Louisiana Tech University
College of Engineering and Science
Integrating Mathematics,
Engineering and the Sciences
in Multivariable Calculus
Bernd S. W. Schröder
Program of Mathematics
and Statistics
College of Engineering and Science
The Motivation Behind
Curriculum Integration
 Knowledge acquired in context is
deeper and more readily applied than
knowledge acquired in isolation.
 You’ll still have to acquire the
knowledge.
College of Engineering and Science
Tech’s Commitment to
Curriculum Integration
 All engineering and science students
are either in the integrated engineering
curriculum or the integrated science
curriculum.
 This is a functioning setup, not a pilot
program.
College of Engineering and Science
Integrated Courses
fall
Freshman Year
winter
math 240
3
Precalc algebra & trig, single
variable differential calculus
engr 120
2
Problem solving, data
analysis, team skills, statistics
chem 100
2
Engineering chemistry
math 241
spring
3
Single variable differential
calculus
engr 121
math 242
Integral calculus, intro
differential equations
2
engr 122
Statics, strengths, report
writing, sketching, design
chem 101
3
2
Circuits, engr economics,
CAD, design project
2
phys 201
Engineering chemistry
Mechanics
Plus 1 additional class -- History, English, Art, ...
Engineering Fundamentals
Design
Teamwork
Computer Skills
Communication Skills
Laboratory Experiences
2 classes/labs (2 hrs each) per week
3
Integrated Courses
fall
Sophomore Year
winter
math 242
3
Basic statistics, multivariable
integral calculus
engr 220
3
Statics and strengths
memt 201
spring
math 244
3
Multivariable differential
calculus, vector analysis
math 245
Sequences, series, differential
equations
engr 221
3
engr 222
EE applications and circuits
2
Engineering materials
3
physics 202
Thermodynamics
3
Electric and magnetic fields,
optics
Plus 1 additional class -- History, English, Art, ...
Engineering Fundamentals
Design
Teamwork
Statistics & Engr Economics
Communication Skills
Laboratory Experiences
3 hours lab & 2.5 hours lecture per week
3
Key Concepts in Multivariable
Calculus
 Multivariable Integration (centers of mass, geometry)
 Multivariable Differentiation (gradients, nabla
operator, divergence, curl)
 Vector fields (gravity, e/m, fluid flow)
 Line integrals (work)
 Surface integrals (throughput)
 Integral Theorems (the connection)
 Think of all the classes that depend on this.
College of Engineering and Science
Goal
 More and deeper exposure to the key
concepts
 Better connection to applications
 As much internal connectivity as possible
 Direct access to three dimensions
 Will decompress the crucial parts
College of Engineering and Science
A Typical Order of Presentation
 Vector valued
functions
 Multivariable
Differentiation,
including definition of
differentiability and
optimization
 Multivariable
Integration
 Vector Fields
 Line Integrals






Green’s Theorem
Surface Integrals
Divergence and Curl
Stokes’ Theorem
Divergence Theorem
End of course
(Danger, Will
Robinson)
College of Engineering and Science
How do we connect better with
applications?
Consider the following order.
 Multivariable Integration (yes, it can be
done before differentiation)
 Vector valued functions (emphasis on
dynamics)
 Partial derivatives up to the gradient
 Now we take off
College of Engineering and Science
Vector Fields
 Vector fields
 Gradient fields, physics
 Line integrals,
fundamental theorem
 Gradient is an underlying
theme
College of Engineering and Science
Surface Integrals (new underlying
theme)
 Emphasis on
throughput
 Visualization with
fluids, electricity,
magnetism
 Gauss’ Law
College of Engineering and Science
Surface Integrals (new underlying
theme)
College of Engineering and Science
Divergence Theorem and Divergence
 Complements the observation that the total flux
of a field over a closed surface is proportional to
the sources enclosed (Gauss’ law)
 Can be used to derive Gauss law from the electric
or gravitational field of a point charge
College of Engineering and Science
 Paste the pictures
Now explain
why
divergence
measures
source
strength. Plus
physics.
Stokes’ Theorem
 Complements the observation that currents are
surrounded by magnetic fields (Ampere’s law).
College of Engineering and Science
Now explain
why the curl
measures
vorticity. Plus
physics.
More on Nabla
 Also: Maxwell equations, div(curl(F))=0, etc.
 Let students do more integrals
College of Engineering and Science
Finishing the Course





Navier-Stokes Equations (just the idea)
Green’s Theorem
Multivariable differentiability, tangent planes
Optimization
The residual differentiability topics go pretty
fast, because the computational side is no
longer a challenge
 In emergencies there are obvious, better
sacrifices than in the standard line up.
College of Engineering and Science
We’re Not Fitting Square Pegs
into Round Holes, but even if …
College of Engineering and Science
College of Engineering and Science
“You have to
hit it real hard,
it will go in.”
College of Engineering and Science
The Path to
Academic Success
When in doubt,
hit it real hard.
When not in
doubt, it can’t
hurt either.
College of Engineering and Science