Sat, 01 Dec 2001
EGM 6611 Continuum Mechanics, Fall 2001
IMPORTANT ANNOUCEMENTS:
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Additional remarks on lectures: (in reverse chronological order)

Wed, 5 Dec 01: No class.
Exam 2 (on-campus students), in class, closed book, closed notes, 1 hour.
There will be no make-up exam for invalid reasons.
* Fri, 30 Nov 2001

Mon, 3 Dec 01:
Lecture 40:
General principles (cont'd): First law of thermodynamics, energy equation. Linear constitutive relation, Lame's constants, relation to Young's modulus and Poisson's ratio, Navier-Stokes equation for compressible flow and governing equation for solids, Stokes equation for incompressible flow and incompressibility constraint.
Application: Biology, microrheology (what is), fluid-solid interaction, polymer.
Handout: Microrheology paper on One- and two-particle microrheology.
Reading assignment: Malvern, Sec 5.4 (to p.231) ; Sec 6.1-6.3 .
* Sat, 01 Dec 2001

WEEK 16

Fri, 30 Nov 01:
Lecture 39:
General principles (cont'd): Balance of angular momentum (cont'd): divergence theorem, use of balance of linear momentum, du-Bois/Reymond lemma, symmetry of Cauchy stress tensor, case of non-polar media. Principle of virtual work (virtual velocity, virtual power).
Reading assignment: Malvern, Sec 5.1-5.3 (to p.220), 5.5, 5.4 (to p.230) .
* Fri, 30 Nov 2001

Wed, 28 Nov 01:
Lecture 38:
General principles (cont'd): Remark on the proof of Leibniz rule for divergence, gradient, material time derivative of products of various tensor fields.
Balance of angular momentum (cont'd): Generalization of divergence theorem to second-order tensors with cross product.
Reading assignment: Malvern, Sec 5.1-5.3 (to p.220), 5.5, 5.4 (to p.230) .
* Wed, 28 Nov 2001

Mon, 26 Nov 01: HW12 due.
Lecture 37:
General principles (cont'd): Road map 2 to continuity equation via a direct proof of the lemma on the time rate of Jacobian determinant, and Reynolds transport theorem. Review balance of linear momentum, Cauchy equation of motion. Balance of angular momentum: Material time derivative of angular momentum, Leibniz rule for material time derivative of the cross product of two vector fields, sum of applied moments, polar and non-polar media, divergence theorem.
Reading assignment: Malvern, Sec 5.1-5.3 (to p.220), 5.5, 5.4 (to p.230) .
* Wed, 28 Nov 2001

WEEK 15

Fri, 23 Nov 01: Thanksgiving. No class.

Wed, 21 Nov 01:
Lecture 36:
Q&A: Solve problem 9, Sec. 4.2, p.136. Review of transformation of coordinates of tensors, definition of tangent vectors.
General principles (cont'd): Direct proof of the lemma on the time rate of Jacobian determinant revisited. Reynolds Transport theorem and generalization to vector field. Balance of linear momentum: material time derivative of linear momentum, sum of forces and divergence theorem, du-Bois/Reymond lemma, Cauchy equation of motion.
Reading assignment: Malvern, Sec 4.3 ; 5.1-5.3 (to p.220) .
* Mon, 26 Nov 2001

Mon, 19 Nov 01: HW12 due.
Lecture 35:
General principles (cont'd): Remark on material time derivative versus partial time derivative. Continuity equation via a direct proof of the lemma on the time rate of Jacobian determinant. Reynolds transport theorem: Leibniz rule for material time derivative of product of scalar functions.
Reading assignment: Malvern, Sec 4.3 ; 5.1, 5.2 .
* Wed, 28 Nov 2001

WEEK 14

Fri, 16 Nov 01:
Lecture 34:
General principles (cont'd): Review: Road map 1 to continuity equation and lemma on time rate of Jacobian determinant of deformation map, condition for incompressible flow (constraint on divergence of velocity field). Concrete example of transformation of coordinates in relation to material time derivative of volume integral in spatial configuration.
Continuity equation via a direct proof of the lemma on the time rate of Jacobian determinant (starting).
Reading assignment: Malvern, Sec 4.3 ; 5.1, 5.2 .

Did you know: Anne Cope, a student in our class, was kind enough to provide me with the following photos of her research on the measurements of wind speed during hurricanes:

Wed, 14 Nov 01:
Lecture 33:
General principles (cont'd): Material time derivative of a scalar field: Density function, Leibniz rule for divergence of the product of a scalar function and a vector field, continuity equation with material time derivative of density function. Conservation of mass revisited: Material time derivative of volume integral, lemma on time rate of change of Jacobian (indirect proof by continuity equation).
Reading assignment: Malvern, Sec 4.3 ; 5.1, 5.2 .
* Wed, 28 Nov 2001

Mon, 12 Nov 01: Veteran Day; no class.

