From f59670c3c661c5e87b6f141c29daa2670dc78b0e Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 17 Jan 2017 20:55:43 -0500 Subject: [PATCH 1/7] lecture 1 notebook and slides --- lecture_01/lecture_01.md | 108 ++++++++++++++++++++++++++++ lecture_01/lecture_01.pdf | Bin 0 -> 319099 bytes lecture_01/output_10_0.svg | 143 +++++++++++++++++++++++++++++++++++++ 3 files changed, 251 insertions(+) create mode 100644 lecture_01/lecture_01.md create mode 100644 lecture_01/lecture_01.pdf create mode 100644 lecture_01/output_10_0.svg diff --git a/lecture_01/lecture_01.md b/lecture_01/lecture_01.md new file mode 100644 index 0000000..b347272 --- /dev/null +++ b/lecture_01/lecture_01.md @@ -0,0 +1,108 @@ + +# Freefall Model +## Octave solution (will run same on Matlab) + + +```octave +%plot --format svg +``` + + +```octave +set (0, "defaultaxesfontname", "Helvetica") +set (0, "defaultaxesfontsize", 18) +set (0, "defaulttextfontname", "Helvetica") +set (0, "defaulttextfontsize", 18) +set (0, "defaultlinelinewidth", 4) +``` + +Define time from 0 to 12 seconds + + +```octave +t=[0,2,4,6,8,10,12]' +``` + + t = + + 0 + 2 + 4 + 6 + 8 + 10 + 12 + + + +Define constants and analytical solution (meters-kilogram-sec) + + +```octave +c=0.25; m=60; g=9.81; v_terminal=sqrt(m*g/c); + +v_analytical = v_terminal*tanh(g*t/v_terminal) +``` + + v_analytical = + + 0.00000 + 18.61630 + 32.45521 + 40.64183 + 44.84646 + 46.84974 + 47.77002 + + + + +```octave +v_numerical=zeros(length(t),1); +for i=1:length(t)-1 + v_numerical(i+1)=v_numerical(i)+(g-c/m*v_numerical(i)^2)*2; +end +v_numerical +``` + + v_numerical = + + 0.00000 + 19.62000 + 36.03213 + 44.83284 + 47.70298 + 48.35986 + 48.49089 + + + +Display time, velocity (analytical) and velocity (numerical) + + +```octave +fprintf('time (s)|vel analytical (m/s)|vel numerical (m/s)\n') +fprintf('-----------------------------------------------') +M=[t,v_analytical,v_numerical]; +fprintf('%7.1f | %18.2f | %15.2f\n',M(:,1:3)'); +``` + + time (s)|vel analytical (m/s)|vel numerical (m/s) + ----------------------------------------------- + 0.0 | 0.00 | 0.00 + 2.0 | 18.62 | 19.62 + 4.0 | 32.46 | 36.03 + 6.0 | 40.64 | 44.83 + 8.0 | 44.85 | 47.70 + 10.0 | 46.85 | 48.36 + 12.0 | 47.77 | 48.49 + + + +```octave +plot(t,v_analytical,'-',t,v_numerical,'o-') +``` + + +![svg](output_10_0.svg) + diff --git a/lecture_01/lecture_01.pdf b/lecture_01/lecture_01.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c54f29efb484d0c493c9867e5083470b86307859 GIT binary patch literal 319099 zcmeFZ1yo#1_Ai_Sf;%C&6G(7(2@)iDaCdjt#x)5LAb4;K!QGtz!QC2Y2u|Zp1I^du z&b@Qz&HcXrtheTUYu;K@yH|IeU3Kcz`PJUL_UW^$UMNXOGO;mpqP*zN>u%|;>&`@B zr(mUUFt$b!6l9Szv$t@yq~L&Y-myqp*}9rJvq;(+xtd9snK+o52@9jRxH_8|*`auT zNr(|u>}Ev^J-o&fRCH$ovJ+L@m`dw?Y3L|TE#Hb7;z`d!P!Q!)PKM9PKQk*j zo}BL7i0`3GroT{$RhTzbBuuH})Tald%U7p*3q*W}&X=%pHb0XUL3clT`mxVh0>#YU z^lwnWau4XCJb?C)Vdvt0DE-Cb`X^|){{?7k)#M$3tZ1z_8r!-Q8PYO!