if vec$(A)=2$vec$(i)-3$vec$(j)-$vec$(k)$ and vec$(B)=i4$vec$(j)-2$vec$(k)$

Find a$)$ vec$(A)$vec$(B),$ b$)$ vec$(B)$vec$(A),$ c$)(vec(A)$ vec$(B))(vec(A)-$vec$(B))$

A particle moves so that its position vector is given by vec$(r)=cosutvec(i)sinatvec(j)$ where ai is a constant. Show that,

a$)$ The velocity vec$(v)$ of the particle is perpendicular to vec$(r),$

b$)$ The acceleration vec$()$ is directed toward the origin and has magnitude proportional to the distance from the origin,

c$)$ vec$(r)$vec$(v)=$ a constant vector.

If vec$(A)=(3x^{2}6y)vec(i)-14$yzvec$(j)20x{z}^2$vec$(k),$ evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(A)*vec(dr)$ from $(0,0,0)$ to $(1,1,1)$ along the following paths $(C)$ :

a$)x=t,y=t^2,z=t^3$

b$)$ the straight lines from $(0,0,0)$ to $(1,0,0),$ then to $(1,1,0)$ and then to $(1,1,1)$

c$)$ the straight line joining $(0,0,0)$ and $(1,1,1)$

Verify the Divergence Theorem for vec$(A)=4$xvec$(i)-2y^2$vec$(j)x^2$vec$(k)$ taken over the region bounded by $x^2{y}^2=4,z=0$ and $z=3.$

Give an explanation and graph each of the following:

a$)6(cos240isin{240}^{\text{'}}),$ b$)4e^{3\frac{i}{5}},$ c$)2e^{-\frac{i}{4}}.$

Using rules of differentiation, find the derivatives of each of the following:

a$)Co{s}^2(2z3i),$ b$)zta{n}^{-1}(lnz),$ c$)[tan{h}^{-1}(iz2){]}^{-4}.$

Evaluate ${}_{S}U(x,y,z)dS$ where $(S)$ is the surface of the parabolold $z=2-(x^2{y}^2)$ above the Oxy plane and $U(x,y,z)$ is equal to: a$)1,$ b$)x^2{y}^2,$ c$)3z.$ Give a physical interpretation in each case.