GG/OCN 312: Advanced Mathematics for Scientists and Engineers I

Instructor: Garrett Apuzen-Ito (, POST 810), Office hours: Thu 1-3:30pm POST 810

TA:  Eric Tong (,  Office hrs: Mon. 10:30am-1:00 pm, Wed. noon-3pm, MSB 305

SOEST Math Tutor:  Alma Trujillo-Costello (, Office hrs: Wed 3-4 pm, Thu 2-3 pm. MSB 306

Classes: MWF, 9:30-10:20 POST 723

Required Textbook: Advanced Engineering Mathematics by Michael. D. Greenberg


By taking GG/OCN 312 students will…

Become comfortable in applying mathematical operations commonly used to solve problems in calculus, vector calculus, and differential equations

Develop familiarity in using computer programs (e.g., Matlab, FreeMat) for solving simple problems, visualization, and applying basic numerical methods

Gain foundational knowledge needed to solve problems in future coursework leading to careers in Earth Sciences, Ocean Sciences, Biology, and Engineering

Be able to learn independently, solve problems creatively, and communicate math clearly and accurately



GG student learning objectives emphasized:

2. Students can apply technical knowledge of relevant computer applications, laboratory methods, field methods, and the supporting disciplines (math, physics, chemistry, biology) to solve real-world problems in geology and geophysics.

3. Students use the scientific method to define, critically analyze, and solve a problem in earth science.

GES student learning outcomes emphasized:

1. Define and explain the basic principles and concepts of chemistry, physics, biology, calculus, geology, geophysics, meteorology, and oceanography. 

2. Apply their understanding of the fundamentals of science and mathematics to the description and quantification of the interactions of the atmosphere, hydrosphere, lithosphere, and biosphere, including humans.

3. Employ the scientific approach to problem solving, and hypothesis formation and testing.

Engineering student learning outcomes emphasized:

An ability to apply knowledge of mathematics, science, and engineering

An ability to identify, formulate, and solve engineering problems

An ability to communicate effectively

A recognition of the need for and an ability to engage in life-long learning


Class format. Course material will be learned by a combination of reading assignments, YouTube lectures, in-class discussions, weekly problem sets, and studying for three exams.  This is a “flipped” class so lectures are to be viewed online via YouTube PRIOR to class and homework will be done in class.


ONLINE LECTURES:  Links to the lecture videos for each data are provided on the syllabus below.  Again the lectures must be viewed prior to class. 


WEEKLY PROBLEM SETS and READING  Reading will reinforce the lectures and will provide formal background to do the problem sets. Problem sets are due on Fridays at 9:30 a.m. at the beginning of class.  Only under extraordinary, incredible, unusually extenuating circumstance can a problem set be turned in late; and you must obtain permission prior to the due date.


Software:  Computer programming literacy is an important part of applied math.  Some of the class exercises will provide practice in this.  For these, you will need Matlab, or a free-ware version such as Freemat or Octave.  Click here for a tutorial on FreeMat and here for a full list of commands and how to use them.


grading:  5% class participation, 45% problem sets, 15% midterm 1, 15% midterm 2, 20% final. 


HONOR CODE:  Our class culture will be built upon respectful and honest interaction.  You are encouraged to work together on your problem sets, but all work turned in for grading (including computer programs) must be yours, and yours alone.  Unless specifically designated, there will be no collaborations during exams.  Cheating will not be tolerated, and everyone is responsible for upholding our honor code.


                                             WORKING SYLLABUS

     Links to videos are blue underlined & should be viewed prior to the day (M,W,F) noted next to them

                                                Reading assignments are red in square brackets                                                                                                          

Introduction and Review

Week 1 [read p 1-6, Schaum’s Outlines]

M 8/21: Class Introduction

W 8/23: Elementary functions:  logarithms, exponentials, trigonometric

F 8/25: Calculus:  Limits and derivatives I                                            PS0, due 8/25:

                          Limits and derivatives II                                          Copies of class notes

                                                                                                            for evaluation

See these tables of functions, limits, derivatives, and integrals

Week 2

M 8/28: Calculus:    Integrals I

                                   Integrals II

W 8/30                     Example of trig. substitution

F 9/1: Taylor Series [pp. 629-636] (see example Matlab script for tips on PS2) and…

            Functions of Multiple Variables and Partial derivatives [pp. 620-624]

                                                                                                                  PS1 due 9/1

Ordinary Differential Equations (ODEs)

Week 3

M 9/4: Happy Labor Day Holiday!

