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Symbolic Mathematics Chapter 12. Objectives. After studying this chapter you should be able to: Create and manipulate symbolic variables Factor and simplify mathematical expressions Solve symbolic expressions Solve systems of equations Determine the symbolic derivative of an expression - PowerPoint PPT Presentation
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and
Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Symbolic MathematicsChapter 12
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Objectives
After studying this chapter you should be able to:
• Create and manipulate symbolic variables• Factor and simplify mathematical expressions• Solve symbolic expressions• Solve systems of equations • Determine the symbolic derivative of an
expression• Integrate an expression symbolically
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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MuPad
• MATLAB’s symbolic capability is based on the MuPad software, • Originally produced by Sciface
Software and purchased by the Mathworks in 2008.
• Previous versions of MATLAB, before 2007b, used Maple as the symbolic engine
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Symbolic Manipulation
• Available in the “Symbolic Math Toolbox” – which is an option in the professional version
• Included with the Student Edition• There are some differences between the
MuPad and Maple implementations of the symbolic toolbox
• If you have Maple installed on your computer it has an interface to MATLAB, which may over-ride the default MuPad engine
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Capabilities
• Manipulate symbolic expressions to • Simplify• Solve symbolically• Evaluate numerically• Take derivatives• Integrate • Perform linear algebraic manipulations
• More advanced features include • LaPlace transforms• Fourier transforms• Variable precision arithmetic
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Notebook Environment
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Section 12.1Symbolic Algebra
• Used regularly in math, engineering and science classes
• Often preferable to manipulate equations symbolically before you substitute in values for variables
96
)3(*22
2
xx
xy
Consider this equation
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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96
)3(*22
2
xx
xy
This looks like a fairly complicated function of x
2)96(
)96(*2
96
)3(*22
2
2
2
xx
xx
xx
xy
If you expand it, it simplifies dramatically
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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However, when you simplify you may lose information
undefinedy
0
0*2
9189
)33(*2 2
When x is equal to -3, the equation is undefined
96
)3(*22
2
xx
xy
Let x equal -3
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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You can choose
• MATLAB’s symbolic capability allows you perform the simplification, or to manipulate the numerator and denominator separately
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Relationships are not always easy to solve
If we know• k0
• Q• R• T
Its easy to solve for k
It’s not easy to solve for T!
/0
Q RTk k e0
0
0
0
ln( ) ln( )
ln
ln
ln( / )
Qk k
RT
k Q
k RT
k Q
k RT
QT
R k k
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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MATLAB’s symbolic capability makes it easy to solve this problem
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Creating Symbolic Variables
• Two approaches• Use the sym command to create
• Single variable• Expression• Equation
• Use the syms command to create • Single variables• Compose expressions and equations
from the variables you’ve defined
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Here’s an example
• Define x as a symbolic variable• x=sym('x') or• syms x
• Use x to create a more complicated expression• y = 2*(x+3)^2/(x^2+6*x+9)
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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x and y are both symbolic variables
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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The syms command can create multiple variables
• syms Q R T k0
• Use these variables to create another symbolic variables
k=k0*exp(-Q/(R*T))
Notice that we used standard algebraic operators – the array operators (.*, ./ and .^) are not used in symbolic algebra
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Create an entire expression with the sym command
• E=sym('m*c^2')
• Since m and c have not been specifically defined as symbolic variables, they are not stored
• E was set equal to a character string, defined by the single quotes inside the function.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Workspace
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Equations vs Expressions
• We can create an entire equation, and give it a name
• ideal_gas_law=sym('P*V=n*R*Temp')
This is an algebraic equationThis is an assignment statement
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Workspace
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Reserved Variable Names
One idiosyncrasy of the implementation of MuPad inside MATLAB is that a number of commonly used variables are reserved. They can be overwritten, however it you try to use them inside expressions or equations you may run into problems.
