625.250.81 - Multivariable Calculus and Complex Analysis

Course Structure

The course materials are divided into 14 modules which can be accessed by clicking Modules on the menu. A module will have several sections including overview and objectives, readings, video lectures and related content, discussions, and assignments. Students should check the Calendar and Announcements regularly for changes in assignment due dates. Modules begin Midnight Monday morning and complete on Sundays evenings @ 11:59pm.

Course Topics

• Matrices and Linear Systems
• Matrix Eigenvalue Problems
• Vector Differential Calculus
• Gradient, Divergence, and Curl
• Green’s Integral Theorem
• Gauss and Stokes Theorems
• Cauchy-Riemann Equations
• Complex Trigonometric Functions
• Cauchy’s Integral Theorem
• Cauchy’s Integral Formula
• Power Series
• Laurent Series
• Residue Integration Method
• Conformal Mappings

Course Goals

This course is designed to provide a comprehensive and thorough treatment of engineering mathematics by introducing students of engineering, physics, mathematics, computer science, and related fields to those areas of applied mathematics that are the most relevant for solving practical problems.

Course Learning Outcomes (CLOs)

• Solve systems of linear equations using various methods including Gaussian and Gauss-Jordan elimination and inverse matrices.
• Construct a basis for the row space, column space, and null space of a matrix and find the rank and nullity of a matrix.
• Examine the dimension of spaces such as those associated with matrices and linear transformations.
• Apply eigenvalues and eigenvectors and use them in applications.
• Apply and interpret the gradient and directional derivatives for a function at a given point.
• Analyze the flow and divergence for a vector field.
• Explain the relationship between multiple and iterated integrals.
• State the definition of a line integral, and use it to find the total change in a function given its gradient field.
• Apply Green’s Theorem and Stokes's Theorem in a region.
• Apply the Divergence Theorem of Gauss to a closed region in space.
• Construct the parametrization of curves, and use it to compute complex line integrals directly.
• Apply Cauchy’s integral theorem and formula to compute line integrals.
• Construct complex-differentiable functions as power series.
• Compute innermost Laurent series at an isolated singularity, and determine the residue.
• Apply the residue theorem to compute complex line integrals and real integrals.
• Apply conformal mappings to determine the properties of functions.

Textbooks

Kreyszig, E. (2011).  Advanced Engineering Mathematics (10th ed.). Hoboken, NJ: John Wiley & Sons Inc.

Other Materials & Online Resources

Webcam, microphone (if provided with computer, this is usually adequate along with adequate lighting and no background sound interference).

Student Coursework Requirements

It is expected that each module will take approximately 10-12 hours per week to complete. Here is an approximate breakdown: reading the assigned sections of the textbook (approximately 4 hours per week), listening to the audio annotated slide presentations (approximately 2 hour per week) as well as the Lightboard Studio demonstrations (approximately 1 hour per week), and working through the module assignments (approximately 4-5 hours per week).

This course will consist of five basic student requirements:

Weekly problem assignments (ungraded)
Weekly problem sets are not graded. Students are encouraged to collaborate on these problems in the public discussion board space provided for each module. Instructor monitors and provides guidance as necessary. In addition, instructor is available via email or weekly synchronous office hours.

Discussion board assignments (10% of final grade)
Weekly discussion board assignments require you to think about real-world applications for the mathematical concepts you will be learning. You are required to respond to each discussion prompt as well as respond to at least one other student’s post. A rubric for how discussion assignments will be evaluated by the instructor is posted in module 1.

Module Assignments (20% of final grade)
Weekly problem assignments are graded. Students may not collaborate on these problems in the public discussion board. Instructor is available via email or weekly synchronous office hours to provide guidance as necessary.

Quizzes (20% of final grade)
Quizzes cover material from previous week and cover material in the weekly problem set, and module assignments. Quizzes are timed, hand written, and submitted online in Canvas. Once you log into a quiz you will have a set amount of time to complete it.

Midterm Exam (20% of final grade)
There is a midterm exam. It is timed, hand written, and submitted online in Canvas. Once you log into the midterm exam you will have a set amount of time to complete it.

Final Exam (30% of final grade)
The final exam is comprehensive. It is timed, hand written, and submitted online in Canvas. Once you log into the midterm exam you will have a set amount of time to complete it. You will not receive feedback from the final exam. You will only receive the letter grade earned on your attempt. Instructor will provide feedback in special session if requested.

Grading Policy

Assignments are due according to the dates posted in the Canvas course site. Students may check these due dates in the Course Calendar or the Assignments in the corresponding modules. Grades will post no later than one week after assignment due dates. A grade of A indicates achievement of consistent excellence and distinction throughout the course—that is, conspicuous excellence in all aspects of assignments and discussion in every week. A grade of B indicates work that meets all course requirements on a level appropriate for graduate academic work. These criteria apply to both undergraduates and graduate students taking the course.

The following grades are used for this course: A+, A, A– (excellent), B+, B, B– (good), C (unsatisfactory), F (failure), I (incomplete). A grade of F indicates the student’s failure to complete or comprehend the course work.

A course for which an unsatisfactory grade (C or F) has been received may be retaken. The original grade is replaced with an R. If the failed course includes laboratory, both the lecture and laboratory work must be retaken unless the instructor indicates otherwise. A grade of W is issued to those who have dropped the course after the refund period but before the drop deadline. The transcript is part of the student’s permanent record at the university. No grade may be changed except to correct an error, to replace an incomplete with a grade, or to replace a grade with an R.

The Whiting School assumes that students possess acceptable written command of the English language. It is proper for faculty to consider writing quality when assigning grades.

For incomplete grades, please see the Graduate Programs catalogue for the Whiting School of Engineering.

The course grading scale is the following:

Academic Policies

Deadlines for Adding, Dropping and Withdrawing from Courses

Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.

Academic Misconduct Policy

All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students.

This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.

Students with Disabilities - Accommodations and Accessibility

Johns Hopkins University values diversity and inclusion. We are committed to providing welcoming, equitable, and accessible educational experiences for all students. Students with disabilities (including those with psychological conditions, medical conditions and temporary disabilities) can request accommodations for this course by providing an Accommodation Letter issued by Student Disability Services (SDS). Please request accommodations for this course as early as possible to provide time for effective communication and arrangements.

For further information or to start the process of requesting accommodations, please contact Student Disability Services at Engineering for Professionals, ep-disability-svcs@jhu.edu.

Student Conduct Code

The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically. 

For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/

Classroom Climate

JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity. 
 
If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).

Course Auditing

When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.