** True or False** The following mathematical operation makes sense and is technically valid.
$$\nabla \cdot \nabla T(x,y,z)$$
1. Yes, it will produce a vector field.
2. Yes, it will produce a scalar field.
3. No, you can not take the divergence of a scalar field.
4. I don't remember what this means.
Note:
* Correct answer: B
## Announcements
* Homework 1 is due Friday in class
* Homework 2 is posted and covers through section 2.1
- It is due next Wednesday
- We will come back to section 1.5 later
* Make sure you have registered your clicker!
- I will start shaming people publically on Friday.
- [https://goo.gl/cVfzS5](https://goo.gl/cVfzS5)
You are trying to compute the work done by a force, $\mathbf{F} = a\hat{x} + x\hat{y}$, along the line $y=2x$ from $\langle 0,0 \rangle$ to $\langle 1,2 \rangle$. What is $d\mathbf{l}$?
1. $dl$
2. $dx\;\hat{x}$
3. $dy\;\hat{y}$
4. $2dx\;\hat{x}$
5. Something else
Note:
* CORRECT ANSWER: E
* It's $dx\;\hat{x}+dy\;\hat{y}$.
* Fall 2016 (given right at the end of class): 8 6 8 58 [20]; 5 0 3 38 [54]
You are trying to compute the work done by a force, $\mathbf{F} = a\hat{x} + x\hat{y}$, along the line $y=2x$ from $\langle 0,0 \rangle$ to $\langle 1,2 \rangle$. Given that $d\mathbf{l} = dx\;\hat{x}+dy\;\hat{y}$, which of the following forms of the integral is correct?
1. $\int_0^1 a\;dx + \int_0^2 x\;dy$
2. $\int_0^1 (a\;dx + 2x\;dx)$
3. $\frac{1}{2} \int_0^2 (a\;dy + y\;dy)$
4. More than one is correct
Note:
* CORRECT ANSWER: D
* All are correct forms, but B and C are ready to integrate.
* Fall 2016: 7 3 0 [90] 0
A certain fluid has a velocity field given by $\mathbf{v} = x\hat{x} + z \hat{y}$. Which component(s) of the field contributed to "fluid flux" integral ($\int_S \mathbf{v}\cdot d\mathbf{A}$) through the x-z plane?
1. $v_x$
2. $v_y$
3. both
4. neither
Note:
* CORRECT ANSWER: B
* Only the vector perpendicular to the surface will contribute
* Fall 2016: 16 [78] 4 0 0
For the same fluid with velocity field given by $\mathbf{v} = x\hat{x} + z \hat{y}$. What is the value of the "fluid flux" integral ($\int_S \mathbf{v}\cdot d\mathbf{A}$) through the entire x-y plane?
1. It is zero
2. It is something finite
3. It is infinite
4. I can't tell without doing the integral
Note:
* CORRECT ANSWER: A
* The velocity field is parallel to the x-y plane every where and hence contributes no flux through the surface.
* Fall 2016: [89] 9 3 0 0
A rod (radius $R$) with a hole (radius $r$) drilled down its entire length $L$ has a mass density of $\frac{\rho_0\phi}{\phi_0}$ (where $\phi$ is the normal polar coordinate).
To find the total mass of this rod, which coordinate system should be used (take note that the mass density varies as a function of angle):
1. Cartesian ($x,y,z$)
2. Spherical ($r,\phi,\theta$)
3. Cylindrical ($s, \phi, z$)
4. It doesn't matter, just pick one.
Note:
* CORRECT ANSWER: C
* It makes the most sense from the geometry of the problem and writing the limits.
* Fall 2016: 0 0 [94] 6 0
Which of the following two fields has zero divergence?
| I | II |
|:-:|:-:|
| <img src ="./images/cq_left_field.png" align="center" style="width: 400px";/> | <img src ="./images/cq_right_field.png" align="center" style="width: 400px";/> |
1. Both do.
2. Only I is zero
3. Only II is zero
4. Neither is zero
5. ???
Note:
* CORRECT ANSWER: B
* Think about dE/dx and dE/dy
* Fall 2016: 7 [34] 13 43 3; (Asked them to consider dvx/dx and dvy/dy) 3 [90] 3 4 0
Which of the following two fields has zero curl?
| I | II |
|:-:|:-:|
| <img src ="./images/cq_left_field.png" align="center" style="width: 400px";/> | <img src ="./images/cq_right_field.png" align="center" style="width: 400px";/> |
1. Both do.
2. Only I is zero
3. Only II is zero
4. Neither is zero
5. ???
Note:
* CORRECT ANSWER: C
* Think about paddle wheel
* Fall 2016: 9 0 [89] 3 0