I am a TA at MIT
A 2-kg mass is attached to a spring with stiffness 40 N/m. The damping constant for the system is 815 N-sec/m. If the mass is pulled 10 cm to the right of equilibrium and given an initial rightward velocity of 2 m/sec, what is the maximum displacement from equilibrium that it will attain?
Find the divergence of the field.The gravitational field in Figure 15.9 and Exercise 38 a in Section 15.3
Use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across the boundary of the region $D$.$-\mathbf{F}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}$a. Cube $\quad D: \quad$ The cube cut from the first octant by the planes $x=1, y=1,$ and $z=1$b. Cube $\quad D: \quad$ The cube bounded by the planes$$x=\pm 1, y=\pm 1, \text { and } z=\pm 1$$c. Cylindrical can $\quad D:$ The region cut from the solid cylinder$$\begin{array}{l}x^{2}+y^{2} \leq 4 \text { by the planes } z=0 \text { and } z=1\end{array}$$
Use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across the boundary of the region $D$.Cylindrical can $\quad \mathbf{F}=\left(6 x^{2}+2 x y\right) \mathbf{i}+\left(2 y+x^{2} z\right) \mathbf{j}+4 x^{2} y^{3} \mathbf{k}$D: The region cut from the first octant by the cylinder $x^{2}+y^{2}=4$ and the plane $z=3$
Use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across the boundary of the region $D$.Wedge $\quad \mathbf{F}=2 x z \mathbf{i}-x y \mathbf{j}-z^{2} \mathbf{k}$D: The wedge cut from the first octant by the plane $y+z=4$ and the elliptic cylinder $4 x^{2}+y^{2}=16$
Use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across the boundary of the region $D$.Thick sphere $\quad \mathbf{F}=\sqrt{x^{2}+y^{2}+z^{2}}(x \mathbf{i}+y \mathbf{j}+z \mathbf{k})$$D:$ The region $1 \leq x^{2}+y^{2}+z^{2} \leq 2$