@B|1reZqU{ z9UU0m_BK4v_5;qV+C}`N-BU~Koe4hHK`4s)`lM!N456y+QCtE#xmNMT#JKOW&cWj6 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of file From 454e500fd3ec0e5b949d165e33903dcb9f9ec019 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Wed, 18 Jan 2017 12:28:12 -0500 Subject: [PATCH 2/7] lecture 1 notebook svg display --- lecture_01/lecture_01.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/lecture_01/lecture_01.md b/lecture_01/lecture_01.md index b347272..ca5a8a5 100644 --- a/lecture_01/lecture_01.md +++ b/lecture_01/lecture_01.md @@ -104,5 +104,6 @@ plot(t,v_analytical,'-',t,v_numerical,'o-') ``` -![svg](output_10_0.svg) +![plot of +velocities](https://github.uconn.edu/rcc02007/ME3255S2017/blob/master/lecture_01/output_10_0.svg) From 97780b0edad4ecf3997961542fab7053659bf0e4 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Wed, 18 Jan 2017 12:30:06 -0500 Subject: [PATCH 3/7] lecture 1 notebook svg display 2 --- lecture_01/lecture_01.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/lecture_01/lecture_01.md b/lecture_01/lecture_01.md index ca5a8a5..807e888 100644 --- a/lecture_01/lecture_01.md +++ b/lecture_01/lecture_01.md @@ -106,4 +106,6 @@ plot(t,v_analytical,'-',t,v_numerical,'o-') ![plot of velocities](https://github.uconn.edu/rcc02007/ME3255S2017/blob/master/lecture_01/output_10_0.svg) + From 3f723f4ec210e8e836bc29f0a63c7fca37b50d73 Mon Sep 17 00:00:00 2001 From: "Ryan C. 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zu|WPGV{k!vA1UJhIR*sG`Ny~rcJ9YodjHUI@ctR&;{Ia{?#I>mXIpL#j>qz4|D!*4 zu1Cj?KVqEFKXs7bb@KkvmK*xdIkQ6`T>qTkW9*N2&jI0jT%CXQhv{IaXKDen`^$hu z$<*!9NQCpZr--!!@NWtmBxz-24g72SJ(gCK6ai|y*4HyM(C32caTpo$LO3{iICu;o qMuv|YP>+L$o6``&5B&cne__7%4tjPDf9)%d$9)9Gq^B2^6Z $\Delta t$=`delta_time` + +## Another example of truncation error is a Taylor series (or Maclaurin if centered at a=0) + +Taylor series: +$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\frac{f'''(a)}{3!}(x-a)^{3}+...$ + +We can approximate the next value in a function by adding Taylor series terms: + +|Approximation | formula | +|---|-------------------------| +|$0^{th}$-order | $f(x_{i+1})=f(x_{i})+R_{1}$ | +|$1^{st}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+R_{2}$ | +|$2^{nd}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\frac{f''(x_{i})}{2!}h^{2}+R_{3}$| +|$n^{th}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\frac{f''(x_{i})}{2!}h^{2}+...\frac{f^{(n)}}{n!}h^{n}+R_{n}$| + +Where $R_{n}=\frac{f^{(n+1)}(\xi)}{(n+1)!}h^{n+1}$ is the error associated with truncating the approximation at order $n$. + +The $n^{th}$-order approximation estimates that the unknown function, $f(x)$, is equal to an $n^{th}$-order polynomial. + +In the Freefall example, we estimated the function with a $1^{st}$-order approximation, so + +$v(t_{i+1})=v(t_{i})+v'(t_{i})(t_{i+1}-t_{i})+R_{1}$ + +$v'(t_{i})=\frac{v(t_{i+1})-v(t_{i})}{t_{i+1}-t_{i}}-\frac{R_{1}}{t_{i+1}-t_{i}}$ + +$\frac{R_{1}}{t_{i+1}-t_{i}}=\frac{v''(\xi)}{2!