we get here? We get fever one X prime off T plus feet too. Why pry off t remember of even veto bills is a constant factor are constant and the dp it becomes integrated This but in a very ex prompted it. He's just ex elf x lt on we probably the end point. So this one we should have won ex off b minus X off, eh? Plus beat you excel. Sorry hero should be Why Off p minus y off, eh? And if you write it in the vector dot product, this should cost Hughes the final former Remember off after his thistle problem B and es and you should be able to too easy See that these two things are the same.