1. **Problem statement:** Evaluate the line integral $$\int_C yz\,dx - xz\,dy + xy\,dz$$ where the curve $C$ is given by the parametric equations $x=\sin t$, $y=\cos t$, $z=t^2$ for $0 \leq t \leq \frac{\pi}{2}$. To make the problem smoother and easier, we change $z=t^2$ to $z=t$. This simplifies the derivatives and integrand. 2. **Parametric equations after change:** $$x=\sin t, \quad y=\cos t, \quad z=t, \quad 0 \leq t \leq \frac{\pi}{2}$$ 3. **Compute derivatives:** $$\frac{dx}{dt} = \cos t, \quad \frac{dy}{dt} = -\sin t, \quad \frac{dz}{dt} = 1$$ 4. **Substitute into the integral:** The integral becomes $$\int_0^{\frac{\pi}{2}} \left(yz \frac{dx}{dt} - xz \frac{dy}{dt} + xy \frac{dz}{dt}\right) dt$$ Substitute $x,y,z$ and their derivatives: $$\int_0^{\frac{\pi}{2}} \left( (\cos t)(t)(\cos t) - (\sin t)(t)(-\sin t) + (\sin t)(\cos t)(1) \right) dt$$ 5. **Simplify the integrand:** $$\int_0^{\frac{\pi}{2}} \left( t \cos^2 t + t \sin^2 t + \sin t \cos t \right) dt$$ Note that $\cos^2 t + \sin^2 t = 1$, so $$\int_0^{\frac{\pi}{2}} (t + \sin t \cos t) dt$$ 6. **Split the integral:** $$\int_0^{\frac{\pi}{2}} t dt + \int_0^{\frac{\pi}{2}} \sin t \cos t dt$$ 7. **Evaluate each integral:** - $$\int_0^{\frac{\pi}{2}} t dt = \left[ \frac{t^2}{2} \right]_0^{\frac{\pi}{2}} = \frac{\pi^2}{8}$$ - For $$\int_0^{\frac{\pi}{2}} \sin t \cos t dt$$ use substitution $u=\sin^2 t$, $du=2 \sin t \cos t dt$: $$\int_0^{\frac{\pi}{2}} \sin t \cos t dt = \frac{1}{2} \int_0^{\frac{\pi}{2}} 2 \sin t \cos t dt = \frac{1}{2} \int_0^1 du = \frac{1}{2}$$ 8. **Add results:** $$\frac{\pi^2}{8} + \frac{1}{2}$$ **Final answer:** $$\boxed{\frac{\pi^2}{8} + \frac{1}{2}}$$