ࡱ> !#  QfPRTo|` 2Qbjbj *y^^^^~#~#~#*d    TX+OPz (R)))Z]L^dJLLLLLL$'hp~#a&UZ@aap^^)) , WWWaj^)")JWaJWW.(n!0>#@)D $~ 7qRVH<0OH~#(^_WR``^^^ppg^^^Oaaaa+++ +++ ***^^^^^^  MATLAB Workbook CME100 Vector Calculus for Engineers Second Edition Authors: Perrine Pepiot Vadim Khayms Table of Contents  TOC \o "1-3" \n \h \z  HYPERLINK \l "_Toc493266731" 1. Saving Workspace  HYPERLINK \l "_Toc493266732" 2. Defining scalar variables  HYPERLINK \l "_Toc493266733" 3. Vector operations ( 1 )  HYPERLINK \l "_Toc493266734" 4. Vector operations ( 2 )  HYPERLINK \l "_Toc493266735" 5. Vector operations ( 3 )  HYPERLINK \l "_Toc493266736" 6. Plots in 1 - D  HYPERLINK \l "_Toc493266737" 7. How to use a script  HYPERLINK \l "_Toc493266738" 8. If elseif else statements  HYPERLINK \l "_Toc493266739" 9. For loops  HYPERLINK \l "_Toc493266740" 10. While loops  HYPERLINK \l "_Toc493266741" 11. More programming examples: car motion  HYPERLINK \l "_Toc493266742" 12. More programming examples: electric circuit  HYPERLINK \l "_Toc493266743" 13. More programming examples: Newtons method  HYPERLINK \l "_Toc493266744" 14. Surfaces in 3 D  HYPERLINK \l "_Toc493266745" 15. Gradient  HYPERLINK \l "_Toc493266746" 16. Motion of a particle  HYPERLINK \l "_Toc493266747" 17. Level curves  HYPERLINK \l "_Toc493266748" 18. Finding local minima of a function  HYPERLINK \l "_Toc493266749" 19. Fitting data  HYPERLINK \l "_Toc493266750" 20. Vectors of random numbers  HYPERLINK \l "_Toc493266751" 21. More random numbers  HYPERLINK \l "_Toc493266752" 22. Performing numerical integration  HYPERLINK \l "_Toc493266753" 23. Computing work done by a force along a path  Getting started 1. Saving Workspace Commands : diary Start saving commands in a file diary(filename) Start saving commands in the file filename diary off Stop the diary The diary command allows saving all command window inputs and outputs (except graphics) in a separate file, which can be opened with most text editors (e.g. Microsoft Word) and printed. Start the diary and save the file as mydiary. Compute the number of seconds in one year. Close the diary. Open the file mydiary using any text editor. SOLUTION In the Command Window : >> diary(mydiary) >> 60*60*24*365 ans = 31536000 >> diary off In Microsoft Word : 60*60*24*365 ans = 31536000 diary off Hint : MATLAB workspace will be saved into a file named diary when no file name is specified. Hint: If you need help with a specific MATLAB command, simply type help command name in the command window. Detailed information will be displayed with everything needed to use this command properly. If you dont know the exact name of the command, type lookfor keyword, which returns all the commands related to that keyword.  YOUR TURN Start a new diary and call it newdiary. Compute the product of all the integers from 1 to 9. Close the diary. Open it with a text editor and print it. 2. Defining scalar variables Commands : + - / * ^ ; Basic algebraic operations clear Clear variables from the workspace % Insert a comment MATLAB can be used as a calculator and can store the inputs or the outputs into variables. If a semicolon ; is placed at the end of a statement, the value of the variable is calculated but not displayed on the screen. To clear variables from the workspace, use the command clear. SOLUTION >> % Define a >> a = 2 a = 2 >> % Define b >> b = 5 b = 5 >> % Compute c >> c = a + b^2 c = 27 >> % Compute a different value for c >> c = a + 2 * b ; >> % Display c >> c c = 12 >> % Clear all variables >> clear >> c ??? Undefined function or variable 'c' Hint: To display the current value of a variable, double-click on its name in the command window. Hint: To avoid retyping previously used command, use up and down arrows  YOUR TURN Set variable a to 12345679, variable b to 9, and assign your favorite number between 1 and 9 to c. Compute the product abc and display it on the screen. Clear all variables at the end. 3. Vector operations ( 1 ) Commands: linspace (x0, x1, N) Create an array of N equally spaced values between x0 and x1 x0 : x : x1 Create an array of values between x0 and x1 spaced by x PART 1 Create the vector  EMBED Equation.3 . Display each of the elements. Multiply the first and second elements. Change the third element to 4 and display the new vector on the screen. SOLUTION >> V = [1 2 3] V = 1 2 3 >> V (1) ans = 1 >> V (2) ans = 2 >> V (3) ans = 3 >> x = V (1)*V (2) x = 2 >> V (3) = 4 V = 1 2 4  YOUR TURN Create a vector containing 6 equally spaced elements from 0 to 10:  EMBED Equation.DSMT4  and display it on the screen. Replace the third element of your vector by the sum of the first and second elements. PART 2 Additional more convenient ways to create arrays with equally spaced elements are illustrated below: >> V = linspace ( 0 , 10 , 6 ) V = 0 2 4 6 8 10 >> W = 0 : 2 : 10 W = 0 2 4 6 8 10  YOUR TURN c) Create a vector containing elements spaced by 0.2 between 1 and 2. d) Create a vector containing 10 equally spaced elements between 1.5 and 2. Hint: You can check the dimension of an array in the command window using the size command. 4. Vector operations ( 2 ) Commands : .* ./ .^ element by element algebraic operations Define two vectors  EMBED Equation.3  and  EMBED Equation.3 . Multiply x and y element by element. Compute the squares of the elements of x. SOLUTION >> % Define x and y >> x=[1 -2 5 3]; >> y=[4 1 0 -1]; >> % Compute z >> z = x . * y z = 4 -2 0 -3 >> % Compute the squares of the elements of x >> x . ^ 2 ans = 1 4 25 9 Hint: A period is needed for element-by-element operations with arrays in front of *, / , and ^ . A period is NOT needed in front of + and - .  