}(t_{i+1}-t_{i})$ + +or the truncation error for a first-order Taylor series approximation is + +$\frac{R_{1}}{t_{i+1}-t_{i}}=O(\Delta t)$ + + +1. digital representation of a number is rarely exact + +2. arithmetic (+,-,/,\*) causes roundoff error + + +```octave +fprintf('%1.20f\n',double(pi)) +fprintf('%1.20f\n',single(pi)) +``` + + 3.14159265358979311600 + 3.14159274101257324219 + + + +```octave + +``` diff --git a/lecture_02/lecture_02.ipynb b/lecture_02/lecture_02.ipynb new file mode 100644 index 0000000..3e0a9e6 --- /dev/null +++ b/lecture_02/lecture_02.ipynb @@ -0,0 +1,597 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Solution to Form #1" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 1 2 3\n", + "\n" + ] + } + ], + "source": [ + "[1,2,3]" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "[1,2,3]*[1;2;3]=?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The first source of error is roundoff error\n", + "## Just storing a number in a computer requires rounding" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "realmax = 1.79769313486231570815e+308\n", + "realmin = 2.22507385850720138309e-308\n", + "maximum relative error = 2.22044604925031308085e-16\n" + ] + } + ], + "source": [ + "fprintf('realmax = %1.20e\\n',realmax)\n", + "fprintf('realmin = %1.20e\\n',realmin)\n", + "fprintf('maximum relative error = %1.20e\\n',eps)\n" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 1\r\n" + ] + } + ], + "source": [ + "s=1;\n", + "for i=1:1000\n", + " s=s+eps/10;\n", + "end\n", + "s==1" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Freefall Model (revisited)\n", + "## Octave solution (will run same on Matlab)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Set default values in Octave for linewidth and text size" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "set (0, \"defaultaxesfontname\", \"Helvetica\")\n", + "set (0, \"defaultaxesfontsize\", 18)\n", + "set (0, \"defaulttextfontname\", \"Helvetica\")\n", + "set (0, \"defaulttextfontsize\", 18) \n", + "set (0, \"defaultlinelinewidth\", 4)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Define time from 0 to 12 seconds with `N` timesteps \n", + "function defined as `freefall`" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "function [v_analytical,v_terminal,t]=freefall(N)\n", + " t=linspace(0,12,N)';\n", + " c=0.25; m=60; g=9.81; v_terminal=sqrt(m*g/c);\n", + "\n", + " v_analytical = v_terminal*tanh(g*t/v_terminal);\n", + " v_numerical=zeros(length(t),1);\n", + " delta_time =diff(t);\n", + " for i=1:length(t)-1\n", + " v_numerical(i+1)=v_numerical(i)+(g-c/m*v_numerical(i)^2)*delta_time(i);\n", + " end\n", + " % Print values near 0,2,4,6,8,10,12 seconds\n", + " indices = round(linspace(1,length(t),7));\n", + " fprintf('time (s)|vel analytical (m/s)|vel numerical (m/s)\\n')\n", + " fprintf('-----------------------------------------------\\n')\n", + " M=[t(indices),v_analytical(indices),v_numerical(indices)];\n", + " fprintf('%7.1f | %18.2f | %15.2f\\n',M(:,1:3)');\n", + " plot(t,v_analytical,'-',t,v_numerical,'o-')\n", + "end\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "time (s)|vel analytical (m/s)|vel numerical (m/s)\n", + "-----------------------------------------------\n", + " 0.0 | 0.00 | 0.00\n", + " 2.0 | 18.76 | 18.82\n", + " 4.0 | 32.64 | 32.80\n", + " 6.1 | 