Notes on Math 375: Topics in Multi-Variable Calculus and Linear Algebra

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# Course Website

# Textbook

*Calculus, Volume II, 2nd Edition,* by Tom M. Apostol

# Calendar

### September

Mon (Dis) | Tue (Lec) | Wed (Dis) | Thu (Lec) | HW |
---|---|---|---|---|

- | - | - | 9/7 | HW1 |

9/11 | 9/12 | 9/13 | 9/14 | HW2 |

9/18 | 9/19 | 9/20 | 9/21 | HW3 |

9/25 | 9/26 | 9/27 | 9/28 | HW4 |

All notes from September in PDF format

### October

Mon (Dis) | Tue (Lec) | Wed (Dis) | Thu (Lec) | HW |
---|---|---|---|---|

10/2 | Exam 1 | 10/4 | 10/5 | HW5 |

10/9 | 10/10 | 10/11 | 10/12 | HW6 |

10/16 | 10/17 | 10/18 | 10/19 | HW7 |

10/23 | 10/24 | 10/25 | 10/26 | - |

10/30 | 10/31 | 11/1 | 11/2 | HW8 |

All notes from October in PDF format

### November

Mon (Dis) | Tue (Lec) | Wed (Dis) | Thu (Lec) | HW |
---|---|---|---|---|

10/30 | 10/31 | 11/1 | 11/2 | HW8 |

11/6 | 11/7 | 11/8 | 11/9 | HW9 |

11/13 | 11/14 | 11/15 | Exam 2 | HW10 |

11/20 | 11/21 | 11/22 | Holiday | - |

11/27 | 11/28 | 11/29 | 11/30 | HW11 |

All notes from November in PDF format

### December

Mon (Dis) | Tue (Lec) | Wed (Dis) | Thu (Lec) | HW |
---|---|---|---|---|

12/4 | 12/5 | 12/6 | 12/7 | HW12 |

12/11 | 12/12 | 12/13 |

# Lecture Outlines

### Chapter 1: Linear Spaces

Week 01 - 9/7: Vector Space

Week 02 - 9/12: Proof Writing

Week 02 - 9/14: Subspace, Span of Vector Spaces, Linear Independence

Week 03 - 9/19: Linear Independence, Basis, Coordinates

Week 03 - 9/21: Theorems about Linear Dependence

Week 04 - 9/26: Inner Product, Length, Angle

Week 04 - 9/28: Distance, Triangle Inequality, Orthogonal, Gramm-Schmidt Process

Week 05 - 10/3: Midterm 1

Week 05 - 10/5: Best Approximation, Fourier Series

### Chapter 2: Linear Transformations and Matrices

Week 06 - 10/10: Linear Transformations, Solving Systems of Equations

Week 06 - 10/12: Injective, Null Space, Range, Rank-Nullity Theorem

Week 07 - 10/17: Algebraic Operations on Linear Transformations, Injective, Inverse

Week 07 - 10/19: Matrix Representation of Linear Transformations, Matrix Multiplication

Week 08 - 10/24: Isomorphism Between Linear Transformations and Matrices, Solving Linear Equations using Matrix

Week 08 - 10/26: Solving Linear Equations

### Chapter 3: Determinants

Week 09 - 10/31: Determinants

Week 09 - 11/2: Uniqueness Theorem, Properties of Determinants

Week 10 - 11/7: Determinant and Area, Inverse of Matrix, Minors and Cofactors

Week 10 - 11/9: Cofactor Expansion, Cramer's Rule, Linear Independence and Determinant

### Chapter 4: Eigenvalues and Eigenvectors

Week 11 - 11/14: Eigenvalues, Eigenvectors

Week 11 - 11/16: Midterm 2

Week 12 - 11/21: Characteristic Polynomial, Trace, Diagonalization

Week 12 - 11/23: Thanksgiving Break

### Chapter 8: Differential Calculus of Scalar and Vector Fields

Week 13 - 11/28: Open Balls, Limits, Continuity, Directional Derivative

Week 13 - 11/30: Partial Derivative, Total Derivative, Linear Approximation Formula

Week 14 - 12/5: Differentiable, Total Derivative, Continuity, Multivariable Chain Rule

Week 14 - 12/7: Differentiable, Chain Rule, Gradient, Level Sets

Week 15 - 12/12: Multivariable Chain Rule, Jacobian Matrix