Multivariable calculus: What is it good for?

Transcription

1 : What is it good for? Department of Mathematics and Statistics, UMBC rouben/ January 2011

2 Purpose The purpose of these slides is to show that the material that one learns in a multivariable calculus course is good for something. In fact, it would be hard to be convinced otherwise after having a look at the slides that follow. holds the key to understanding the world around us. Make it your business to master that key!

3 Notation = i x + j y + k z the gradient operator 2 = 2 x y z 2 the Laplacian operator i, j, k orthogonal unit vectors x, y, z Cartesian coordinates

4 Fluid Mechanics The Navier-Stokes equations Good for simulation and analysis of incompressible fluid flow in ducts and tubes, rivers, ocean, blood in arteries,... u + ( u)u = ν 2 u 1 ρ p + f u = 0 balance of momentum incompressibility u velocity p pressure ν kinematic viscosity f body force per unit mass ρ density

5 Gas Dynamics Equations of gas dynamics Good for understanding flight dynamics, supersonic flight, jet engines, rocket propulsion, orbital reentry vehicles,... ρ ( u + ( u)u) = µ 2 u + λ ( u) p + ρf balance of momentum ρ + (ρu) = 0 ρ = ϕ(p) conservation of mass equation of state u velocity p pressure µ dynamic viscosity ρ density λ 2nd viscosity f body force per unit mass

6 Elasticity Navier s equations of linear elasticity Good for analysis and design of bridges, towers, machine components, auto and aircraft body,... ρ 2 u = S + f balance of momentum 2 E = 1 2( u + ( u) T ) the infinitesimal strain S = C[E] the constitutive equation u displacement C the elasticity tensor S stress tensor ρ density E strain tensor f body force per unit volume

7 Chemical kinetics Reaction-diffusion equations Good for analysis of chemical reactions, combustion, physiology, morphogenesis, blood clotting,... u 1 u 2 u n = D 1 2 u 1 + f 1 (u 1, u 2,... u n ) = D 2 2 u 2 + f 2 (u 1, u 2,... u n ) = D n 2 u n + f n (u 1, u 2,... u n ) u i D i f i concentration of the i th species diffusion rate of the i th species reaction rate of the i th species

8 Electromagnetism Maxwell s equations Good for study of propagation of light, electric, and magnetic fields, design of radio and radar antennas, cell phones, electric motors, fiber optics,... E = ρ ɛ 0 Gauss s law, electric field B = 0 E = B E B = µ 0 J + µ 0 ɛ 0 Gauss s law, magnetic field Faraday s law of induction Ampère s circuital law E electric field J current density B magnetic field ρ charge density ɛ 0 electric constant µ 0 magnetic constant

9 Quantum Mechanics Schrödinger s equation for a single particle Good for understanding the structures of molecules, atoms, subatomic systems,... i Ψ = 2 2m 2 Ψ + V(x)Ψ Ψ the wave function V(x) the potential function Planck s constant m particle mass i = 1

10 Heat conduction The heat equation Good for design of heat exchangers, radiators, power plants, computer CPU heat sinks,... cρ T = (κ T) T temperature c specific heat κ conductivity ρ density

11 Heat conduction The heat equation Good for design of heat exchangers, radiators, power plants, computer CPU heat sinks,... cρ T = (κ T) T temperature c specific heat κ conductivity ρ density... and coupled with the equations of gas dynamics Good for atmospheric dynamics, weather forecasting, global warming,...

12 Anything else? What you have seen in the preceding slides is far from exhaustive. I put together these after only a few minutes of thought and naturally included stuff that I know a thing or two about. Certainly there is a slew of applications which I have failed to mention. Don t be discouraged if you don t see your favorite subject included here. If you know about interesting applications and can supply brief descriptions with enough details, I will consider adding them here with due acknowledgment. Let me know.