40.79 | 40.97\n", + " 8.0 | 44.80 | 44.94\n", + " 10.0 | 46.84 | 46.93\n", + " 12.0 | 47.77 | 47.82\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t30\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t50\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + 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"source": [ + "# Types of error\n", + "## Freefall is example of \"truncation error\"\n", + "### Truncation error results from approximating exact mathematical procedure" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We approximated the derivative as $\\delta v/\\delta t\\approx\\Delta v/\\Delta t$\n", + "\n", + "Can reduce error by decreasing step size -> $\\Delta t$=`delta_time`" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Another example of truncation error is a Taylor series (or Maclaurin if centered at a=0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Taylor series:\n", + "$f(x)=f(a)+f'(a)(x-a)+\\frac{f''(a)}{2!}(x-a)^{2}+\\frac{f'''(a)}{3!}(x-a)^{3}+...$\n", + "\n", + "We can approximate the next value in a function by adding Taylor series terms:\n", + "\n", + "|Approximation | formula |\n", + "|---|-------------------------|\n", + "|$0^{th}$-order | $f(x_{i+1})=f(x_{i})+R_{1}$ |\n", + "|$1^{st}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+R_{2}$ |\n", + "|$2^{nd}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\\frac{f''(x_{i})}{2!}h^{2}+R_{3}$|\n", + "|$n^{th}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\\frac{f''(x_{i})}{2!}h^{2}+...\\frac{f^{(n)}}{n!}h^{n}+R_{n}$|\n", + "\n", + "Where $R_{n}=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}h^{n+1}$ is the error associated with truncating the approximation at order $n$.\n", + "\n", + "The $n^{th}$-order approximation estimates that the unknown function, $f(x)$, is equal to an $n^{th}$-order polynomial. \n", + "\n", + "In the Freefall example, we estimated the function with a $1^{st}$-order approximation, so \n", + "\n", + "$v(t_{i+1})=v(t_{i})+v'(t_{i})(t_{i+1}-t_{i})+R_{1}$\n", + "\n", + "$v'(t_{i})=\\frac{v(t_{i+1})-v(t_{i})}{t_{i+1}-t_{i}}-\\frac{R_{1}}{t_{i+1}-t_{i}}$\n", + "\n", + "$\\frac{R_{1}}{t_{i+1}-t_{i}}=\\frac{v''(\\xi)}{2!}(t_{i+1}-t_{i})$\n", + "\n", + "or the truncation error for a first-order Taylor series approximation is\n", + "\n", + "$\\frac{R_{1}}{t_{i+1}-t_{i}}=O(\\Delta t)$\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "1. digital representation of a number is rarely exact\n", + "\n", + "2. arithmetic (+,-,/,\\*) causes roundoff error" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "3.14159265358979311600\n", + "3.14159274101257324219\n" + ] + } + ], + "source": [ + "fprintf('%1.20f\\n',double(pi))\n", + "fprintf('%1.20f\\n',single(pi))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": 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Cooper" Date: Thu, 19 Jan 2017 12:19:28 -0500 Subject: [PATCH 5/7] HW 1 add collaborator --- HW1/README.md | 4 ++++ HW1/g9.png | Bin 0 -> 78564 bytes 2 files changed, 4 insertions(+) create mode 100644 HW1/g9.png diff --git a/HW1/README.md b/HW1/README.md index 426c9c5..bb1c7b4 100644 --- a/HW1/README.md +++ b/HW1/README.md @@ -55,3 +55,7 @@ vii. Paste this into the Homework #1 Google form reponse. (https://goo.gl/forms/jINjsioRLwQOiYtl2)[https://goo.gl/forms/jINjsioRLwQOiYtl2] +![Step 9](g9.png) + + viii. Click on the gear for "Settings" then "Collaborators" on the left menu. Add + `rcc02007` (Ryan C. 