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Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where $C$ is given by the vector function $\mathbf{r}(t) .$

$$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+x y \mathbf{k}$$

$$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}, \quad 0 \leqslant t \leqslant \pi$$

$\int_{C} \mathbf{F} \cdot d \mathbf{r}=0$

Vector Calculus

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