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(a) Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where

$\mathbf{F}(x, y, z)=x \mathbf{i}-z \mathbf{j}+y \mathbf{k}$ and $C$ is given by

$\mathbf{r}(t)=2 t \mathbf{i}+3 t \mathbf{j}-t^{2} \mathbf{k},-1 \leqslant t \leqslant 1 .$

(b) Illustrate part (a) by using a computer to graph $C$ and the vectors from the vector field corresponding to $t=\pm 1$ and $\pm \frac{1}{2}($ as in Figure 13$) .$

a) $\int_{C} \mathbf{F} \cdot d \mathbf{r}=-2$

b) $F(r(t))=\left\langle 2 t, t^{2}, 3 t\right\rangle$ so $ F(r(1))=\langle, 2,1,3\rangle, F\left(r\left(\frac{1}{2}\right\rangle\right) \rangle )=\left\langle 1, \frac{1}{4}, \frac{3}{2}\right\rangle$ $F(r(1))=\langle 2,1,4\rangle$

Vector Calculus

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Johns Hopkins University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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