YOUR TURN Divide y by x element by element. Display the result. 5. Vector operations ( 3 ) Evaluate the function y given below between  EMBED Equation.3  and  EMBED Equation.3 :  EMBED Equation.3  To do so, first create a vector x containing equally spaced elements in the domain of the function. Then compute the corresponding vector y. Each element of y will be equal to the value of the function evaluated at the corresponding elements of x. SOLUTION >> x = 0 : 0.1 : pi; >> y = sin(2*x) . * exp(x) + x .^ 2 / 3 y = Columns 1 through 9 0 0.1203 0.2827 0.4889 0.7409 1.0404 1.3888 1.7873 2.2365 Columns 28 through 32 13.6493 13.3488 12.7582 11.8345 10.5330  YOUR TURN Compute the value of the following function of x between 1 and 3, using equally spaced points separated by 0.2:  EMBED Equation.3  6. Plots in 1 - D Commands : plot Create a scatter/line plot in 1-D title Add a title to the current figure xlabel, ylabel Add x and y labels to the current figure legend Add a legend to the current figure hold on / hold off Plot multiple curves in the same figure subplot Make several plots in the same window The displacement of an object falling under the force of gravity is given by:  EMBED Equation.3  where g is the acceleration of gravity [m/s2], y is the distance traveled [m], and t is time [s]. The following table gives the values of g for three different planets: PlanetgEarth9.80 m/s2Jupiter24.79 m/s2Mercury3.72 m/s2 PART 1 Plot the distance fallen as a function of time near the surface of the Earth between 0 and 10 seconds, with an increment of 1 second. Add a title and x and y labels. SOLUTION >> % Define time values >> t = 0:1:10; >> % Define acceleration of gravity >> g = 9.80; >> % Compute displacement y >> y = g * t.^2; >> % Plot >> plot(t,y,'b--') >> % Create labels >> grid on >> xlabel('Time [s]') >> ylabel('Distance fallen [m]') >> title('Distance fallen as a function of time on Earth')  Hint: You can define the color and the style of a curve when using the plot command. In the example below, b- - stands for blue dashed line. Other possible options are : for a dotted line, -. for a dash-dotted line, - for a solid line. The colors are designated by the first letter of the desired color (b,r,g,y k = black). If the style or the color is not specified, MATLAB uses its default settings. See help plot for additional information. Hint: You can zoom in and out, draw arrows, add titles, labels, legends, or write text directly in the figure using the menu bar.  YOUR TURN Repeat the same exercise for Mercury. PART 2 Multiple functions can be plotted on the same figure using the hold on and hold off commands. Plot the displacement curve for both the Earth and Jupiter in the same figure and add a legend. SOLUTION >> % Compute displacement for Earth >> g = 9.80; >> t = 0:1:10; >> y = g * t.^2; >> % Plot displacement for Earth >> plot(t,y,'b--') >> grid on >> hold on >> % Both functions are plotted in the same figure >> % Compute displacement for Jupiter >> g = 24.79; >> y = g * t.^2; >> % Plot displacement for Jupiter >> plot(t,y,'r-') >> % Create labels >> title('Distance fallen as a function of time') >> xlabel('Time [s]') >> ylabel('Distance fallen [m]') >> % Add a legend >> legend('Earth','Jupiter') >> % End of the hold command >> hold off  YOUR TURN Plot all three curves for the three planets in the same figure. Use different styles for each of the planets and include a legend PART 3 The three curves can also be plotted on the same figure in three different plots using the subplot command. Plot the curves for Earth and Jupiter using subplot. SOLUTION >> % Compute displacement for Earth >> t = 0:1:10; >> g = 9.80; >> y = g*t.^2; >> % Plot >> subplot (2,1,1) >> plot (t,y,'b--') >> xlabel ('Time [s]') >> ylabel ('Distance fallen [m]') >> title ('Distance fallen as a function of time on Earth') >> grid on >> % Define displacement for Jupiter >> g=24.79; >> y = g*t.^2; >> % Plot >> subplot(2,1,2) >> plot(t,y,'r-') >> xlabel('Time [s]') >> ylabel('Distance fallen [m]') >> title('Distance fallen as a function of time on Jupiter') >> grid on  YOUR TURN Using subplot, create three different displacement plots in the same window corresponding to each of the three planets. Programming 7. How to use a script Commands : clear Clear all variables close Close the current figure Open a new script (or .m) file. Plot the function  EMBED Equation.3  between 0 and 15 using 1000 points. Include labels and a title. Save your file as niceplot.m. In the command window, select the appropriate directory and execute the script by typing niceplot in the command window SOLUTION Open new .m file Type commands % Creating a nice plot % Clear all the variables clear % Close current window close % Defining the function x = linspace ( 0 , 15 , 1000 ); y = sin(x) .