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\centerline
{
\bf Solutions to the ``QUIZ'' for Lecture 26
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Which of the following answers is nonsense. Explain!

1. $\int_0^{\pi} \int_{-1}^{1} r \cos \theta \, dr \, d \theta$ (in polar coordinates).
 Nonsense. The limit of integration
of $r$ can never be negative numbers, since $r$ is {\bf distance} from the origin, and distance is never negative.

2. $\int_0^{\pi} \int_{0}^{1} \int_{-3}^{3} r \cos \theta \, \, dz dr \, d \theta$ (in cylindrical coordinates).
OK, the limits-of-integration for the $z$ integration are allowed to be negative.

3. The tangent plane to the surface $z=x^2+y^2$ at $(1,1,2)$ is $z=2+ 2x(x-1)+2y(y-1)$ .
Nonsense: the format of equation of a plane is z=ax+by+c where $a$ $b$ and $c$ are numbers.

4. For some vector field ${ \bf F}$, $div \, {\bf F}= \langle x, y, z \rangle $.
Nonsense: $div \, {\bf F}$ is always a {\bf function}, never a vector-field.

5. For some vector field ${ \bf F}$, $curl \, {\bf F}= \sin(x+y+z) $.
Nonsense: $curl \, {\bf F}$ is another vector-field, never a function.

6. The arclength of a certain curve equals $-5$. Nonsense: length is never negative.

7. The volume of a certain solid body equals $-11.1$. Nonsense: volume is never negative.

8. The volume-integral of a certain function over a certain solid-body equals $-11.1$. OK.

9. The line-integral of a certain scalar function over a certain curve equals $-7$. OK.

10. $\sin^{-1} (2)$. Nonsense: $sin x$ is always between $-1$ and $1$.




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