Instructor Information

Tony Johnson

Cell Phone: 240-636-2708

Anthony (Tony) Johnson is a professional staff member and senior research scientist at the Johns Hopkins University Applied Physics Laboratory.
Tony is a former US Army Officer and was an Academy Professor in the Department of Mathematical Sciences at the United States Military Academy. He is a graduate of the Naval Postgraduate Applied Mathematics Doctoral program and has over 15 years of experience in computational analysis. He served as the Program Director and Chief Analyst of the Mathematical Sciences Research Program at the United States Military Academy. In addition, he held positions as the Assistant Dean for Research as well as the Director and Chief Analyst of both the Mathematical Sciences Center and the Network Science Center at West Point prior to joining the Johns Hopkins University Applied Physics Laboratory.
He specializes in finite element modeling, has done extensive research using the complex boundary element method to model fluid flow problems, and developed complex variable applications to network science problems.

Course Information

Course Description

This course covers fundamental mathematical tools useful in all areas of applied mathematics, including statistics, data science, and differential equations. The course covers basic principles in linear algebra, multivariate calculus, and complex analysis. Within linear algebra, topics include matrices, systems of linear equations, determinants, matrix inverse, and eigenvalues/eigenvectors. Relative to multivariate calculus, the topics include vector differential calculus (gradient, divergence, curl) and vector integral calculus (line and double integrals, surface integrals, Green’s theorem, triple integrals, divergence theorem and Stokes’ theorem). For complex analysis, the course covers complex numbers and functions, conformal maps, complex integration, power series and Laurent series, and, time permitting, the residue integration method. Prerequisite(s): Differential and integral calculus. Course Note(s): Not for graduate credit.

Course Goal

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 Objectives

  • (Linear Algebra) By the end of the course, students should be able to:
    Solve systems of linear equations using various methods including Gaussian and Gauss-Jordan elimination and inverse matrices.
    Perform matrix algebra, invertibility, and the transpose and understand vector algebra.
    Determine relationship between coefficient matrix invertibility and solutions to a system of linear equations and the inverse matrices.
    Create orthogonal and orthonormal bases: Gram-Schmidt process and use bases and orthonormal bases to solve application problems.
    Understand determinants and their properties.
    Find a basis for the row space, column space and null space of a matrix and find the rank and nullity of a matrix.
    Find the dimension of spaces such as those associated with matrices and linear transformations.
    Find eigenvalues and eigenvectors and use them in applications.
  • (Vector Calculus) By the end of the course, students should be able to:
    Identify, describe, and visualize equations in 3-space.
    Use contour maps for functions of two or three variables to analyze the functions.
    Find and interpret the unit tangent and unit normal vectors and curvature.
    Find and interpret the gradient and directional derivatives for a function at a given point.
    Find the total differential of a function of several variables and use it to approximate incremental change in the function.
    Analyze and solve constrained and unconstrained optimization problems.
    Explain the relationship between multiple and iterated integrals.
    Evaluate multiple integrals either by using iterated integrals or approximation methods.
    Relate rectangular coordinates in 3-space to spherical and cylindrical coordinates, and use spherical and cylindrical coordinates as an aid in evaluating multiple integrals.
    Define a line integral, and use it to find the total change in a function given its gradient field.
    Calculate and interpret the flow and divergence for a vector field.
  • (Complex Analysis) By the end of the course, students should be able to:
    Express complex-differentiable functions as power series.
    Find parametrizations of curves, and compute complex line integrals directly.
    Use antiderivatives to compute line integrals.
    Use Cauchy’s integral theorem and formula to compute line integrals.
    Identify the isolated singularities of a function and determine whether they are removable,
    poles, or essential.
    Compute innermost Laurent series at an isolated singularity, and determine the residue.
    Use the residue theorem to compute complex line integrals and real integrals.
    Understand the theory of integration of analytic functions and the applications of the Cauchy-Goursat theorem. 
    Understand the simpler examples of conformal mappings.

When This Course is Typically Offered

This course is typically offered in the fall term at APL.


  • Matrices, Vectors, 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 Equations
  • Cauchy's Integral Formula
  • Power Series
  • Laurent Series
  • Residue Integration Method
  • Final Exam

Student Assessment Criteria

Class Preparation and Participation 10%
Quizzes 30%
Midterm Exam 30%
Final Exam 30%

Timely feedback on students' performance is an established learning tool. I will endeavor to return all graded assignments back to you, as quickly as possible.

Class preparation and participation involves demonstration that one has completed the homework problems and can articulate problem solving strategy when called upon in class.

Quizzes will normally be graded and returned to you before the start of the next graded assignment. Homework is your responsibility and will be address only to the extent that is required to prepare for quizzes and exams.

A final grade of A indicates achievement of consistent excellence and distinction throughout the course—that is, conspicuous excellence in all aspects of assignments and discussions every week.

A final 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.

Final grades will be determined by the following weighting:

100-90 = A

89-80 = B

79-70 = C

<70 = F

The policy on exams and quizzes is that the work will be your own.  However, collaboration for homework is encouraged, but with caution. Be certain that you are learning the concepts and not simply copying the answers.

Computer and Technical Requirements

Mathematica and Matlab will be used to illustrate some principles used in this course.  Students may use Mathematica, Matlab or another type of mathematical software package or other software language, such as Java, C++ etc. Links to other mathematical packages (SciLab, Octave) will be provided as an alternative to Matlab. Computer software will not be used on any graded assignment.

Participation Expectations

Blackboard will be used for this course and students are expected to participate in any discussion posted on the course website.  Students will also be expected to participate in class discussion of concepts and example problems.

Homework will be given at the end of each class. Quizzes based on the material covered in the last class will be given at the beginning of class and should take no more that 30 minutes.


Textbook information for this course is available online through the MBS Direct Virtual Bookstore.

Course Notes

There are no notes for this course.

Final Words from the Instructor

The textbook for this course is Kreyszig, E. (2011).  Advanced Engineering Mathematics (10th ed.). Hoboken, NJ: John Wiley & Sons Inc.

(Last Modified: 07/22/2019 06:27:40 PM)