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Skills Review

This section lays out, in a structured manner, the mathematics skills required of undergraduate students in 16.06.

Also provided is a table in which each mathematical knowledge topic is mapped to core courses within the Department of Aeronautics and Astronautics.

Readings listed in the table below are from the textbooks for 18.01, 18.02, and 18.03 and three online textbooks:

(S) Amazon logo Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York, NY: McGraw-Hill, 1995. ISBN: 0070576424.

(E) Amazon logo Edwards, C. Multivariable Calculus. 6th ed. Upper Saddle River, NJ: Prentice Hall, 2002. ISBN: 0130339679.

(EP) Amazon logo Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems. 4th ed. Upper Saddle River, NJ: Prentice Hall, September 29, 1999. ISBN: 0130113018.

Online textbooks:

Amazon logo Strang, Gilbert. Calculus. Wellesley, MA: Wellesley-Cambridge Press, 1991. ISBN: 0961408820.

Kleitman, Daniel. Calculus for Beginners and Artists.

Kleitman, Daniel. Calculus with Applications. (used in 18.013A)

16.06 Skills Review

Further information on the Mathematical Knowledge Topics for each 16.06 lecture may be found in the Supplementary Math Notes (PDF), which are organized by 16.06 lecture topics and the associated Mathematical Knowledge Topics.

The following list of topics link to the corresponding entry in the table below.

  1. Course Introduction
  2. Introduction to Control Systems
  3. Control System Analysis and Design
  4. Disturbances and Sensitivity
  5. Steady-State Errors
  6. The s-Plane, Poles and Zeroes
  7. Transient Response Characteristics and System Stability
  8. Dominant Modes
  9. Transient Performance and the Effect of Zeroes
  10. The Effect of Zeroes
  11. State Space
  12. State Space Modeling
  13. More State Space Modeling and Transfer Function Matrices
  14. Quanser Model and State Transition Matrices
  15. Solutions of State Space Differential Equations
  16. Controllability
topics mathematical knowledge topics ocw resources readings other resources

Module 1: Control System Analysis
1. Course Introduction        
2. Introduction to Control Systems Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Linearization

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 12: Applications of Differentiation: Direct Use of Linear Approximation

 EP: Section 3.1: Introduction and Review of Power Series Strang, Section 10.4: The Taylor Series for e^x, sin x, and cos x (PDF)
  First-Order Differential Equations

18.03: Lecture Notes

Linear Equations (Lec #3) (PDF)

 EP: Section 1.5: Linear First-Order Equations

S: Section 14.4: Taylor Series and Taylor's Formula		  
  Laplace Transforms

18.03: Lecture Notes

Laplace Transform: Basic Properties (Lec #21) (PDF)

Application to ODEs & Partial Fractions (Lec #22) (PDF)

Completing the Square & Transforms of Delta and Time Translated Functions (Lec #23) (PDF)

Convolution and Laplace Transform & The Pole Diagram (Lec #24) (PDF)

Numerical Methods (Lec #25) (PDF)

18.03: Readings

Application to ODEs & Partial Fractions (Lec #22) (PDF)

18.03: Recitation Problems

Laplace Transform & Poles (Recitation Problems #14) (PDF); Solutions (PDF)

Laplace Transform and ODEs (Recitation Problems #15) (PDF); Solutions (PDF)

Laplace Transform: Second-order Equations (Completing the Square), t-shift Formula, Step and Delta Signals, Weight and Transfer, Convolution, and Poles (Recitation Problems #16) (PDF); Solutions (PDF)

Laplace Transform: Meaning of the Pole Diagram & Euler's Method (Recitation Problems #17) (PDF); Solutions (PDF)

 EP: Chapter 4: Laplace Transform Methods  
3. Control System Analysis and Design Functions 18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Second-Order Differential Equations

18.03: Lecture Notes

Solutions of Spring-mass-dashpot Models (Lec #9) (PDF)

18.03: Recitation Problems

Second-order Linear Equations: Constant Coefficient, Homogeneous (Recitation Problems #6) (PDF); Solutions (PDF)

 EP: Section 2.1: Second-Order Linear Equations  
4. Disturbances and Sensitivity Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Differentiation

18.01: Lecture Notes

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 8: Calculation of Derivatives by Rule

 S: Chapters 2-4: The Derivative of a Function, The Computation of Derivatives, Applications of Derivatives

Strang, Chapter 2: Derivatives (PDF)

Strang, Chapter 3: Linear Functions (PDF)