W 9/6: Complex numbers I [pp.1108-1113, 116-1121, 1125-1129]

               Complex numbers II

               Also, review cylindrical and spherical coordinates (e.g., pages from Simmons)

F9/8      Introduction to Differential Equations [Ch1, pp. 1-16]            PS2 due 9/8


Week 4

M9/11   Linear 1st-order ODEs:  [Ch2.2 pp. 18-31 ]

               The homogeneous case

               Non-homogeneous case I: Integrating factor method 


W9/13:Non-homogeneous case II: Variation of parameters

F9/15    Linear first-order ODEs: Applications [Ch2.3 pp. 33-43 ]      PS3 due 9/15

               The RL Electrical circuit                                                        

               Chemical mixing

               Radioactive decay

Week 5

M9/18   Separable Equations [§ 2.4]

W9/20   Solving ODE’s as Exact Differentials [§ 2.5.1]

               Using Integrating Factors to make Exact Differentials [§ 2.5.2]        

F9/22    Numerical Methods [§ 6.1 & 6.2]                                             PS4 due 9/22

               Euler’s Method (20 min)                                                         

               Euler’s Method: Errors and error analysis

Week 6

M9/25   Mid-Point rule (§ 6.3.1)


W9/27   Runge-Kutta (§ 6.3.2)

F9/29    Higher Order ODEs [§ 3.1-3.2]

               Summary and Linear Dependence vs Linear Independence (27 min).

               Review for Exam                                                                       PS 5 due 9/29


Week 7

•M10/2   Midterm 1 on material for Problem Sets 1-4  (midterm 1 study problems)

W10/4   Higher Order ODEs [§ 3.3]

               General solution to the initial value problem (28 min.)

               Solution to Homogeneous Equation:  Constant coefficients (36 min) [§ 3.4]

F 10/6   Harmonic Oscillator, Free Oscillation;

               Application to Harmonic Oscillator (26 min.) [§ 3.5]             PS 6 due 10/6

               2-minute video of a harmonic oscillator in the lab


Matrices and Linear Algebra

Week 8

M10/9 Higher Order Homogeneous Equations with  Non-Constant Coeffs (31 min) [§ 3.6]

W10/11 Higher Order Nonhomogeneous Equations (43 min):  [§ 3.7]

F10/13  Non-damped, Forced Oscillator (33 min) [§ 3.8]                     PS7 due 10/13


Week 9

M10/16 System of Linear Equations [Ch. 8]

               Gauss Elimination (36 min)

               Matrices & Gauss-Jordan Reduction (42 min)

W10/18 Matrices & Matrix Arithmetic (42 min) [§ 10.1-10.3]

•F10/20 Determinants (30 min) [§ 10.4]

              No Class: ***Please volunteer for SOEST Open House       PS8 due 10/20


Week 10

M10/23 Vector Space of a Matrix (17 min) [§9.9.2, §10.5.1]

                 Rank of a Matrix (16 min) [§10.5.1]

                 Rank:  Application to Stoichiometry (11 min) [§10.5.1]

                 Rank, Linear Dependence, and Solution to Ax=c (30 min) [§10.5.2]                

W10/25 Inverse Matrix (31 min) [§10.6.1] and

               Cramer’s Rule (9 min) [§10.6.2]         

F10/27 Basis of a vector (15 min) [§9.9]

              Vector Transformations and Change of Basis (39 min) [§10.7]           PS9 due 10/27


Vectors, Tensors, and Vector Calculus

Week 11:

•M10/30 Eigenvalue Problem [§11.1-11.2]

              Solving the Eigenvalue problem (34 min.)

W11/1 Applications to solving systems of ODEs and Marcov Processes (21 min)

              Eigenvectors of Symmetric Matrices (7 min.) [§11.3]

F11/3 Midterm 2 Study Questions                                                         PS10 due 11/3


Week 12

•M11/6      Midterm 2 on material for Problem Sets 6-9

W11/8  Diagonalization [§11.4]

F11/10 No Class:  Veteran’s Day Holiday                                            


Week 13

M11/13 Vectors in 3D space (36 min) [§ 14.1-14.5]

W11/15 Scalar and Vector Fields (10 min) [§ 16.1-16.2]

                 Divergence (40 min) [§16.3]

F11/17 Gradient (27 min) [§16.4]                                                          PS11 due 11/17


Week 14

M11/20 Curl (29 min) [§16.5]

W11/22 Combinations of and with the “del” operator (31 min) [§16.6]         

F11/24 No Class:  Happy Thanksgiving                                                

Week 15

M11/27 Non-Cartesian Coordinates [§16.7]                                         

               Cylindrical Coordinates I:  Definition (9 min)                        

               Cylindrical Coordinates II:  Div, grad, curl, and Laplacian (28 min)

•W11/29 Non-Cartesian Coordinates [§16.7]                                         

               Spherical Coordinates Definition (7 min)

               Divergence theorem  [§ 16.8.1-16.8.2]: derivation (20 min)

F12/1 Divergence theorem  [§ 16.8.1-16.8.2]:                                       PS12 due 12/1

                                    Application to fluid continuity (17 min)

                                    Application to conservation of heat (18 min)


Week 16:

M12/4 Stokes theorem:  Definition and derivation (21 min) [§ 16.9.1-16.9.2]

                                         Application to Ampere’s law and Maxwell’s Equations (14 min)


W12/6 Final Exam Prep                                                                         PS 13 due Wed 12/6

Study Questions for PS10-11 and 12-13


Final Exam Friday Dec. 15, 9:45-11:45.  Cumulative for ALL problem sets, except Problem Set 5.