* x; % Plotting y plot ( x , y ) grid on title ( nice plot ) xlabel ( x ) ylabel ( y ) Save file as niceplot.m Execute script : >> niceplot  Hint: It is a good idea to always start your script with a clear command. This will help eliminate a lot of unexpected errors associated with using previously computed values when you execute the script multiple times Hint: To insert comments within a script, start each line with a % symbol. The comments will appear green. Hint: If you type help niceplot in the command window, the comments contained at the beginning of the file will be displayed. This is how all help information is stored and displayed for existing MATLAB functions  YOUR TURN Modify the above script by adding a second curve to the same plot:  EMBED Equation.3 . Use different styles for the two functions. Add a legend. 8. If elseif else statements Command : if condition statement elseif condition statement else statement end a) Suppose you are buying apples at a supermarket. If you buy more than 5 apples, you get a 10% discount. Write a program computing the amount owed given the price per apple (assume it to be $1) and the number of apples bought. b) Suppose now the store offers a super discount of 40% if you buy more than 20 apples. Modify the code in part a) to implement this new condition using the elseif statement. SOLUTION a) % Number of apples bought n = 6; % Price of one apple price = 1; if n >= 5 cost = 90/100 * price * n; else cost = price * n; end % Show the result cost >> apple cost = 5.4000 b) % Number of apples bought n = 21; % Price of one apple price = 1; if n >= 5 & n < 20 cost = 90/100 * price * n; elseif n >= 20 cost = 60/100 * price * n; else cost = price * n; end % Show the result cost >> apple cost = 12.6000 YOUR TURN If you really love apples, you can purchase a pre-packaged box with 30 apples for $15. Modify the code above to include this offer. Test your code for each of the following cases: n = 3, 11, 22 and 30. For each n, compute the cost by hand and then compare it to the result obtained by running your script. 9. For loops Command : for variable = initial value : increment : final value statement statement end a) Compute 10!, i.e. the product of all the integers from 1 to 10. b) Compute the sum of all even numbers from 0 to 100. c) Define the following piecewise function between  EMBED Equation.3  and  EMBED Equation.3  using 1000 points and plot it.  EMBED Equation.3  SOLUTION a) In a script called part a: % Initialization result = 1; % Computation of the successive products for i = 1:1:10 result = result * i; end % Show the result result In the command window : >> part1 result = 3628800 b) In a script called part b: % Initialization result = 1; % Computation of the sum for i = 0:2:100 result = result + i; end % Show the result result In the command window : >> part2 result = 2551 c) In a script called part c: % Define vector x x = linspace(-3,5,1000); % Test each value of x and compute the % appropriate value for y for i = 1:length(x) if x(i) < 0 y(i) = x(i); elseif x(i)>1 y(i) = log(x(i)); else y(i) = sin (pi * x(i)); end end % Plot y as a function of x plot(x,y)   YOUR TURN Write a script to compute the sum of all components of a vector. Your script should work regardless of the size of the vector. Test your script with using the following vector:  EMBED Equation.3 . Define and plot the function  EMBED Equation.3  for x ranging between 1 and 1, with x coordinates spaced by 0.05. 10. While loops Command : while condition statement end abs (x) returns the absolute value of x sqrt (x) returns the square root of x If x is an approximate value of the square root of a number a, then  EMBED Equation.3  is generally a better approximation. For example, with a = 2, starting with x = 1, we obtain the following successive approximations:  EMBED Equation.3  approaching in the limit the true value of  EMBED Equation.3  Using the above formula, write a script to compute iteratively the value of  EMBED Equation.3 . At each step, compare your solution to the exact value of  EMBED Equation.3  and stop your iteration when the difference between your solution and the exact one is less than  EMBED Equation.3 . Start the iteration with the initial guess x = 1. Plot the absolute value of the error versus the current iteration number. SOLUTION % Value of a a = 10; % First guess for the square root of a x(1) = 1; i = 1; % If the error is greater than 10^-4, continue iterating while abs (x (i) sqrt (10) ) >= 10^-4 i = i + 1; x ( i ) = 1/2 * (x (i-1) + a / x (i-1)); end % Show the result x ( i ) plot ( abs ( x-sqrt(10) ) ) grid on title('Error between sqrt(10) and the numerical approximation x') xlabel('Number of iterations') ylabel('error')   YOUR TURN In the above algorithm, x is a vector containing all the successive approximations of  EMBED Equation.3 . We are usually interested only in the final result and dont need to store each intermediate step. Modify the above script so that x is now a scalar iterate that overwrites the previous one. Compute  EMBED Equation.3 to an error of less than  EMBED Equation.3 , starting with x = 1. Using the modified code, determine how many iterations are necessary to reach the required accuracy? [Hint: add a counter whose value is incremented by one each time you perform an iteration] 11. More programming examples: car motion Starting from rest, a car (mass = 900 kg) accelerates under the influence of a force of 2000 Newtons until it reaches 13.89 m/s (50 km/h). What is the distance traveled ? How long does it take to travel this distance ? Newtons law:  EMBED Equation.3  (1) This equation determines the velocity of the car as a function of time. To obtain the position of the car, we use the following kinematic