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Course Contents Updated Dec. 10, 2007
Mechanics 1
Covered Sunday
9/16
Distances and Sizes 1
Covered Sunday 9/16
Forces and Translational Equilibrium 3
Covered Sunday 9/16
Rotational Equilibrium 4
Covered Sunday 9/16
Vector Product 5
Covered Sunday 9/16
Force in the Achilles Tendon 6
Covered Sunday 9/16 + Section
Forces on the Hip 7
Covered Sunday 9/16 + Section
The Use of a Cane 9
Covered Sunday 9/16 + Section
Work 10
Covered Sunday 9/23
Stress and Strain 12
Covered Sunday 9/23
Shear 13
Covered Sunday 9/23
Hydrostatics 13
Covered Sunday 9/23
Buoyancy 14
Covered Sunday 9/23
Compressibility 15
Covered Sunday 9/23
Viscosity 15
Covered Sunday 9/23
Viscous Flow in a Tube 15 Covered Sunday 9/30
+ Section
Pressure-Volume Work 18
The Human Circulatory System 19 Covered Sunday
9/30
Turbulent Flow and the Reynolds Number 21
Covered Sunday 9/23 (only Reynold's number
calculations)
Exponential Growth and Decay 31
Covered Sunday 9/16
Exponential Growth 31
Covered Sunday 9/16
Exponential Decay 33
Covered Sunday 9/16
Semilog Paper 34
Covered Sunday 9/16
Variable Rates 35
Covered Sunday 9/30
Clearance 36
Covered Sunday 9/16
Multiple Decay Paths 37
Covered Sunday 9/30
Decay Plus Input at a Constant Rate 38
Covered Sunday 9/30
Decay with Multiple Half-Lives and Fitting Exponentials 38
Covered Sunday 9/30
The Logistic Equation 39
Covered Sunday 9/16
Log-log Plots, Power Laws, and Scaling 39
Covered Sunday 9/30
Log-log Plots and Power Laws 39
Covered Sunday 9/30
Food Consumption, Basal Metabolic Rate, and Scaling 40
Covered Sunday 9/30 + Section
Systems of Many Particles 49
Covered Sunday 9/30 (Basic introduction to the
topic)
Gas Molecules in a Box 50
Covered Sunday 10/21
Microstates and Macrostates 51
Covered Sunday 10/21
The Energy of a System: The First Law of Thermodynamics 53
Covered Sunday 10/21
Ensembles and the Basic Postulates 54
Covered Sunday 10/21
Thermal Equilibrium 56
Covered Sunday 10/28
Entropy 58
Covered Sunday 10/28
The Boltzmann Factor 58
Covered Sunday 10/28
The Nernst Equation 59
Covered Sunday 10/28
The Pressure Variation in the Atmosphere 60
Covered Sunday 10/28
Equipartition of Energy and Brownian Motion 60
Covered Sunday 10/28
Heat Capacity 61
Equilibrium When Particles Can Be Exchanged: The Chemical Potential 61
Concentration Dependence of the Chemical Potential 62
Systems That Can Exchange Volume 63
Extensive Variables and Generalized Forces 64
The General Thermodynamic Relationship 64
The Gibbs Free Energy 65
Gibbs Free Energy 65
An Example: Chemical Reactions 66
The Chemical Potential of a Solution 67
Transformation of Randomness to Order 69
Transport in an Infinite Medium 81
Flux, Fluence, and Continuity 81
Covered Sunday 11/4
Definitions 81
Covered Sunday 11/4
The Continuity Equation in One Dimension 82
Covered Sunday 11/4
The Continuity Equation in Three Dimensions 82
The Integral Form of the Continuity Equation 83
The Differential Form of the Continuity Equation 84
The Continuity Equation with a Chemical Reaction 85
Drift or Solvent Drag 85
Covered Sunday 11/4
Brownian Motion 85
Covered Sunday 11/4
Motion in a Gas: Mean Free Path and Collision Time 85
Covered Sunday 11/11
Motion in a Liquid 86
Diffusion: Fick's First Law 87
Covered Sunday 11/4
The Einstein Relationship Between Diffusion and Viscosity 89
Fick's Second Law of Diffusion 91
Covered Sunday 11/11 -
INCLUDED IN MIDTERM EXAM
Time-Independent Solutions 92
Example: Steady-State Diffusion to a Spherical Cell and End Effects 94
Diffusion Through a Collection of Pores, Corrected 95
Diffusion from a Sphere, Corrected 95
How Many Pores Are Needed? 96
Other Applications of the Model 96
Example: A Spherical Cell Producing a Substance 96
Drift and Diffusion in One Dimension 98
A General Solution for the Particle Concentration as a Function of Time 99