D, E, I, O, beta, zeta, theta, psi, gamma, Ci, Si, Ei
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Manipulating Symbolic Expressions and Equations
• Equations are set equal to something
• Expressions are not
• Both expressions and equations can be assigned a name, using the MATLAB assignment operator
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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The variable ideal_gas_law has been assigned to an equation
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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The variable E as been assigned to an expression
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Working with Equations and Expressions
• Some MATLAB functions work on both expressions and equations
• Others are limited in their scope
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Extracting Numerators and Denominators
• These functions work on expressions
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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The numden function extracts the numerator and denominator from an expression
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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You can combine symbolic variables using standard algebraic operators
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Expanding and Factoring
num is an expression
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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w is an equation
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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The factor function used with an expression, den
The factor function used with an equation, w
The collect function is similar, and collects common terms
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Simplifying
• The expand, factor and collect functions can be used to “simplify” an expression and sometimes an equation
• What constitutes a simplification is not always obvious
• The simplify function uses a set of built in rules
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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simplify used on an expression
simplify used on an equation
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Simple
• The simple function is different from simplify
• It tries all of the different simplification techniques, and chooses the result that is the shortest
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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All of the possibilities evaluated are reported, however there is only one actual answer
Both simple and simplify work on expressions and equations
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Hint
• Use the poly2sym function as a shortcut to create a polynomial
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Hint
• Extract the coefficients from a polynomial, using the sym2poly function
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Section 12.2Solving Equations and Expressions
• Use the solve function
• Automatically sets expressions equal to 0 and solves for the roots
• Uses the equality specified in equations
• Solves for the variables in systems of equations
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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You can define your expression or equation in the solve function
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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If x has previously been defined as a symbolic, the quotes aren’t
necessary
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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The answer from the solve function is not necessarily a number
You can specify what variable you want to solve for
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Example using solve
Sometimes it’s useful to redefine a variable for latter use
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Solving Systems of Equations
This result is a structure array. These arrays are described in Chapter 11
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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There are several different approaches to find the actual values of x, y, and z
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Give the result a name, such as answer, and then specify the field name inside the structure array to retrieve the values for x, y, and z
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Assign individual variable names. Notice that x, y and z are symbolic variables
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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If you need to use the value of x, y or z in a function that needs a double as input, you’ll need to change the variable type from symbolic to double
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Hint
• Using the solve function for multiple equations has both advantages and disadvantages over using the linear algebra techniques described in chapter 9.
• In general, if a problem can be solved using matrices the matrix solution will take less computer time.
• However, linear algebra is limited to first order equations.
• The solve function may take longer, but it can solve non-linear problems and can solve problems with symbolic variables.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Example 12.1
• Use MATLAB’s symbolic capability to solve an equation
• k = k0*exp(-Q/RT)• Solve for Q
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Hand solution
0 expQ
k kRT
0
expk Q
k RT
0
lnk Q
k RT
0lnk
Q RTk
Solutions
MATLAB Solution
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Substitution
• Once we have a symbolic expression we’ll probably want to substitute numbers into it.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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We could substitute in a new variable – in this case we’ll put a y everywhere there used to be an x
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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We could substitute in a number – in this case 3 for x
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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If the variables inside the expression have been explicitly defined as symbolics we don’t need the single quotes
To substitute into multiple variables group them with curly braces
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Example 12.2
• Using Symbolic Math to solve a ballistics problem
vertical distance
horizontal distance
Range
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Equations of Motion
)cos(**0_ tvxdist
2**2/1)sin(**0_ tgtvydist
Find the equation for the distance traveled when the projectile hits the ground
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Strategy
• Find the time the projectile hits the ground
• Substitute back into the horizontal distance equation
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB solution M-file
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Results
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Section 12.3Symbolic Plotting
• The symbolic toolbox includes a group of functions to create symbolic plots
• The most basic is the ezplot
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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EZPlot
• Allows you to plot symbolic expressions
• ezplot(S)• Defaults to a range of -2to +2
• ezplot(S, [xmax, xmin])
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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-2 +2
Note this plot was created with the student version –The symbolic functionality is included in the student version
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Add your own titles, axis labels and other annotations using the same functions described for numeric plotting in Chapter 5
Notice that ezplot creates a title and axis labels automatically
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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ezplot supports implicit plotting
• The equation for a circle can be expressed implicitly as:• x2 + y2 = 1
• You could solve for y, but it’s not necessary with ezplot
• ezplot('x^2 + y^2 =1',[-1.5,1.5])
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Ezplot supports parametric equation graphs
• The equation for a circle can be expressed parametrically as:• x=sin(t)• y=cos(t)
• To create the graph use…• ezplot(‘sin(x)’,’cos(x)’)
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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x
y
x2 + y2 =1 = 0
-1 0 1-1.5
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-1 -0.5 0 0.5 1
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x
y
x = sin(x), y = cos(x)
Implicit and Parametric plots of a circle
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Hint
Most symbolic functions will allow you to either enter a symbolic variable that represents a function, or to enter the function itself enclosed in single quotes. For exampley=sym(‘x^2-1’)ezplot(y)is equivalent to ezplot(‘x^2-1’)
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Other Symbolic Plots
• Additional symbolic plotting functions are available, which mirror the functions used in numeric MATLAB plotting options
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Symbolic Plot Types
ezplot Function plotter if z is a function of x ezplot(z)
ezmesh Mesh plotter if z is a function of x and yezmesh(z)
ezmeshc Combined mesh and contour plotter
if z is a function of x and yezmeshc(z)
ezsurf Surface plotter if z is a function of x and yezsurf(z)
ezsurfc Combined surface and contour plotter
if z is a function of x and yezsurfc(z)
ezcontour Contour plotter if z is a function of x and yezcontour(z)
ezcontourf Filled contour plotter if z is a function of x and yezcontourf(z)
ezplot3 3-D parametric curve plotter
if x is a function of tif y is a function of tif z is a function of tezplot3(x,y,z)
ezpolar Polar Coordinate plotter if r is a function of ezpolar(r)
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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To demonstrate these plot types create a symbolic version of “peaks”
We broke this function up into three parts to make it easier to enter into the computer. Notice that there are no “dot” operators used in these expressions, since they are all symbolic.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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When we created the same plots using a standard MATLAB approach it was necessary to define an array of both x and y values, mesh them together, and calculate the values of z based on the two dimensional arrays. The symbolic plotting capability contained in the symbolic toolbox makes creating these graphs much easier.