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zn_jctKh>l=yLw2vvbgw44x@cReXnRr2}$3Xors4-XxZzp+0N^*ZIQ@ix#Qkj^u0Qk z%~m~`X(gR=o-*ELLBCxV4RQO(k0Fk}YEmjGABbzy6nR-XUflT~@Fzl_W_Q2K`l90} zNGbaJdmKd3{ZBK={9iGg|7E)W57s(fIGdvI1M}|NpeTOs+?V|%@%!4GJ@JDCUpM<< z`ssM|!yXhb``<)Z8z!m^F){DbzrSDJT8IijOzh{4KZEl<4}NCO&*~68Y{h=Ij{lJM ZDCKpW1}v&+^!{F*U{mWWB;)Ie{{{b$@e}|6 literal 0 HcmV?d00001 From 88f2c758bdfad34a22abe59fb52de2fbb7d0d12e Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Thu, 19 Jan 2017 15:15:00 -0500 Subject: [PATCH 6/7] office hours updated --- README.md | 2 +- lecture_02/lecture_02.pdf | Bin 0 -> 126338 bytes 2 files changed, 1 insertion(+), 1 deletion(-) create mode 100644 lecture_02/lecture_02.pdf diff --git a/README.md b/README.md index a7c930e..ca1565e 100644 --- a/README.md +++ b/README.md @@ -19,7 +19,7 @@ matlab/octave functions and programming best practices. **Instructor**: Prof. Ryan C. Cooper (ryan.c.cooper@uconn.edu) -**Office hours**: Fridays 10am-12pm in Engineering II room 315 +**Office hours**: Mon 2:30-4:30pm and Thur 11am-1pm in Engineering II room 315 ## Teaching Assistants: - Graduate: **TBD** diff --git a/lecture_02/lecture_02.pdf b/lecture_02/lecture_02.pdf new file mode 100644 index 0000000000000000000000000000000000000000..edad8fc5e4a8695f0f4c2e61bb7b05af44edab0d GIT binary patch literal 126338 zcmb5Vb9AN8(l$J?ZQI7g<|Ns%ZBOiEV%xTD+vdc!ZA|oLe&?L$d(OAsf8M?Jt?RC? z?&|8Qu6?bw$>c=D=$PogLz6XLRJ}s85Hb?l7+65_@DMTpjIB)_%?Me)AO&bbLP7>H z^Pi5!_FvbZ`i{mT#)dXV#?X9x&<>9F#`;#!t{E5FlTrH|QC+W8yI`~V-ckZY2p|*2 zHZZ6)!9)1W_0*=dBkFo4TLfwfyV1nwwnnR`VstZcV9N2D7AD_l(IG_7;&=pON z7>J5FFD^P`BpfRjXGf-Xy^(- zZ8Reuk`pd6=-$SYuRc>gZa<8&wui=_o=%!=hA?5g;XmH);s@ZNVraxcaw~D$wy-8&dhmmo6v=2ddVeG_qmU%$RGcArd zlMoF{w|?ZNtl+foON9}J3~C~Zl-GAw-QC4oD(}EdWB4)<>yoyNxYph)*L5kA6!P?Y z(D%|iRP;#2jA0>mXe)G7Ff_2uXgSzPaJ=s!_xl|O-U21?6vqT23#VFnzA`h6Y8Pec 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zF|q)4`tNy}8G$YQ!^XtO3iz`;U^>P>@-i|o{LRmdOpF}Bcm==J0qhUPe^VX@>!0_= z%*@L4w|ajz=J)a#n3(|_z)G zp2eT(IDQ^Xf6vRz%JAp?vT?8h{+5@6js4HO>)I|Jb7Qu42H^Nx`67o{v1QV^@`=^X8L=1 zz(wLOd7T{efpW^>*P{$2a}VR6`_WI)p=fLCMDTMH06ri3c`adMVoUIAm;Lp8!O= Date: Tue, 24 Jan 2017 11:01:13 -0500 Subject: [PATCH 7/7] welcome to Peiyu --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index ca1565e..eebd665 100644 --- a/README.md +++ b/README.md @@ -22,8 +22,8 @@ matlab/octave functions and programming best practices. **Office hours**: Mon 2:30-4:30pm and Thur 11am-1pm in Engineering II room 315 ## Teaching Assistants: -- Graduate: **TBD** -- Office hours: 2 hours / week in office **TBD** +- Graduate: Peiyu Zhang +- Office hours: 2 hours / week **Prerequisite:** CE 3110, MATH 2410Q