Strang, Chapter 4: The Chain Rule (PDF)
  Partial Differentiation     Strang, Chapter 13: Partial Derivatives (PDF)
5. Steady-State Errors Functions  

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Chapter 19: Partial Derivatives Strang, Chapter 13: Partial Derivatives (PDF)
  Limits   S: Section 2.5: The Concept of Limit Strang, Section 2.6: Limits (PDF)
6. The s-Plane, Poles and Zeroes Functions  

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

  S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Complex Numbers 18.03: Readings

Notes and Exercises C.1-C.3: Complex Numbers (Ses #5) (PDF)

Supplementary Notes 5: The Algebra of Comlex Numbers (Ses #5) (PDF)

Supplementary Notes 6: The Complex Exponential (Ses #5) (PDF)


18.03: Lecture Notes

Complex Numbers, Complex Exponential (Ses #5) (PDF)		    
  Partial Fractions   S: Section 10.6: The Method of Partial Fractions

EP: Section 4.3: Translation and Partial Fractions		 Strang, Section 7.4: Partial Fractions (PDF)
7. Transient Response Characteristics
and System Stability		 Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

  S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Complex Numbers 18.03: Readings

Notes and Exercises C.1-C.3: Complex Numbers (Ses #5) (PDF)

Supplementary Notes 5: The Algebra of Comlex Numbers (Ses #5) (PDF)

Supplementary Notes 6: The Complex Exponential (Ses #5) (PDF)


18.03: Lecture Notes

Complex Numbers, Complex Exponential (Ses #5) (PDF)		    
  Partial Fractions   S: Section 10.6: The Method of Partial Fractions

EP: Section 4.3: Translation and Partial Fractions		 Strang, Section 7.4: Partial Fractions (PDF)
8. Dominant Modes Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Partial Fractions   S: Section 10.6: The Method of Partial Fractions

EP: Section 4.3: Translation and Partial Fractions		 Strang, Section 7.4: Partial Fractions (PDF)
9. Transient Performance and the Effect of Zeroes Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions 
  Linearization

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 12: Applications of Differentiation: Direct Use of Linear Approximation

 EP: Section 3.1: Introduction and Review of Power Series Strang, Section 10.4: The Taylor Series for e^x, sin x, and cos x (PDF)
  Second-Order Differential Equations

18.03: Lecture Notes

Solutions of Spring-mass-dashpot Models (Lec #9) (PDF)

18.03: Recitation Problems

Second-order Linear Equations: Constant Coefficient, Homogeneous (Recitation Problems #6) (PDF); Solutions (PDF)

 EP: Section 2.1: Second-Order Linear Equations  
10. The Effect of Zeros Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
11. State Space Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  First-Order Differential Equations

18.03: Lecture Notes

Linear Equations (Lec #3) (PDF)

 EP: Section 1.5: Linear First-Order Equations

S: Section 14.4: Taylor Series and Taylor's Formula		  
  Second-Order Differential Equations

18.03: Lecture Notes

Solutions of Spring-mass-dashpot Models (Lec #9) (PDF)

18.03: Recitation Problems

Second-order Linear Equations: Constant Coefficient, Homogeneous (Recitation Problems #6) (PDF); Solutions (PDF)

 EP: Section 2.1: Second-Order Linear Equations  
  Vector-Matrix Equations 18.03: Readings

Notes and Exercises LS.1: Linear Systems (Ses #32) (PDF)


18.03: Lecture Notes

Linear Systems and Matrices (Ses #32) (PDF)


18.03: Recitation Problems

Systems of First-order Equations (Rec #21) (PDF); Solutions (PDF)

Linear Systems (Rec #15) (PDF); Solutions (PDF)		 EP: Section 5.3: Matrices and Linear Systems Strang, Section 11.4: Matrices and Linear Equations (PDF)
12. State Space Modeling Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Nth Order Differential Equations   EP: Section 2.2: General Solutions of Linear Equations  
  Vector-Matrix Equations 18.03: Readings

Notes and Exercises LS.1: Linear Systems (Ses #32) (PDF)

18.03: Lecture Notes

Linear Systems and Matrices (Ses #32) (PDF)

18.03: Recitation Problems

Systems of First-order Equations (Rec #21) (PDF); Solutions (PDF)

Linear Systems (Rec #15) (PDF); Solutions (PDF)		 EP: Section 5.3: Matrices and Linear Systems Strang, Section 11.4: Matrices and Linear Equations (PDF)
  Second-Order Differential Equations

18.03: Lecture Notes

Solutions of Spring-mass-dashpot Models (Lec #9) (PDF)