relation:  EMBED Equation.3  (2) Although this problem can be solved analytically, in this exercise we would like to come up with a numerical solution using MATLAB. We first need to discretize the equations, i.e. evaluate the position and velocity at equally spaced discrete points in time, since MATLAB doesnt know how to interpret continuous functions. The solution for  EMBED Equation.3  and  EMBED Equation.3  will then be represented by vectors containing the values of the velocity and the position of the car respectively at these discrete points in time separated by EMBED Equation.3 . The time derivative for the velocity in (1) can be approximated by:  EMBED Equation.3  Equation (1) then becomes:  EMBED Equation.3  or, if we solve for the velocity at step n+1:  EMBED Equation.3  , where the unknown is  EMBED Equation.3 . Similarly, equation (2) becomes:  EMBED Equation.3  and the position at step n+1 will given by:  EMBED Equation.3  Solve the above equations for position and velocity and plot the position as a function of time. Display the distance traveled and the time it takes to reach 50 km/h. SOLUTION % Force applied to the car in Newtons F = 2000; % Mass of the car in kg m = 900; % Time step in seconds deltat = 0.1 % Initialization: at t=0, the car is at rest t(1) = 0; v(1) = 0; x(1) = 0; n = 1; % Iterate until the velocity reaches 50 km/h = 13.89 m/s. For that, we need a WHILE loop which tests the value of the velocity at each step. If it's greater than 13.89 m/s, we exit the loop. while v ( n ) < 13.89 % Advance time t(n+1) = t(n) + deltat; % Compute the new velocity v(n+1) = v(n) + F/m*deltat; % Compute the new position x(n+1) = x(n) + v(n)*deltat; % Increment n n = n+1; end % Plot the position as a function of time plot(t,x) grid on title('position of the car as a function of time') xlabel('time [s]') ylabel('position [m]') % Display the position reached by the car (last element of the x vector) x(length(x)) % Time needed to go from 0 to 50 km/h (last element of the t vector) t(length(t)) In the command window: % Position reached by the car ans = 43.4000 % Time needed to reached 50 km/h ans = 6.3000   YOUR TURN When the speed of the car reaches 50 km/h, it stops accelerating and begins to slow down. At the beginning, the brakes are applied slowly, and as the time increases, the braking force increases. The breaking force exerted on the car can be modeled as follows:  EMBED Equation.3  in Newtons, with t in seconds. The equations of motion become:  EMBED Equation.3  and  EMBED Equation.3  The discretized equations are then written as follows:  EMBED Equation.3  and  EMBED Equation.3  The goal of this exercise is to compute the distance driven and the time needed for the car to slow down from 50 km/h to 10 km/h. Write a script to solve these equations numerically. Choose zero for both the initial position and time and 13.89 m/s for the initial velocity. Use the same time step as in the above example. Stop the iteration when the velocity reaches 2.78 m/s (10 km/h). Plot the position and velocity as a function of time in the same figure, using subplot. Add a title, labels, and a grid. Determine the time and the distance traveled to slow down to 10 km/h. 12. More programming examples: electric circuit A circuit has the following current-voltage characteristic (i.e. a curve giving the output current in Amps as a function of the input voltage V in Volts: Given the input voltage  EMBED Equation.3 , write a script that computes the output current  EMBED Equation.3 . Test your code for the following input:  EMBED Equation.3  for  EMBED Equation.DSMT4  Use the time step  EMBED Equation.3 . Plot  EMBED Equation.3  and  EMBED Equation.3  on the same set of axes for t ranging between 0 and 2 seconds. SOLUTION clear close % Define the input voltage as a function of t t = 0:0.1:2*pi; V = 3+3*sin(3*t); % Test each component of V and compute the % current output for i=1:length(V) if V(i)<1 I(i) = 10; elseif V(i)>1 & V(i)<5 I(i) = -9/4*V(i)+49/4; else I(i) = 1; end end % Plot the results plot(t,V,'-',t,I,'b-.') xlabel('Time') legend('Input Voltage','Output Intensity') title('Input and output of a circuit') grid on  YOUR TURN A physical circuit does not respond instantaneously. Suppose there is a delay of 0.3 seconds (3 time steps) between the input and the output, i.e. the current I at time t corresponds to the voltage V at time t -3t ). Modify the code above to include this delay. If the current is undefined at a certain time, assume it to be zero. Plot V and I on the same set of axes. Use different styles. Add a grid, a title, and a legend. 13. More programming examples: Newtons method A diode is a semiconductor component whose property is its tendency to conduct electric current in one direction only. When the voltage reaches a threshold value (typically around 700 mV), the diode conducts current in the forward direction. The current-voltage (I-V) characteristic of the diode used in this example is given by:  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4   EMBED Equation.DSMT4  at 300K Compute the current in the following circuit.         