Question

Question 1. Determine whether or not \(\mathrm{F}(x, y)=e^{x} \sin y \mathbf{i}+e^{x} \cos y_{\mathbf{j}}\) is a conservative field. If it is, find its potential function \(f\).

Question 2. Find the curl and the divergence of the vector field \(\mathbf{F}=\sin y z \mathbf{i}+\sin z x \mathbf{j}+\sin x y \mathbf{k}\)

Question 3. Find the flux of the vector field \(\mathbf{F}=z \mathbf{i}+y \mathbf{j}+x \mathbf{k}\) across the surface \(r(u, v)=\langle u \cos v, u \sin v, v\rangle, 0 \leq u \leq 1,0 \leq v \leq \pi\) with upward orientation.

Question 4. Use Stokes" theorem to calculate \(\iint_{S}\) curl F * \(d S\), where \(F=2 y \cos z i+e^{x} \sin =j+x e^{y} k, S\) is the hemisphere \(x^{2}+y^{2}+z^{2}=9, z \geq 0\) oriented upward.

Question 5. Using Laplace transforms solve the following initial-value problems:

1) \(x^{\prime \prime}+2 x^{\prime \prime}+x=e^{-t}, x(0)=1, x^{\prime}(0)=0\);

2) \(x^{\prime \prime}+4 x=\cos 2 t, x(0)=1, x^{\prime}(0)=-2\).

Question 6. An electric circuit consists of an inductor (with inductance \(L=1\) henry), a resistor (with resistance \(R=9\) ohms), a capacitor (with capacitance \(\mathrm{C}=0.05\) farads), and a power supply \(c(t)=6+40 t\) volts at time \(t\) seconds, \(t \geq 0\). No current is flowing in the circuit at time \(t=0\). Using Kirchoff's Second Law (p.120 in the manual) and Laplace transforms find \(I(t)\) - the current in this circuit at time \(\mathrm{t}\).

Answer 1