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)
# Let the shaming begin
## 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