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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 $$ \int_C \textbf{v} \cdot d \textbf{r} = \textbf{v} \cdot [ \textbf{r}(b) - \textbf{r}(a)] $$

$$

\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}

$$

Vector Calculus

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Johns Hopkins University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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