18.03: Recitation Problems

Second-order Linear Equations: Constant Coefficient, Homogeneous (Recitation Problems #6) (PDF); Solutions (PDF)

 EP: Section 2.1: Second-Order Linear Equations  
13. More State Space Modeling and Transfer Function
Matrices		 Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Second-Order Differential Equations

18.03: Lecture Notes

Solutions of Spring-mass-dashpot Models (Lec #9) (PDF)

18.03: Recitation Problems

Second-order Linear Equations: Constant Coefficient, Homogeneous (Recitation Problems #6) (PDF); Solutions (PDF)

 EP: Section 2.1: Second-Order Linear Equations  
  Vector-Matrix Equations 18.03: Readings

Notes and Exercises LS.1: Linear Systems (Ses #32) (PDF)

18.03: Lecture Notes

Linear Systems and Matrices (Ses #32) (PDF)

18.03: Recitation Problems

Systems of First-order Equations (Rec #21) (PDF); Solutions (PDF)

Linear Systems (Rec #15) (PDF); Solutions (PDF)		 EP: Section 5.3: Matrices and Linear Systems Strang, Section 11.4: Matrices and Linear Equations (PDF)
  Laplace Transforms

18.03: Lecture Notes

Laplace Transform: Basic Properties (Lec #21) (PDF)

Application to ODEs & Partial Fractions (Lec #22) (PDF)

Completing the Square & Transforms of Delta and Time Translated Functions (Lec #23) (PDF)

Convolution and Laplace Transform & The Pole Diagram (Lec #24) (PDF)

Numerical Methods (Lec #25) (PDF)

18.03: Readings

Application to ODEs & Partial Fractions (Lec #22) (PDF)

18.03: Recitation Problems

Laplace Transform & Poles (Recitation Problems #14) (PDF); Solutions (PDF)

Laplace Transform and ODEs (Recitation Problems #15) (PDF); Solutions (PDF)

Laplace Transform: Second-order Equations (Completing the Square), t-shift Formula, Step and Delta Signals, Weight and Transfer, Convolution, and Poles (Recitation Problems #16) (PDF); Solutions (PDF)

Laplace Transform: Meaning of the Pole Diagram & Euler's Method (Recitation Problems #17) (PDF); Solutions (PDF)

 EP: Chapter 4: Laplace Transform Methods  
  Linear Algebra 18.03: Readings

Notes and Exercises LS.1: Review of Linear Algebra (PDF)		    
14. Quanser Model and State Transition Matrices Functions

18.013A: Readings

Kleitman, Calculus with Applications, Chapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of Functions Kleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
  Second-Order Differential Equations

18.03: Lecture Notes

Solutions of Spring-mass-dashpot Models (Lec #9) (PDF)

18.03: Recitation Problems

Second-order Linear Equations: Constant Coefficient, Homogeneous (Recitation Problems #6) (PDF); Solutions (PDF)

 EP: Section 2.1: Second-Order Linear Equations  
  First-Order Differential Equations

18.03: Lecture Notes

Linear Equations (Lec #3) (PDF)

 EP: Section 1.5: Linear First-Order Equations

S: Section 14.4: Taylor Series and Taylor's Formula		  
  Linear Algebra 18.03: Readings

Notes and Exercises LS.1: Review of Linear Algebra (PDF)		    
  Power Series 18.03: Readings

Notes and Exercises LS.6.3: The Exponential Matrix (PDF)		 EP: Section 5.7: Matrix Exponentials and Linear Systems

S: Chapter 14: Power Series

Strang, Section 10.5: Power Series (PDF)

Kleitman, Calculus for Beginners and Artists, Section 15.5: Power Series
  Laplace Transforms

18.03: Lecture Notes

Laplace Transform: Basic Properties (Lec #21) (PDF)

Application to ODEs & Partial Fractions (Lec #22) (PDF)

Completing the Square & Transforms of Delta and Time Translated Functions (Lec #23) (PDF)

Convolution and Laplace Transform & The Pole Diagram (Lec #24) (PDF)

Numerical Methods (Lec #25) (PDF)

18.03: Readings

Application to ODEs & Partial Fractions (Lec #22) (PDF)

18.03: Recitation Problems

Laplace Transform & Poles (Recitation Problems #14) (PDF); Solutions (PDF)

Laplace Transform and ODEs (Recitation Problems #15) (PDF); Solutions (PDF)