All of these graphs can be annotated using the standard MATLAB functions such as title, xlabel, text, etc.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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These contour plots are a two-dimensional representation of the three-dimensional peaks function
The polar graph requires us to define a new function
Any of these ezplot graphs can handle parameterized equations
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Example 12.3
• Plot the trajectory of a ballistic projectile• Assume 100 m/sec initial velocity• Assume g = 9.8 m/sec2
vertical distance
horizontal distance
Range
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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12.4 Calculus
• MATLAB’s symbolic toolbox supports• Symbolical differentiation• Symbolic integration
• This makes it possible to find analytical solutions for many problems, instead of numeric approximations.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Differentiation
• Concept introduced in Calculus I
• However… a derivative is really just the slope of an equation
• A common application of derivatives is to find velocities and accelerations
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Consider a race car…
• Assume that during a race the car starts out slowly, and reaches its fastest speed at the finish line
• To avoid running into the stands, the car must then slow down until it finally stops
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Model
• We might model the position of the car using a sine wave
)20/)10(*sin(*2020 tdist
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Create a plot of position vs time using ezplot
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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5
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time, sec
Car position
Dis
tanc
e fr
om S
tart
ing
Line
Finish Line
ezplot of position
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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diff function
• The diff function finds a symbolic derivative
• The velocity is the derivative of the position, so to find the equation of the velocity of the car we’ll use the diff function, then plot the result
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Find the symbolic derivative, which corresponds to the velocity
Create a plot of velocity and time
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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The velocity is the derivative of the position with respect to time
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0.5
1
1.5
2
2.5
3
time, sec
Race Car Velocity
velo
city
, di
stan
ce/t
ime
Finish Line
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Acceleration
• The acceleration is the derivative of the velocity, so to find the equation of the acceleration of the car we’ll use the diff function, then plot the result
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Determine the equation for the acceleration
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Acceleration is the derivative of the velocity
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-0.2
0
0.2
0.4
0.6
time, sec
Race Car Acceleration
acce
lera
tion,
vel
ocity
/tim
e
Finish Line
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Symbolic Differentiationdiff(f) Returns the derivative of the
expression f with respect to the default independent variable
y=sym('x^3+z^2')diff(y)ans =3*x^2
diff(f,’t’)
Returns the derivative of the expression f with respect to the variable t.
y=sym('x^3+z^2')diff(y,'z')ans =2*z
diff(f,n)
Returns the nth derivative of the expression f with respect to the default independent variable
y=sym('x^3+z^2')diff(y,2)ans =6*x
diff(f,’t’,n)
Returns the nth derivative of the expression f with respect to the variable t.
y=sym('x^3+z^2')diff(y,'z',2)ans =2
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Partial Derivatives
• If you have multiple variables, MATLAB takes the derivative with respect to x – unless you specify otherwise
• All the other variables are kept constant
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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To find the derivative with respect to some variable other than x, you must specify it in the diff function
Notice that t is enclosed in single quotes, since we haven’t specified it as a symbolic variable
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Higher order derivatives
• To find higher order derivatives we can either • nest the diff function• or specify the derivative order in the
diff function
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Example 12.4
• Find the angle where the maximum range occurs
• Find the maximum range
horizontal distance
Range
Vertical distance
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Angle for Max Range
Max Height
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.
