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Skills Review



Skills Review

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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.

topicsmathematical knowledge topicsmit ocw resourcesreadingsother resources


Module 1: Control System Analysis


1. Course Introduction
2. Introduction to Control Systems Functions

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, 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 SeriesStrang, 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 DesignFunctions18.013A: Readings

Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, 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 SensitivityFunctions

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 FunctionsKleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
Differentiation



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 DifferentiationStrang, Chapter 13: Partial Derivatives (PDF)
5. Steady-State ErrorsFunctions 

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Chapter 19: Partial DerivativesStrang, Chapter 13: Partial Derivatives (PDF)
 Limits S: Section 2.5: The Concept of LimitStrang, Section 2.6: Limits (PDF)
6. The s-Plane, Poles and ZeroesFunctions 

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
 Complex Numbers18.03: 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 ApplicationsChapter 1: Philosophy, Numbers and Functions

 S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
 Complex Numbers18.03: 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 ModesFunctions

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, 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 ZeroesFunctions

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, 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 SeriesStrang, 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 ZerosFunctions

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

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

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, 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 Equations18.03: 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 SystemsStrang, Section 11.4: Matrices and Linear Equations (PDF)
12. State Space ModelingFunctions

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, Calculus for Beginners and Artists, Chapter 3: Linear Functions
 Nth Order Differential Equations EP: Section 2.2: General Solutions of Linear Equations 
 Vector-Matrix Equations18.03: 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 SystemsStrang, 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 ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, 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 Equations18.03: 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 SystemsStrang, 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 Algebra18.03: Readings

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

18.013A: Readings


Kleitman, Calculus with ApplicationsChapter 1: Philosophy, Numbers and Functions

S: Sections 1.5-1.6: The Concept of a Function, Graphs of FunctionsKleitman, 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 Algebra18.03: Readings

Notes and Exercises LS.1: Review of Linear Algebra (PDF)
  
 Power Series18.03: 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 ArtistsSection 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 EquationsFirst-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)
 Convolution18.03: 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 Algebra18.03: 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. ControllabilityFirst-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 Series18.03: 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 ArtistsSection 15.5: Power Series
 Linear Algebra18.03: Readings

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

 

Curriculum Map
Mathematical Knowledge TopicsUtilizedReviewedTaught
1 Calculus
1.1 FunctionsUFl, UTh, Dyn, USig, Therm, Cont
1.2 DifferentiationUFl, UTh, Dyn, USig, Therm, Cont
1.3 IntegrationUFl, Dyn, USig, Therm, ContUTh
1.4 Series and sums: Taylor, FourierUFl, Dyn, USig, Therm, Cont
1.5 Vector CalculusUFlUFl
2 GeometryDyn
2.1 Analytical GeometryDyn, UFl
2.2 Trigonometry
3 Differential Equations
3.1 ODEsUTh, ThermDyn, ContUSig, Cont
3.2 PDEsThermThermUFl
3.3 Integrad EquationsUFl
4 Linear Algebra
4.1 Matrix AlgebraUSig, ContDyn, USig
4.2 Linearization, Linear SystemsContDyn
4.3 State (discrete)USig, Cont
4.4 Tensors (multidimensional objects)UFl
5 Complex Analysis
5.1 Complex VariablesContUFlUSig
5.2 Frequency domain, variables and plotsUSig, Cont

5.3 Transforms: Fourier, Laplace

USigUSig, Cont
6 Probability and StatisticsTBDTBDTBD
7 Discrete MathematicsTBDTBDTBD

 








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