200 Calculus 1

Info

Grading:

• 3 quizzes, 15
• Midsem, 35
• Endsem, 40
• class assignment, 10

Lecture notes

These notes have been reorganized because I lost count of the lectures at some point, and do not reflect the exact sequence of topics covered in class.

• CAL1_L1 ✅
  • The p-norm
• CAL1_L2 ✅
  • Equivalence of p-norms
• CAL1_L3 ✅
  • Equivalence of all norms on a finite dimensional normed linear space
• CAL1_L4 ✅
  • Example of complete normed linear space, inner product spaces (mostly stuff covered in ALG1)
• CAL1_L5 ✅
  • Examples of finding limits in ${\mathbb{R}}^2$

Metric space topology

Ref: Royden & Fitzpatrick

• CAL1_L6 ✅
  • Cantor intersection theorem, constructing the completion of a metric space
• CAL1_L7 ✅
  • Finite intersection property, total boundedness, characterization of compact sets
• CAL1_L8 ✅
  • Extreme value theorem, Lebesgue covering theorem
• CAL1_L9
  • Separable metric spaces, Lindelöf covering theorem, Baire category theorem
• CAL1_L10
  • Banach contraction principle

Multivariable functions

Ref: Ghorpade & Limaye

• CAL1_L11 ✅
  • Order properties of ${\mathbb{R}}^n$, Intervals, disks, and bounded sets
• CAL1_L12
  • Monotonicity and Bimonotonicity, bounded variation and bivariation
• CAL1_L13 ✅
  • Convexity and Concavity, paths, local extrema and saddle points
• CAL1_L14
  • Intermediate value property

Multivariable calculus

Ref: Apostol (Mathematical Analysis) ch12 & ch13, Rudin ch9

• CAL1_L15 ✅
  • Partial and directional derivatives
• CAL1_L16 ✅
  • Total derivatives, the Jacobian matrix
• CAL1_L17 ✅
  • The chain rule
• CAL1_L18 ✅
  • The mean value theorem for differentiable functions
• CAL1_L19 ✅
  • Sufficient conditions for differentiability and equality of mixed partial derivatives
• CAL1_L20 ✅
  • Taylor’s formula for functions from ${\mathbb{R}}^n$ to $\mathbb{R}$
• CAL1_L21
  • Inverse function theorem
• CAL1_L22
  • Implicit function theorem
• CAL1_L23
  • Extrema of real valued functions
• CAL1_L24
  • Extremum problems with side conditions