SOLUTION Using Kirschoffs voltage law:  EMBED Equation.DSMT4  We can substitute the current I by its expression as a function of  EMBED Equation.DSMT4 using the given by the I-V characteristic:  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  As a result we obtain an equation for  EMBED Equation.DSMT4 of the form  EMBED Equation.DSMT4 , which can be solved using the Newtons method. The current is then obtained using the diode I-V characteristic giving the current in the circuit as a function of  EMBED Equation.DSMT4 . The algorithm is quite simple: starting from an initial guess  EMBED Equation.DSMT4 , a better approximation of  EMBED Equation.DSMT4 is obtained from the following expression:  EMBED Equation.DSMT4  or, in general:  EMBED Equation.DSMT4 . In our case, the derivative of f is easily computed:  EMBED Equation.DSMT4  Perform several iterations of the Newtons method using the for loop. Find the diode voltage and the current in the circuit using 20 iterations. In a script: clear % Parameters of the problem E0 = 3; % Input voltage Is = 7*10^-14; % Value of Is R = 200; % Value of the resistor C = 0.026; % Value of KT/q % Initial guess for Vdiode Vdiode = 1; % Iterations for i = 1:20 % computation of the derivative of f evaluated at Vdiode fprime = -1 - Is * R / C * exp( Vdiode/C ); % computation of f evaluated at Vdiode f = E0 - Vdiode - Is * R * ( exp( Vdiode/C ) - 1); % New value of Vdiode Vdiode = Vdiode - f/fprime; end % Display solution % Vdiode Vdiode % Current in the circuit I = Is*(exp( Vdiode/C ) - 1) In the command window: >> newton Vdiode = 0.6718 I = 0.0116 YOUR TURN A for loop was used in the above script to perform the Newtons iteration. The iteration was terminated after a specified number of loops. In this case, we have performed a certain number of iterations without concerning ourselves with the accuracy of the computed solution. A more efficient approach would be to test for convergence and stop the iteration when the required level of accuracy has been achieved. This can be done using a while loop. The convergence criterion often used is the difference between the solutions obtained in two successive iterations. The loop is terminated when the difference is below a specified threshold value. a) Rewrite the above script using a while loop. Terminate the iteration when the absolute value of the difference between any two successive iterates for  EMBED Equation.DSMT4  is less than 10-6. Note: youll have to store the old value of  EMBED Equation.DSMT4  before computing the new one. b) How many iterations were required to achieve the specified accuracy? 14. Surfaces in 3 D Command : meshgrid Create a two-dimensional grid of coordinate values surface Create a 3D surface on the domain defined by meshgrid shading (interp) Interpolate colormap colorbar Add a colorbar Plot the function  EMBED Equation.DSMT4  for x ranging between -3 and 3, and y ranging between 4 and 4. Use an increment of 0.1 in both directions. Include labels and a title. Once the figure appears on the screen, rotate it using one of the menu buttons. Use the command shading(interp) to remove the discrete mesh. Add a colorbar to the figure. Note: The  EMBED Equation.DSMT4  in the denominator is needed to avoid division by zero when  EMBED Equation.DSMT4  and to get the correct value in the limit:  EMBED Equation.DSMT4  SOLUTION clear close all [x,y] = meshgrid(-3:0.1:3,-4:0.1:4); z = sin(x.^2+y.^2)./(x.^2+y.^2+10^-16); surface(x,y,z) grid on shading('interp') xlabel('x') ylabel('y') zlabel('z') title('Nice 3D plot') colorbar   YOUR TURN Plot the function  EMBED Equation.DSMT4  for x and y both ranging between  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Use an increment of 0.1. Do not use the shading option. Add labels and a title. Note: the plot you obtain represents a diffraction pattern produced by a small rectangular aperture illuminated by a monochromatic light beam. The function z represents the intensity of the light behind the aperture. Hint: You can change the limits by selecting the arrow in the figure menu and double-clicking the figure itself. A window will appear in which you can modify the key parameters. Try to re-define the limits on z to go from 0 to 0.1 to better resolve surface features. 15. Gradient Commands : gradient Computes the numerical derivative of a function Find the derivative of the following function:  EMBED Equation.3  for x ranging between 0 and 1, with an increment of 0.1. Plot both the analytical and the numerical derivatives in the same figure. SOLUTION clear close x=0:0.5:10; y=x.*exp(-x); derY = gradient(y,0.5); derYex = (1 - x).*exp(-x); plot(x,derY,'-') hold on plot(x,derYex,':') grid on title('derivative of the function y = xe^{-x}') legend('numerical derivative','exact derivative') xlabel('x') ylabel('y')  Hint: The spacing between points corresponds to the difference between the adjacent elements of vector x. YOUR TURN Compute analytically the derivative of the function EMBED Equation.3  Using the gradient function, compute the derivative of the function in a) for x ranging between 0 and 1, with an increment of 0.1. Plot the analytical and the numerical derivatives in the same figure. 