Calculus (2)

Instructor: Tsung-Ju Lee（李宗儒）

Term: Spring（春季）

Location: 經緯廳

Lecture Time: Wednesdays 10:10AM - 12:00PM and Fridays 9:00AM - 9:50AM

TA Recitation: Fridays 8:00AM - 8:50AM

Course Overview

This course provides a comprehensive introduction to Calculus on multivariate functions.

Prerequisites

  • Calculus (1)

Textbooks

  • Calculus: one and several variables, Saturnino L. Salas, Einar Hille, and Garret J. Etgen, 2021, 10E, 978-1-119-77067-1, John Wiley & Sons Singapore Pte. Ltd.

Grading

  • Quiz: 16%
  • Midterm: 32%
  • Final: 32%
  • Problem solving in TA session: 20%

Schedule

Week Date Topic Materials
1 Feb 25 Curves in Euclidean space

Calculus for vector-valued univariate functions; curves in space; arc length; curvature.
  • 13.1 - 13.5
2 Mar 4, Mar 6 Functions of several variables

Functions of several variables; partial derivatives; open and closed sets; limits and continuity.
  • Chap. 14
3 Mar 11, Mar 13 Differential Calculus for multivariate functions (I)

Differentiability and gradient; directional derivatives; mean-value theorem; the chain rule.
  • 15.1 - 15.3
4 Mar 18, Mar 20 Differential Calculus for multivariate functions (II)

Tangent lines and tangent planes; local extreme values; absolute extreme values.
  • 15.4 - 15.6
5 Mar 25, Mar 27 Differential Calculus for multivariate functions (III)

Maxima and minima with side conditions; Lagrange multiplier; reconstructing a function from its gradient.
  • 15.7 - 15.9
6 Apr 1 Integral Calculus for multivariate functions (I)

Double integrals and triple integrals; evaluation of integrals by repeated integrals.
  • 16.2 - 16.3 & 16.6 - 16.7
7 Apr 10 Integral Calculus for multivariate functions (II)

Double and triple integrals as limits of Riemann sums; polar coordinate revisited; cylindrical coordinates and spherical coordinates.
  • 16.4 & 16.8 - 16.9
8 Apr 15, Apr 17 Integral Calculus for multivariate functions (III)

Jacobians; change of variable formulae.
  • 16.10
9 Apr 22, Apr 24 Vector Calculus (I) and (II)

Line integrals; the fundamental theorem of line integrals; line integrals with respect to arc length.
  • 17.1-17.2 & 17.4
10 Apr 29 Midterm

The midterm exam covers material from Week 2 to Week 8.
11 May 6, May 8 Vector Calculus (III)

Green’s theorem; parametrized surfaces; surface area.
  • 17.5 - 17.7
12 May 13, May 15 Vector Calculus (IV)

Surface integrals; Stokes’ theorem.
  • 17.7 & 17.10
13 May 20, May 22 Quiz (II). Vector Calculus (V)

Stokes’ theorem; divergence theorem.
  • Quiz (II) covers materials from Week 9 to Week 11.
  • 17.10 & 17.8 - 17.9
14 May 27, May 29 Power series for univariate functions (I)

Infinite series; the integral test; the comparison test; the root test; the ratio test.
  • 11.1 - 11.4
15 Jun 3, Jun 5 Power series for univariate functions (II)

Absolute convergence and conditional convergence; alternating series; Taylor series.
  • 11.5 - 11.7
16 Jun 10, Jun 12 Power series for univariate functions (III)

Power series; differentiation and integrations of power series.
  • 11.8 - 11.9
17 Jun 17 Final Exam

It will covers all the materials from Week 9 to Week 16.