Laplace Transform: Second-order Equations (Completing the Square), t-shift Formula, Step and Delta Signals, Weight and Transfer, Convolution, and Poles (Recitation Problems #16) (PDF); Solutions (PDF)

Laplace Transform: Meaning of the Pole Diagram & Euler's Method (Recitation Problems #17) (PDF); Solutions (PDF)

 EP: Chapter 4: Laplace Transform Methods  
15. Solutions of State Space Differential Equations First-Order Differential Equations

18.03: Lecture Notes

Linear Equations (Lec #3) (PDF)

 EP: Section 1.5: Linear First-Order Equations

S: Section 14.4: Taylor Series and Taylor's Formula		  
  Integration   E: Chapter 15: Vector Calculus

S: Chapters 5,6,7,10		 Strang, Chapter 5: Integrals (PDF)

Strang, Sections 15.1-15.2: Vector Fields, Line Integrals (PDF)
  Convolution 18.03: Readings

Notes and Exercises I.4: Convolution (PDF)


18.03: Recitation Problems

Impulse Response = Weight Function and Convolution (Rec #13) (PDF); Solutions (PDF)		 EP: Section 4.4: Derivatives, Integrals, and Products of Transforms  
  Linear Algebra 18.03: Readings

Notes and Exercises LS.1: Review of Linear Algebra (PDF)		    
  Inverse Laplace Transforms

18.03: Lecture Notes

Laplace Transform: Basic Properties (Lec #21) (PDF)

Application to ODEs & Partial Fractions (Lec #22) (PDF)

Completing the Square & Transforms of Delta and Time Translated Functions (Lec #23) (PDF)

Convolution and Laplace Transform & The Pole Diagram (Lec #24) (PDF)

Numerical Methods (Lec #25) (PDF)

18.03: Readings

Application to ODEs & Partial Fractions (Lec #22) (PDF)

18.03: Recitation Problems

Laplace Transform & Poles (Recitation Problems #14) (PDF); Solutions (PDF)

Laplace Transform and ODEs (Recitation Problems #15) (PDF); Solutions (PDF)

Laplace Transform: Second-order Equations (Completing the Square), t-shift Formula, Step and Delta Signals, Weight and Transfer, Convolution, and Poles (Recitation Problems #16) (PDF); Solutions (PDF)

Laplace Transform: Meaning of the Pole Diagram & Euler's Method (Recitation Problems #17) (PDF); Solutions (PDF)

 EP: Chapter 4: Laplace Transform Methods  
16. Controllability First-Order Differential Equations

18.03: Lecture Notes

Linear Equations (Lec #3) (PDF)

 EP: Section 1.5: Linear First-Order Equations

S: Section 14.4: Taylor Series and Taylor's Formula		  
  Power Series 18.03: Readings

Notes and Exercises LS.6.3: The Exponential Matrix (PDF)		 EP: Section 5.7: Matrix Exponentials and Linear Systems

S: Chapter 14: Power Series		 Strang, Section 10.5: Power Series (PDF)

Kleitman, Calculus for Beginners and Artists, Section 15.5: Power Series
  Linear Algebra 18.03: Readings

Notes and Exercises LS.1: Review of Linear Algebra (PDF)		    

 

Curriculum Map
Mathematical Knowledge Topics Utilized Reviewed Taught
1 Calculus      
1.1 Functions UFl, UTh, Dyn, USig, Therm, Cont    
1.2 Differentiation UFl, UTh, Dyn, USig, Therm, Cont    
1.3 Integration UFl, Dyn, USig, Therm, Cont UTh  
1.4 Series and sums: Taylor, Fourier UFl, Dyn, USig, Therm, Cont    
1.5 Vector Calculus UFl UFl  
2 Geometry Dyn    
2.1 Analytical Geometry Dyn, UFl    
2.2 Trigonometry      
3 Differential Equations      
3.1 ODEs UTh, Therm Dyn, Cont USig, Cont
3.2 PDEs Therm Therm UFl
3.3 Integrad Equations     UFl
4 Linear Algebra      
4.1 Matrix Algebra USig, Cont Dyn, USig  
4.2 Linearization, Linear Systems   Cont Dyn
4.3 State (discrete)     USig, Cont
4.4 Tensors (multidimensional objects)     UFl
5 Complex Analysis      
5.1 Complex Variables Cont UFl USig
5.2 Frequency domain, variables and plots     USig, Cont

5.3 Transforms: Fourier, Laplace

 USig   USig, Cont
6 Probability and Statistics TBD TBD TBD
7 Discrete Mathematics TBD TBD TBD