16. Motion of a particle Command : plot3 Plot a curve in 3D The motion of a particle in three dimensions is given by the following parametric equations:  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  a) plot the particles trajectory in 3D using the plot3 function b) compute the velocity components  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  and the magnitude of the velocity vector c) compute the acceleration components  EMBED Equation.DSMT4  , EMBED Equation.DSMT4  and the magnitude of the acceleration vector d) compute the components of the unit tangent vector  EMBED Equation.DSMT4  e) compute the angle  EMBED Equation.DSMT4  between the velocity vector and the acceleration vector clear % time vector dt = 0.1; t = 0:dt:2*pi; % x, y and z vectors x = cos(t); y = 2*sin(t); z = 0*t; a) % plot trajectory in 3D plot3(x,y,z) grid on xlabel('x') ylabel('y') zlabel('z') b) % Computation of the velocity components % Vx = dx/dt Vx = gradient(x,dt); % Vy = dy/dt Vy = gradient(y,dt); % Speed speed = sqrt(Vx.^2+Vy.^2); c) % Computation of the acceleration components % Ax = dVx/dt Ax = gradient(Vx,dt); % Ay = dVy/dt Ay = gradient(Vy,dt); % Acceleration acceleration = sqrt(Ax.^2+Ay.^2); d) % Computation of the unit tangent vector components % Tx = Vx/|V| Tx = Vx./speed; % Ty = Vy/|V| Ty = Vy./speed; e) % Computation of the angle between the velocity and acceleration vectors % cos(theta) = V.A/(|V||A|) theta = acos((Vx.*Ax+Vy.*Ay)./speed./acceleration);  YOUR TURN The equations of motion of the particle are given below:  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  Redo all 5 steps illustrated in the example above Verify that the magnitude of your tangent vector at  EMBED Equation.DSMT4  is unity. Plot the velocity, acceleration and the angle between the velocity and the acceleration vectors using the subplot command. Comment on the values obtained for the acceleration for t equal to 0 and 2. Hint: To create several figures, type figure in the command window. A new window will appear. 17. Level curves Command : meshgrid Create two-dimensional arrays to be used by meshc meshc Plot a surface with level curves contour Plot level curves shading(interp) Remove mesh from a surface plot colorbar Add a colorbar Consider the following function :  EMBED Equation.3  EMBED Equation.DSMT4  a) Plot the function z for x ranging between 0 and 2, with an increment of 0.1 in both directions using the meshc command. This command will draw the three-dimensional surface along with a contour plot underneath. b) Make a contour plot of the function above using the contour command. This will only display the level curves in a horizontal plane. Include labels and a title. Hint: Using the contour command, you can specify the number of levels you would like to be displayed. The default number is 10. SOLUTION close all clear % Define the grid [x,y]=meshgrid(0:0.1:2*pi); % Define the function of two variables z = y.*cos(x).*sin(2*y); % Plot both surface and level curves meshc(x,y,z) grid on xlabel('x') ylabel('y') zlabel('z') title('meshc command') figure % Draw 20 level curves contour(x,y,z,20) grid on xlabel('x') ylabel('y') title('contour command')    YOUR TURN The temperature in the interior of a rectangular plate is given by the following function of x and y:  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  Plot the temperature as a function of x and y using meshc and contour. Use an increment of 0.05. Display 30 level curves. In both figures add a colorbar, labels, and a title. 18. Finding local minima of a function Command : fminsearch find a local minimum of a function a) For functions of a single variable: find the minimum of the function  EMBED Equation.DSMT4 , first analytically and then numerically using the fminsearch command. Hint : Unlike most MATLAB operations, you dont need to define a vector x before using fminsearch. Simply type in the function as shown below. fminsearch looks for the local minimum which is closest to the initial guess you specify as the third parameter.  EMBED Equation.DSMT4  SOLUTION Analytically: EMBED Equation.DSMT4  Numerically: >> fminsearch ( ' 1-x.*exp (-2 * x) ' , 0 ) ans = 0.5  b) For functions of several variables: find the minimum of the function  EMBED Equation.DSMT4  over the domain  EMBED Equation.DSMT4  using the fminsearch command. Follow these steps: plot the function over the given domain to approximately locate the minimum. - use these approximate values of the coordinates as your initial guess Hint: The syntax for functions of several variables is slightly different. Define  EMBED Equation.DSMT4 . Then in MATLAB, x and y will be represented by the two components:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , and the command becomes:  EMBED Equation.DSMT4  SOLUTION >> % Plot the surface to visualize the minimum >> [ x , y ] = meshgrid ( 0:0.1:pi , 0:0.1:pi ) ; >> f = cos(x.*y).*sin(x); >> surface(x,y,f); >> % The minimum is around the point [ 1.5 2 ] >> % Transform x and y in X(1) and X(2) >> fminsearch( ' cos( x(1) * x(2)) * sin(x(1)) ',[1.5 2]) ans = 1.57082350280598 1.99997995837548   YOUR TURN Consider the following function of two variables:  EMBED Equation.DSMT4  a) Analytically find the minimum of this function b) Plot the function on a suitably chosen domain to clearly see the minimum. Add a colorbar, labels, and a title c) Using fminsearch, locate the minimum numerically 19. Fitting data Command : polyfit find a polynomial that best fits a set of data Reynolds number is a non-dimensional parameter frequently used in fluid mechanics:  EMBED Equation.DSMT4  where V is the characteristic velocity of the flow  EMBED Equation.DSMT4 is the kinematic viscosity L is the characteristic length scale of the flow (such as the length or the diameter of a body) The Reynolds number indicates the relative