Question

# \text { Let } \mathbf{v} \text { be the vector field } \mathbf{v}=\left(y z^{2}-\sin (x), x z^{2}+2 y, 2 x y z\right) a. Evaluate (by direct integration), the line integralv·

dr where \int_{C_{1}} \mathbf{v} \cdot d \mathbf{r}= b. Calculate the curl of v. Enter your answer as a vector using thematrix button \nabla \times \mathbf{v}= \begin{aligned} &\text { d. Find a scalar field } \phi \text { with } \mathbf{v}=\nabla \phi, \text { and calculate }\\ &\phi(0,1,0)-\phi(0,0,1) \end{aligned} \phi(0,1,0)-\phi(0,0,1)= \text { e. Evaluate } \int_{C_{2}} \mathbf{v} \cdot d \mathbf{r}, \text { where } C_{2} \text { is the path parameterised by } x(t)=\pi\left(-t^{7}+1\right) y(t)=-97 t^{4}-67 t^{3}+38 t^{2}-84 t+1 z(t)=t \mathrm{e}^{t-1}, \quad(0 \leq t \leq 1) \int_{C_{2}} \mathbf{v} \cdot d \mathbf{r}= \text { Let } \mathbf{v} \text { be the vector field } \mathbf{v}=\left(y z^{2}-\sin (x), x z^{2}+2 y, 2 x y z\right) C1is the straight line from the point with position vector (0,0, 1) tothe point with position vector (0, 1,0). C1is the straight line from the point with position vector (0,0, 1) tothe point with position vector (0, 1,0).

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