Applied Mathematics (Ⅰ)#
This is a short sequence of lectures on Applied Mathematics (Ⅰ) for undergraduate students at the Department of Atmospheric Sciences, National Taiwan University, written by Yu-Chiao Liang.
Note
Some of the course materials were modified from Professor Gilbert Strang’s, Professor Arthur Mattuck’s, and Professor Steven Strogatz’s course materials, and many online materials.
Class information#
Curriculum Number: AtmSci2011
Curriculum Id Number: 209 27110
Credit: 4
Time: Monday 3,4 (10:20 am - 12:10) and Wednesday 6,7 (1:20 pm - 3:10 pm)
Place: 大氣B105
Co-teach with Prof. Min-Hui Lo
Requirements: basic knowledge in calculus and programming skills
Office hours: TA office hour (Wednesday 12:00-1:20 pm at B105) or by appointment
Grading (100%): homework assignments (50%); final exam (25%); midterm exam (20%); class interaction (5%)
Homework assignments:
Homework due is every Wednesday at 1:20 pm.
50% off before Wednesday’s class ends at 15:10 pm.
No credit after Wednesday’s class.
借別人參考的作業也強制適用以上規則！
Course website: here
Teaching assistants#
Yih Wang (王逸)
Shih-Ni Zhou (周詩倪)
Yi-An Feng (馮以安)
Ya-Fan Chung (鍾雅帆)
Course description#
This course gives students an overview of first and second order linear ordinary differential equations (OEDs) and linear algebra. Basic numerical methods for solving ODEs will also be introduced. These math skills will be applied to atmospheric sciences problems.
Class schedule#
First-order and second-order ODEs:#
Week1 (9/2 & 9/4): responses to (complex) exponential, oscillating, and other inputs
Week2 (9/9 & 9/11): integrating factor, separable euqations and exact solution
Week3 (9/16 & 9/18): phase line, fundamental equation of mechanics, unforced damped and harmonic motions
Week4 (9/23 & 9/25): method of undetermined coefficients, exponential response function, variation of parameters
Numerical ODE and vector calculus#
Week5 (9/30 & 10/2): finite difference,
Week6 (10/7 & 10/9): vector and tensor, gradient, divergence, curl, Gauss’ theorem, Stoke’s theorem
Linear algebra:#
Week7 (10/14 & 10/16): row and column view of matrix multiplication, independence, basis, and dimension
Week8 (10/21): buffer, four fundamental subspaces
Week8 (10/23) midterm exam#
Linear algebra:#
Week9 (10/28 & 10/30): four fundamental subspaces factorization
Week10 (11/4 & 11/6): CR decomposition and LU decomposition:
$$A = LU$$
Week11 (11/11 & 11/13): orthogonalization (Gram-Schmidt) and eigenvalues and diagnolization: $$A = QR$$ $$S = Q\Lambda Q^{T}$$ $$A = X\Lambda X^{T}$$
Week12 (11/18 & 11/20): singular value decomposition (SVD): $$A = U\Sigma V^{T}$$
Numerical methods for linear algebra:#
Week13 (11/25 & 11/27): numerical linear algebra
Solving linear system:#
Week14 (12/2 & 12/4): matrix exponential, and plane: source, sink, saddle
Week15 (12/9 & 12/11): buffer