importance of the viscous effects compared to the inertia effects. If the Reynolds number is low, viscous effects are predominant and if its large, viscosity can be ignored. Lets consider the force exerted on a sphere moving in the air. This drag force depends on the viscosity of the fluid, the velocity of the sphere and its dimension. Therefore, it is directly related to the Reynolds number. The drag force can be expressed as follows:  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the density of air  EMBED Equation.DSMT4  is the radius of the sphere  EMBED Equation.DSMT4  is the velocity of the sphere  EMBED Equation.DSMT4  is drag coefficient (non-dimensional) In 1851 Stokes had demonstrated that the drag coefficient at low Reynolds numbers was inversely proportional to Re:  EMBED Equation.DSMT4  (low  EMBED Equation.DSMT4 ) (1) A group of students have measured this coefficient for a sphere of radius r in a wind tunnel. Here are their results. Confirm that the data satisfies the Stokes law.  EMBED Equation.DSMT4 0.058750.10030.15850.47863.020 EMBED Equation.DSMT4 492256.2169.858.8810.86 Taking the logarithm of equation (1), we obtain:  EMBED Equation.DSMT4  (2) To check whether the Stokes law is satisfied, we can plot  EMBED Equation.DSMT4 versus  EMBED Equation.DSMT4  and see whether the data points fall along a straight line. Follow these steps: a) Create two vectors containing the logarithm of the experimental data  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  b) Find the line that fits best these points. Display its coefficients on the screen and compare them with the theoretical result. c) In the same figure plot the experimental data, the best fit line, and the theoretical line. Add a legend. Hint: The polyfit command finds a polynomial of a specified order that best fits the given data. In this example, we want to see if the experimental data falls along a straight line. We therefore choose 1 as the degree of the polynomial. Hint: The polyfit command returns a vector containing the coefficients of the polynomial:  EMBED Equation.DSMT4  corresponding to  EMBED Equation.DSMT4  Hint: To suppress the line connecting the experimental data points, select the big arrow in the figures menu and double-click on the line. A property editor window will appear. Change the line style to no line and choose circles for the marker style. SOLUTION close all clear all % Experimental data Re = [0.05875 0.1003 0.1585 0.4786 3.020 7.015]; Cd = [492 256.2 169.8 58.88 10.86 5.5623]; % Take the logarithm of these data lnRe = log(Re); lnCd = log(Cd); % Construct the line that fits best data. As we want a % line, the degree of the polynomial is set to 1. P = polyfit(lnRe,lnCd,1) % Define the fitting line, so it can be plotted lnRefit = linspace(log(0.05875),log(7.015),20); lnCdfit = P(1)*lnRefit + P(2); % Define the theoretical line : ln(Cd) = - ln(Re) + ln(24) lnRetheo = linspace(log(0.05875),log(7.015),20); lnCdtheo = log(24) - lnRetheo; plot(lnRe,lnCd) hold on plot(lnRefit,lnCdfit,'--') plot(lnRetheo,lnCdtheo,'-.') xlabel('ln(Re)') ylabel('ln(Cd)') title('drag coefficient on a sphere as a function of the Reynolds number') grid on P = -0.92980905266427 3.45384753973349 If the experimental data were exactly following Stokes law, P would be : Ptheo = -1.0000 3.17805383 The experiment are thus in good agreement with the theory   YOUR TURN The same students have repeated their experiments at Reynolds numbers between 104 and 105 and have found a constant drag coefficient of 0.4775. Assume the density of air EMBED Equation.DSMT4 is 1.25 kg/m3. The following table give the drag force on the sphere as a function of the air speed.  EMBED Equation.DSMT4 (m/s)46789 EMBED Equation.DSMT4  (N)0.0360.092980.106820.15680.193 Find the radius of the sphere used in the experiments. Follow these steps: a) In the definition of the drag force:  EMBED Equation.DSMT4  the group  EMBED Equation.DSMT4 is now a constant, since the drag coefficient is assumed to be constant. That means that the plot of  EMBED Equation.DSMT4  versus  EMBED Equation.DSMT4 should be close to a straight line. Using polyfit, find an equation of a line that best fits the data given in the table b) Using the results in part a) extract the value of  EMBED Equation.DSMT4  c) Using the numerical values provided, estimate the radius of the sphere 20. Vectors of random numbers Command : randn generate a vector whose components are chosen at random hist plot a histogram At the assembly line of a big aircraft company, bolts are stored in boxes containing 1000 parts each. The bolts are supposed to have a nominal diameter of 5 millimetres. But due to the uncertainty in the manufacturing process, some turn out to be larger and some turn out to be smaller than nominal. a) Create a random vector whose components are the diameters of each bolt in a box using the randn command. The syntax of the randn command is as follows: randn ( M , N ) * A + B, where: - M and N determine the size of the output vector, i.e. the number of random values generated. M corresponds to the number of rows and N to the number of columns. In our example, we want random diameters for 1000 bolts, so the output will be a vector with M = 1000 and N = 1 - A (called standard deviation) is related to the uncertainty in the diameter of the bolts. Here, A = 0.25 mm. This means that about 67% of the bolts have diameters within 0.25 mm of the nominal diameter. - B is the nominal diameter of the bolts. Here, B = 5 mm b) Plot a histogram showing the number of bolts as a function of their diameter SOLUTION a) >> x = randn(1000,1) * 0.25 + 5; b) >> hist(x,30) >> title(Bolts contained in a box) >> xlabel(Diameter in mm) >> ylabel(Number of bolts)  Hint : The hist command takes the vector generated by the randn command as input and counts the number of bolts having a diameter within a certain range (a bin). The number of bins is defined by the second parameter.  