What is the divergence in the boxed region?
<img src ="./images/pt_charge_red_box.png" align="right" style="width: 400px";/>
1. Zero
2. Not zero
3. ???
Note:
* CORRECT ANSWER: A
* Lines in; lines out - harder to see dE/dx and dE/dy
* One of those curious ones where the 2D picture might get in the way; think 3D
Consider a vector field defined as the gradient of some well-behaved scalar function:
$$\mathbf{v}(x,y,z) = \nabla T(x,y,z).$$
What is the value of $\oint_C \mathbf{v} \cdot d\mathbf{l}$?
1. Zero
2. Non-zero, but finite
3. Can't tell without a function for $T$
Note:
* CORRECT ANSWER: A
* Closed loop integral of a gradient is zero.
* Fall 2016: [92] 4 4 0 0
## Announcements
* Homework 1 solutions posted immediately after class
* Graded Homework 1 returned next Friday
* Homework 2 posted (due next Friday)
* WAMPS is organizing GRE study sessions
* First session: Wed., Sept. 12 4-5 pm (BPS 1400)
## Register your clicker!
- Adams, Joe
- Allen, Grant
- Bensley, Justin
- Bertus, Thomas
- Boyd, Brendan
- Briseno, Robert
- Brook, Evan
- Byrd, Benjamin
- Czyzewski, Austin
- Dara, Jacob
- Ding, Fang
- Evasic, Jacob
- Fowler, David
## And it cotinues...
- Hindenach, John
- Jiang, Shan
- Lewis, Jim
- Li, Xingyu
- Li, Zihan
- Maestrales, Sarah
- Myers, Cody
- Osella, Anna
- Patel, Shivang
- Smith, Dylan
- Wallace, Ian
- Ward, Jenny
## And yet, we are not done...
- Wicklund, Courtney
- Wilks, Gavin
- Williams, Brandon
- Xu, Fu
- Zuzelski, Joel
## Now we are done.
Register clicker here:
[https://goo.gl/nrebCr](https://goo.gl/nrebCr)
For me, the first homework was ...
1. entirely a review.
2. mostly a review, but it had a few new things in it.
3. somewhat of a review, but it had quite a few new things in it.
4. completely new for me.
I spent ... hours on the first homework.
1. 1-2
2. 3-4
3. 5-6
4. 7-8
5. More than 9
## Numerical Integration
<img src="./images/numerical_midpoint.gif" align="center" style="width: 600px";/>
Consider this trapezoid
<img src="./images/trapezoid_shape.png" align="left" style="width: 300px";/>
What is the area of this trapezoid?
1. $f(c)h$
2. $f(d)h$
3. $f(c)h + \frac{1}{2}f(d)h$
4. $\frac{1}{2}f(c)h + \frac{1}{2}f(d)h$
5. Something else
Note:
* Correct Answer: D
The trapezoidal rule for a function $f(x)$ gives the area of the $k$th slice of width $h$ to be,
$$A_{k} = \frac{1}{2}h\left(f(a+(k-1)h) + f(a+kh)\right)$$
What is the approximate integral, $I(a,b) = \int_a^b f(x) dx$, $I(a,b) \approx$
1. $\sum_{k=1}^N \frac{1}{2}h\left(f(a+(k-1)h) + f(a+kh)\right)$
2. $h\left(\frac{1}{2}f(a) + \frac{1}{2}f(b) + \frac{1}{2}\sum_{k=1}^{N-1}f(a+kh)\right)$
3. $h\left(\frac{1}{2}f(a) + \frac{1}{2}f(b) + \sum_{k=1}^{N-1}f(a+kh)\right)$
4. None of these is correct.
4. More than one is correct.
Note:
* Correct Answer: D (both A and C)
The trapezoidal rule takes into account the value and slope of the function. The next "best" approximation will also take into account:
1. Concavity of the function
2. Curvature of the function
3. Unequally spaced intervals
4. More than one of these
5. Something else entirely