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Integration
• Usually introduced in Calculus II
• Often visualized as the area under a curve
• MATLAB has built in symbolic integration capability.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Consider a piston cylinder device
• Work done by a piston cylinder device as it moves up or down, can be calculated by taking the integral of P with respect to V
2
1PdVW
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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To perform the integration we need to know how P changes with V
• If P is constant the problem becomes
2
1W P dV
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Model of the behavior of a piston cylinder device
0 1 2 3 4 599
99.5
100
100.5
101Pressure Profile in a Piston Cylinder Device
Volume, cm3
Pre
ssur
e, p
sia
gas
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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10
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100
110
Volume, cm3
Pre
ssur
e, p
sia
Pressure Profile in a Piston Cylinder Device
Initial volume
Final volume
The area under the curve corresponds to the work
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Hand Calculation
psiacmW
psiaPif
VPPVPVPVdVPPdVW
100*3
100 3
14
4
1
4
1
4
1
Read this as: Work is equal to the integral of P with respect to V, from V=1 to V=4
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB Solution
Work is equal to the integral of P with respect to V, from V=1 to V=4
Substitute in 100 as the value of P
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Symbolic Integration
int(f) Returns the integral of the expression f with respect to the default independent variable
y=sym('x^3+z^2')int(y)ans =1/4*x^4+z^2*x
int(f,’t’)
Returns the integral of the expression f with respect to the variable t.
y=sym('x^3+z^2')int(y,'z')ans =x^3*z+1/3*z^3
int(f,a,b)
Returns the integral with respect to the default variable, of the expression f between the numeric bounds, a and b.
y=sym('x^3+z^2')int(y,2,3)ans =65/4+z^2
int(f,’t’,a,b)
Returns the integral with respect to the variable t, of the expression f between the numeric bounds, a and b.
y=sym('x^3+z^2')int(y,'z',2,3)ans =x^3+19/3
int(f,’t’,a,b)
Returns the integral with respect to the variable t, of the expression f between the symbolic bounds, a and b.
y=sym('x^3+z^2')int(y,'z','a','b')ans =x^3*(b-a)+1/3*b^3-1/3*a^3
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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12.5 Differential Equations
• Differential equations contain both
• the derivative of the dependent variable with respect to the independent variable
• the dependent variable
dyy
dt is a differential equation
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Default variable
• Although any symbol can be used for either the independent or the dependent variable, the default independent variable is t in MATLAB (and is the usual choice for most ordinary differential equation formulations.)
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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dsolve
• When we solve a differential equation, we are looking for an expression for y in terms of t
• dsolve requires the differential equation as input• use the symbol D to
specify derivatives with respect to the independent variable
dsolve is a “function function”
Using a single input results in a family of results
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Specify an initial or boundary condition in the second field
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Here’s a more complicated example
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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You can specify the independent variable in the third field
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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To specify a higher order derivative in the dsolve function put the order immediately after the D
Higher Order Derivatives
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Hint
• Don’t use the letter D in your variable names in differential equations.
• It will confuse the function into thinking you are trying to specify a derivative
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Use the dsolve function to solve systems of equations
• dsolve('eq1,eq2,...', 'cond1,cond2,...', 'v')
The result is a structure array
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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• For complicated (or ill behaved) systems of equations you may find it easier to use MuPad • Remember that MATLAB’s symbolic
capability is based on the MuPad engine
• There are many differential equations that can’t be solved analytically at all • The numerical techniques described in
Chapter 13 can be used to solve many of these equations.
MATLAB can not solve every differential equation symbolically.
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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12.6 Converting Symbolic Expressions to MATLAB functions
• It is often useful to manipulate expressions symbolically … but then to perform numeric calculations using more traditional MATLAB functions
• matlabFunction converts a symbolic expression to an anonymous function
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Symbolic Manipulation
An anonymous function
matlabFunction
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Summary
• MATLAB uses MuPad as its symbolic engine
• The symbolic toolbox is an optional component of the professional version
• A subset is included with the student version
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Summary – Variable Definition
• Use either• sym• syms
• The sym command can be used to create symbolic expressions or equations
• The syms command can create multiple symbolic variables in one step
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Summary – Composition of expressions
• Once symbolic variables have been created they can be used to create more complicated expression
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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SummaryEquations vs Expressions
• Equations are set equal to something
• Expressions are not
• If you set one expression equal to another, you’ve created an equation
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Summary – Symbolic functions
• numden
• expand
• factor
• collect
• simplify
• simple
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Summary – Solve
• If the input to solve is an expression MATLAB sets it equal to 0 and solves
• If the input is an equation, MATLAB solves the equation for either the default variable, or a user defined variable
• solve can also solve systems of equations
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Summary - dsolve
• Used to solve differential equations
• D signifies a derivative
• Can be used to solve systems of equations
• Not all differential equations can be solved analytically
MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,
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Summary - Calculus
• diff - finds the derivative
• int - takes the integral
