College of Engineering and Science 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

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 fallspringwinter math 240 3math 241 3math Precalc algebra & trig, single variable differential calculus Single variable differential calculus Integral calculus, intro differential equations engr 120 2engr engr Problem solving, data analysis, team skills, statistics Statics, strengths, report writing, sketching, design Circuits, engr economics, CAD, design project Freshman Year chem chem Engineering chemistry phys Mechanics Plus 1 additional class -- History, English, Art,... Engineering Fundamentals Teamwork Communication Skills Design Computer Skills Laboratory Experiences 2 classes/labs (2 hrs each) per week Engineering Class

Integrated Courses fallspringwinter math 242 3math 244 3math Basic statistics, multivariable integral calculus Multivariable differential calculus, vector analysis Sequences, series, differential equations engr 220 3engr engr Statics and strengthsEE applications and circuitsThermodynamics memt physics Engineering materialsElectric and magnetic fields, optics Sophomore Year Plus 1 additional class -- History, English, Art,... Engineering Fundamentals Teamwork Communication Skills Design Statistics & Engr Economics Laboratory Experiences 3 hours lab & 2.5 hours lecture per week Engineering Class

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

Now explain why divergence measures source strength. Plus physics. §Paste the pictures

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

“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