Diffusion as a Random Walk 100
Transport Through Neutral Membranes 111
Covered Sunday 11/11 -
NOT INCLUDED IN EXAM
Membranes 111
Covered Sunday 11/11
Osmotic Pressure in an Ideal Gas 112
Covered Sunday 11/11
Osmotic Pressure in a Liquid 114
Covered Sunday 11/11
Some Clinical Examples 115
Covered Sunday 11/25
Edema Due to Heart Failure 116
Covered Sunday 11/25
Nephrotic Syndrome, Liver Disease, and Ascites 116
Covered Sunday 11/25
Edema of Inflammatory Reaction 116
Covered Sunday 11/25
Headaches in Renal Dialysis 116
Covered Sunday 11/25
Osmotic Diuresis 116
Covered Sunday 11/25
Osmotic Fragility of Red Cells 117
Covered Sunday 11/25
Volume Transport Through a Membrane 117
Solute Transport Through a Membrane 119
Example: The Artificial Kidney 120
Covered Sunday 11/25
Countercurrent Transport 121
(Concept Only) Covered Sunday 11/25
A Continuum Model for Volume and Solute Transport in a Pore 122
Volume Transport 123
Solute Transport 124
Summary 127
Reflection Coefficient 127
The Effect of Pore Walls on Diffusion 128
Net Force on the Membrane 128
Impulses in Nerve and Muscle Cells 135
Physiology of Nerve and Muscle Cells 135
Covered Sunday 12/02
Coulomb's Law, Superposition, and the Electric Field 137
Covered Sunday 12/02
Gauss's Law 138
Covered Sunday 12/02
Potential Difference 141
Covered Sunday 12/02
Conductors 142
Covered Sunday 12/02
Capacitance 143
Covered Sunday 12/02
Dielectrics 143
Covered Sunday 12/02
Current and Ohm's Law 145
Covered Sunday 12/02
The Application of Ohm's Law to Simple Circuits 146
Covered Sunday 12/02
Charge Distribution in the Resting Nerve Cell 148
Covered Sunday 12/02
The Cable Model for an Axon 149
Covered Sunday 12/02
Electrotonus or Passive Spread 153
Covered Sunday 12/09
The Hodgkin-Huxley Model for Membrane Current 154
Voltage Clamp Experiments 154
Covered Sunday 12/02
Potassium Conductance 156
Sodium Conductance 157
Leakage Current 158
Voltage Changes in a Space-Clamped Axon 158
Propagating Nerve Impulse 159
Covered Sunday 12/09
Myelinated Fibers and Saltatory Conduction 160
Covered Sunday 12/09
Membrane Capacitance 163
Rhythmic Electrical Activity 164
The Relationship Between Capacitance, Resistance, and Diffusion 165
Capacitance and Resistance 165
Capacitance and Diffusion 165
The Exterior Potential and the Electrocardiogram 177
The Potential Outside a Long Cylindrical Axon 177
The Exterior Potential is Small 179
The Potential Far From the Axon 180
The Exterior Potential for an Arbitrary Pulse 181
Electrical Properties of the Heart 184
The Current-Dipole Vector of the Heart as a Function of Time 186
The Electrocardiographic Leads 186
Some Electrocardiograms 189
Refinements to the Model 189
The Axon Has a Finite Radius 190
Nonuniform Exterior Conductivity 191
Anisotropic Conductivity: The Bidomain Model 191
Electrical Stimulation 192
The Electroencephalogram 196
Biomagnetism 203
The Magnetic Force on a Moving Charge 203
The Magnetic Field of a Moving Charge or a Current 205
The Divergence of the Magnetic Field Is Zero 205
Ampere's Circuital Law 205
The Biot-Savart Law 206
The Displacement Current 207
The Magnetic Field Around an Axon 208
The Magnetocardiogram 209
The Magnetoencephalogram 211
Electromagnetic Induction 213
Magnetic Stimulation 214
Magnetic Materials and Biological Systems 214
Magnetic Materials 215
Measuring Magnetic Properties in People 216
Magnetic Orientation 217
Detection of Weak Magnetic Fields 218
Electricity and Magnetism at the Cellular Level 227
Donnan Equilibrium 227
Potential Change at an Interface: The Gouy-Chapman Model 229
Ions in Solution: The Debye-Huckel Model 231
Saturation of the Dielectric 233
Ion Movement in Solution: The Nernst-Planck Equation 234
Zero Total Current in a Constant-Field Membrane: The Goldman Equations 236
Membrane Channels 238
Noise 242
Shot Noise 242
Johnson Noise 242
