Contents. 1 Matlab basics How to start/exit Matlab Changing directory Matlab help... 2

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1 Contents 1 Matlab basics How to start/exit Matlab Changing directory Matlab help Symbolic variables Declaration Substitution Differentiation Integration Solving systems of equations Simplifying results Easier format Built-in mathematical functions 4 4 Visualization Plotting surfaces z = f(x, y) Plotting more than one surface Matlab m-files Comments in an m-file Suppressing output on screen Numeric calculations (approximations) Integration Solving systems

2 Matlab introduction for Math 2204 Scope This is not an extensive Matlab guide. This is only an overview of some basic Matlab commands that are useful for Math2204: Multivariable Calculus. 1 Matlab basics 1.1 How to start/exit Matlab Click on the Matlab icon if there is any, or activate Matlab via Start-Applications- Matlab (for PCs) or Finder-Applications-Matlab (for Macs). A Matlab Command Window should appear in which you can type commands. 1.2 Changing directory If you didn t start Matlab from the directory in which your m-files are, set the correct directory through the menu bar. 1.3 Matlab help If you don t know exactly how to use a built-in Matlab function, say plot, type in your Matlab window: help plot. This will tell how to use it. 2 Symbolic variables 2.1 Declaration To declare symbols as a symbolic variable, use syms. To declare x and y as a symbolic variable, use Then you can use x and y to define functions. For example, for f(x, y) = 2x + y, use 2.2 Substitution To substitute a symbolic variable in an expression by a value, use subs. To substitute x = 1 and y = 2 in the function f(x, y) = 2x + y, use f0 = subs(f, [x y], [1 2]) f = 4 2

3 2.3 Differentiation Matlab s diff can be used to differentiate a (symbolic) function. To find the partial derivatives f x and f xy of f(x, y) = 2x + y, use: fx = diff(f, x) fxy = diff(fx, y) fx = 2 fxy = Integration Matlab s int can be used to integrate a (symbolic) function. x To find f(x, y) dy for f(x) = 2x + y and f(x, y) dy, use: 0 I1 = int(f, y) I2 = int(f, y, 0, x) I1 = y*(4*x + y) / 2 I2 = 5*x^2 / Solving systems of equations Matlab s solve can be used to solve a system of equations, if an analytic solution can be found. To solve f 1 (x, y) = (x 1)(y 3) = 0 f 2 (x, y) = x(y 2) = 0 for x and y, use: f1 = (x-1)*(y-3) f2 = x*(y-2) sol = solve(f1, f2, x, y) sol.x sol.y Then sol.x has all solutions for x, i.e. 0 and 1 In sol.y are the corresponding y values, i.e. 3 and 2. Thus the solutionbs are (0, 3) and (1, 2). 3

4 2.6 Simplifying results Matlab doesn t always simplify expressions. You can enforce this using simplify f = (x^2 - y^2) / (x + y) f = simplify(f) gives the result in simplified form f = x - y 2.7 Easier format To obtain results in a format that is easier to read, use Matlab s pretty. I1 = int(f, y) pretty(i1) y(2x + y) gives. 2 3 Built-in mathematical functions You can use the following built-in mathematical functions in Matlab. abs(x) sin(x) cos(x) sqrt(x) log(x) log10(x) exp(x) absolute value of x sine of x cosine of x square root of x natural logarithm of x 10 base logarithm of x exponential of x 4 Visualization 4.1 Plotting surfaces z = f(x, y) For plotting a surface z = f(x, y) use ezsurf. ezsurf(f, [xmin xmax ymin ymax]) plots the symbolic function f over the domain x min x x max and y min y y max. To plot the function f(x, y) = 2x + y over the interval 1 x 1 and 0 y 2, use ; 4

5 ezsurf(f, [ ]) 4.2 Plotting more than one surface If you need to plot more than 1 surface in one figure, use hold on after the first plotcommand. To plot the functions f(x, y) = 2x+y and g(x, y) = x+2y over the interval 1 x 1 and 0 y 2, use ; g = x + 2*y ezsurf(f, [ ]) hold on; ezsurf(g, [ ]) Matlab will keep on adding to the same plot until you use hold off. 5 Matlab m-files You can open a new Matlab m-file by using from the menu bar File/New/Script which will open a new window in which you can write the Matlab M-file. If you prefer to use another editor make sure that the m-file has extension.m (and make sure it is in the correct directory). When the script is written, save it ( Save from menu bar). Then run the script: use Run from menu bar or type the filename of the script in the Command Window. 5.1 Comments in an m-file Comment lines start with a %. Everything after the % will be treated as a comment. 5.2 Suppressing output on screen Ending a line with a ; suppresses output on the screen. For example, for 2 symbolic variables x and y, ; will not show any output on your screen, while will show f on the screen. 5

6 6 Numeric calculations (approximations) 6.1 Integration Matlab s integral2 can be used to approximate a double integral. To approximate 1 x 0 0 yx 2 + x dy dx, use: f y) y.*x.^2 + x xmin = 0 xmax = 1 ymin = 0 ymax y) x I = integral2(f, xmin, xmax, ymin, ymax) I = The variables f, xmin etc are only for clarity. You may substitute the right-hand sides directly in integral2. Note the. which indicates elementwise operations with vectors x and y. This is necessary for multiplication.*, division./, and powers.^. Matlab s integral3 can be used to approximate a triple integral. 6.2 Solving systems Matlab s fsolve can be used to find an approximate solution to a system of algebraic equations, close to some intial guess x 0. To approximate x 2 e x 1 = 0 2x 1 e x 2 = 0 write a small Matlab m-file specifying the system of equations fun fsolve.m. Then choose an appropriate initial guess, say (x 1, x 2 ) = (1, 2) and use fsolve: x0 = [1; 2]; fsolve(@fun fsolve, x0) 6