** 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