WEEK 13

Fri, 9 Nov 01:
Lecture 32:
General principles (cont'd): Proof of divergence theorem (cont'd). Material time derivative of a vector field: Velocity, material and spatial forms; acceleration, material and spatial forms.
Exam1 returned; statistics.
Reading assignment: Malvern, Sec 4.3 ; 5.1, 5.2 .
HW12: Please download.

Wed, 7 Nov 01:
Lecture 31:
General principles (cont'd): Conservation of mass: Equation of continuity by physical reasoning of balance of mass, divergence theorem (proof), and du-Bois/Reymond lemma (proof).
Reading assignment: Malvern, Sec 4.3 ; 5.1, 5.2 .

Mon, 5 Nov 01:
Lecture 30:
Material finite-strain tensor (cont'd): Angle change, interpretation, reduction to small-deformation case; volume change: recall the relationship between infinitesimal volumes due to transformation of coordinates, determinant of Jacobian matrix (deformation gradient), conservation of mass, relationship between volumes, mass densities, and determinant of deformation gradient, eigenvalue problem, principal directions, preservation of right angles, volume change from the material finite-strain viewpoint.
Handout: Penetration mechanics (EGM6611 30.pdf, 645KB)
General principles: Conservation of mass: Motivation with penetration mechanics (Lagrangian or material modeling, mesh distortion; Eulerian or spatial modeling, fixed mesh), balance of mass of spatial control volume.
Reading assignment: Malvern, Sec 4.3 ; 5.1, 5.2 .

WEEK 12

Fri, 2 Nov 01: Homecoming; no class.

Wed, 31 Oct 01:
Lecture 29:
Material finite-strain tensor (cont'd): Matrix of components in terms of displacements; Geometrical measures of strain, 1-D case, some tensor calculus involving deformation gradient, motivation with small-deformation case, stretch, unit extension, reduction to small-deformation case, angle change.
Reading assignment: Malvern, Sec 4.5 .

Mon, 29 Oct 01: HW7 and HW8 due.
Lecture 28:
Finite-strain tensors (cont'd): Material (Green-Lagrange) finite-strain tensor (cont'd); expression of deformation gradient in terms of displacement gradient, tensor form, component form; expression of material finite-strain tensor in terms of displacement gradient, and then in terms of small strain tensor; case of small deformation, approximation, motivation with beam theory, slope and curvature.
Reading assignment: Malvern, Sec 4.5 .

WEEK 11

Fri, 26 Oct 01:
Lecture 27:
Finite-strain tensors (cont'd): Deformation gradient, expression in terms of displacement (material) gradient, relative change of square of infinitesimal length, motivation, the factor 2, motivation with small strain, material (Green-Lagrange) finite-strain tensor, component form.
Reading assignment: Malvern, Sec 4.5 .

Wed, 24 Oct 01:
Lecture 26:
Strain and deformation (cont'd): Meaning of first invariant (volumetric strain); parallelism with Cauchy stress tensor, strain vector, tangential (shear) strain; Mohr circles in 3-D, domain of allowable strain states, range of values of normal strains and maximum shear strains relative to principle strains. Finite-strain tensors: Deformation gradient (Jacobian of deformation map), component form.
Reading assignment: Malvern, Sec 4.2, 4.5 .

Mon, 22 Oct 01: No class.
Exam 1, closed book, closed notes, 2 hours in the evening (for on-campus students). There will be no make-up exam for invalid reasons.
Room 101 NEB, period E2-E3 (8:20pm-10:10pm).

WEEK 10

Fri, 19 Oct 01:
Lecture 25:
Strain and deformation (cont'd): More on decomposition of displacement-gradient matrix, engineering shear strain, the naive approach, detailed test of tensor character, uniqueness of decomposition, the factor 1/2. Invariants of small-strain tensor.
Reading assignment: Malvern, Sec 2.5 ; 4.2 .

Wed, 17 Oct 01:
Lecture 24:
Strain and deformation (cont'd): More on tensorial shear-strain and engineering-shear strain via decomposition of matrix of components of the displacement gradient tensor, test of tensor character; decomposition of displacement gradient tensor, anti-symmetric tensor, small rotations, relation to cross product and anti-symmetric tensors.
Reading assignment: Malvern, Sec 2.5 ; 4.2 .