YOUR TURN On the same assembly line, nuts are produced in boxes containing 5000 parts each. All nuts have a nominal diameter of 5.5 mm, however, the nuts in each box have been machined to different tolerances. Using subplot, plot the distributions of nuts as a function of their size for each of the four boxes for which the standard deviations are respectively A=0.45 mm, 0.50 mm, 0.55 mm and 60 mm. 21. More random numbers Command : mean Compute the average of the components of a vector stem Create a stem plot Highway patrol uses a radar to measure the speed of the cars on a freeway. The speed limit is 55 mph. The officers pull over all cars whose speed is greater than 80 mph and write a ticket for $150 each. If 8000 vehicles are observed on the highway each day, on the average, how much money does the highway patrol collect each day ? Compute the average over one full year and assume A = 10 in the randn command. SOLUTION clear all close all % the average of the daily amount is done over a year for i = 1:365 % create a vector whose components are the speed of % each car taking the freeway x = randn(8000,1)*10 + 55; % Initialize the amount collected each day S(i) = 0; % test the speed of each vehicle. If it's greater than 80, add % 150 to S for k = 1:length(x) if x(k) > 80 S(i) = S(i) + 150; end end end % Compute the average of vector S containing the amount % collected each day during a year Savg = mean(S) % Plot the ammount of money collected each day stem(S) xlabel('days') ylabel('amount') Savg = 7374.65753424658   YOUR TURN Modify the above script compute the average percentage of cars that have a speed of more than 10 mph over the speed limit. Compute your daily average over one year. 22. Performing numerical integration Command : trapz Computes the integral of a function over a given domain using the trapezoidal method Compute the integral of the function  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  using the trapezoidal method. Compare the result to the exact analytical solution. SOLUTION clear close x = linspace(0,1,100) y = x.^2+x+1; I = trapz(x,y) I = 1.8333 Analytical result : I = 11/6 = 1.8333  YOUR TURN a) Compute the following integral analytically:  EMBED Equation.DSMT4  b) Compute the same integral numerically using the trapz command. Use an increment of 0.01 c) Repeat part b), this time with an increment of 0.2. Whats your conclusion? 23. Computing work done by a force along a path Our sun continuously radiates energy in the form of electromagnetic waves. When these waves are reflected off of or absorbed by a surface, pressure is applied. These infinitesimal forces cause a small acceleration, which, in the case of a satellite in orbit, modifies its path. In this example, well compute the work done by the force due to solar pressure acting on a satellite travelling in an elliptical orbit around the sun. The force due to solar radiation can be written as  EMBED Equation.DSMT4  where: c is the speed of light. c = 3.108 m/s G is the incident solar power flux  EMBED Equation.DSMT4 Watt/m2 A is the area of the spacecraft normal to the sun. The area is assumed to be constant A = 8 m2  EMBED Equation.DSMT4  unit vector directed from the spacecraft toward the sun The orbit is planar and is given parametrically by the following equation:  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4  Find the work done by the force of solar radiation on the spacecraft over the following interval ( EMBED Equation.DSMT4 ). Use the time step of  EMBED Equation.DSMT4 . SOLUTION The work done by a force is computed using the following formula:  EMBED Equation.DSMT4   is the angle between the velocity vector and the force. Each component of the integral is computed separately. Then the integral is computed using the trapz command. clear % parameters G = 1350; A = 8; c = 3*10^8; % definition of the orbit t = 0:0.001:pi/2; % parameters of the ellipse a = 1.5*10^11; b = 1.3*10^11; % definition of the parametric curve x = a*cos(t); y = b*sin(t); % velocity of the spacecraft velx = gradient(x,0.001); vely = gradient(y,0.001); % magnitude of the velocity velmag = sqrt(velx.^2+vely.^2); % acceleration of the spacecraft accx = gradient(velx,0.001); accy = gradient(vely,0.001); % magnitude of the acceleration accmag = sqrt(accx.^2+accy.^2); % magnitude of the force due to solar radiations f = -1/c*G*A; % cosine of the angle between the force and the velocity % vector costheta = (velx.*accx+vely.*accy)./velmag./accmag; % computation of the work done for t between 0 and pi W = trapz(f*velmag.*costheta,t) In the command window: W = 7.1195e+005 YOUR TURN Suppose that the spacecraft is now travelling in an orbit specified by the following parametric equation:  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4 . 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