Sensory Transducers 243
Possible Effects of Weak External Electric and Magnetic Fields 244
Introduction 244
Effects of Strong Fields 244
Fields in Homes are Weak 244
Epidemiological Studies 245
Laboratory Studies 245
Reviews and Panel Reports 245
Electric Fields in the Body 246
Electric Fields in a Spherical Cell 246
Electrical Interactions and Noise 247
Magnetic Interactions and Noise 247
Feedback and Control 255
Steady-State Relationships Among Variables 256
Determining the Operating Point 257
Regulation of a Variable and Open-Loop Gain 257
Approach to Equilibrium without Feedback 259
Approach to Equilibrium in a Feedback Loop with One Time Constant 259
A Feedback Loop with Two Time Constants 262
Models Using Nonlinear Differential Equations 263
Describing a Nonlinear System 264
An Example of Phase Resetting: The Radial Isochron Clock 265
Stopping an Oscillator 268
Difference Equations and Chaotic Behavior 268
The Logistic Map: Period Doubling and Deterministic Chaos 269
The Bifurcation Diagram 270
Quasiperiodicity 271
A Feedback Loop with a Time Constant and a Fixed Delay 272
Negative Feedback Loops: A Summary 273
Additional Examples 274
Cheyne-Stokes Respiration 274
Hot Tubs and Heat Stroke 274
Pupil Size 274
Oscillating White-Blood-Cell Counts 275
Waves in Excitable Media 275
Period Doubling and Chaos in Heart Cells 276
The Method of Least Squares and Signal Analysis 285
The Method of Least Squares and Polynomial Regression 285
The Simplest Example 285
A Linear Fit 286
A Polynomial Fit 287
Variable Weighting 288
Nonlinear Least Squares 288
The Presence of Many Frequencies in a Periodic Function 289
Fourier Series for Discrete Data 290
Introducing the Fourier Series 290
Equally Spaced Data Points Simplify the Equations 290
The Standard Form for the Discrete Fourier Transform 291
Complex Exponential Notation 291
Example: The Square Wave 292
Example: When the Sampling Time Is Not a Multiple of the Period of the Signal
292
Example: Spontaneous Births 293
Example: Photosynthesis in Plants 294
Pitfalls of Discrete Sampling: Aliasing 294
Fast Fourier Transform 295
Fourier Series for a Periodic Function 295
The Power Spectrum 296
Correlation Functions 298
Cross-Correlation of a Pulse 298
Cross-Correlation of a Nonpulse Signal 299
Cross-Correlation Example 299
Autocorrelation 299
Autocorrelation Examples 299
The Autocorrelation Function and the Power Spectrum 300
Nonperiodic Signals and Fourier Integrals 301
Introduce Negative Frequencies and Make the Coefficients Half as Large 301
Make the Period Infinite 302
Complex Notation 303
Example: The Exponential Pulse 303
The Delta Function 304
The Energy Spectrum of a Pulse and Parseval's Theorem 304
Parseval's Theorem 305
Example: The Exponential Pulse 305
The Autocorrelation of a Pulse and Its Relation to the Energy Spectrum 305
Noise 306
Correlation Functions and Noisy Signals 308
Detecting Signals in Noise 308
Signal Averaging 308
Power Spectral Density 309
Units 309
Frequency Response of a Linear System 310
Example of Calculating the Frequency Response 311
The Decibel 311
Example: Impulse Response 312
The Frequency Spectrum of Noise 312
Johnson Noise 312
Shot Noise 315
1/f Noise 315
Testing Data for Chaotic Behavior 316
Embedding 316
Surrogate Data 316
Stochastic Resonance 317
Threshold Detection 317
Feynman's Ratchet 318
Images 325
The Convolution Integral and its Fourier Transform 325
One Dimension 325
Two Dimensions 326
The Relationship Between the Object and the Image 327
Point-Spread Function 327
Optical-, Modulation-, and Phase-Transfer Functions 328
Line- and Edge-Spread Functions 329
Spatial Frequencies in an Image 329
Summary 331
Two-Dimensional Image Reconstruction from Projections by Fourier Transform 331
Reconstruction from Projections by Filtered Back Projection 332
An Example of Filtered Back Projection 335