Mon, 15 Oct 01:
Lecture 23:
Q&A: Remark on a previous HW problem.
Strain and deformation (cont'd): Remark on directional derivative of a scalar function, gradient and its relation to optimization (steepest ascent/descent direction); normal strain, small-strain tensor, small-rotation tensor and its effects on the normal strain; components of small-strain tensor, tensorial shear-strain and engineering-shear strain, physical meaning (angle change).
Reading assignment: Malvern, Sec 2.5 ; 4.2 .

WEEK 9

Fri, 12 Oct 01:
Lecture 22:
Strain and deformation (cont'd): Derivative (rate of change) of displacement vector along a curve in initial configuration (directional derivative), motivation (small strain in 1-D continuum, rate of change of scalar function along a curve, gradient, level sets, tangent vectors); normal strain; gradient of displacement vector and its transpose, decomposition into symmetric tensor and anti-symmetric tensor, component form.
Reading assignment: Malvern, Sec 2.5 ; 4.2 .
HW8: Please download.

Wed, 10 Oct 01:
Lecture 21:
Q&A: Remarks on, and solution for, some HW problems. (Case of uniaxial tension, plane of maximum shear stress, 45 degree angle, failure surface in triaxial tests, verification with Mohr circle).
Strain and deformation: Kinematics; initial (undeformed) configuration, coordinates and basis vectors; current (deformed) configuration, coordinates and basis vectors; deformation map; displacement vector, tangent vector (tensor calculus concept, component form), derivative (rate of change) of displacement vector along a curve in initial configuration (component form, tensor form, decomposition), normal strain, parallelism with normal stress.
Reading assignment: Malvern, Sec 2.5 ; 4.1-4.2 .

Did you know: The "Vandermonde determinant" is actually a misnomer (a credit wrongly assigned to Vandermonde), probably due to a historical mistake of someone misreading the notation used in Vandermonde's work. In other words, Vandermonde, who made important contributions to the theory of determinants, is the wrong eponym for this particular matrix and its determinant; but the name stuck.

Mon, 8 Oct 01: HW6 due.
Lecture 20:
Q&A: Remarks on, and solution, for some previous HW problems.
Reading assignment: Malvern, Sec 2.5 ; 4.1-4.2 .

WEEK 8

Fri, 5 Oct 01:
Lecture 19:
Cauchy stress tensor (cont'd): von Mises yield condition (an important application of stress decomposition and stress invariants) (cont'd): Review the yield condition in physical space and in the coordinates along the principal directions; Principal stress space: Coordinate axes, principal stress vector, deviatoric plane, decomposition of principal stress vector, normal stress and tangential stress on deviatoric plane, comparison with octahedral normal stress and octahedral shear stress, yield criterion as cylinder in stress space, comparison with cylinder in physical space with principal directions, difference in cylinder radii.
Q&A: Remarks on, and solution for, some previous HW problems.
Handouts:


Reading assignment: Malvern, Sec 6.5, pp.334-338, pp.352-353 (Fig.6.15) .
HW7: Please download.

Did you know: Richard von Mises wrote the classic book Theory of Flight, and was the Founding Editor of the Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM) (Journal of Applied Mathematics and Mechanics), which debuted in 1921. In his classic book The Mathematical Theory of Plasticity [1950], Hill wrote "Most of the various yield criteria that have been suggested for metals are now only of historic interest, since they conflict with later experiments in predicting that a hydrostatic stress always influences yielding. The two simplest which do not have this fault are the criterion of Tresca ... and the criterion due to von Mises [1913]."

Wed, 3 Oct 01:
Lecture 18:
Q&A on previous lecture; remark on Mohr circles for 2-D stress states.
Cauchy stress tensor (cont'd): von Mises yield condition (an important application of stress decomposition and stress invariants): Geometric interpretation, octahedral normal stress and octahedral shear stress, criterion of maximum octahedral shear stress, yield surface (cylinder, radius; difference with cylinder in stress space).
Reading assignment: Malvern, Sec 3.4-3.5 ; 6.5, pp.334-338, p.353 (Fig.6.15) .
Handout: You can now download the article The creation and unfolding of the concept of stress by C. Truesdell (EGM6611 14.pdf, 20.1MB; thanks to Shani).

Did you know: Otto Mohr was "one of Europe's most decorated engineers of the 19th century". (S.P. Timoshenko, History of Strength of Materials, 1953).

Mon, 1 Oct 01: HW5 due.
Lecture 17:
Cauchy stress tensor (cont'd): Mohr circles for 3-D stress states: Expression for the components of normal in terms of principal stresses, normal and tangential stresses; Vandermonde matrix and its determinant; radius inequalities and domain of allowable stress states; important conclusion on the bounds of the magnitude of normal stress and tangential stress.
Reading assignment: Malvern, Sec 3.4-3.5 ; 6.5, pp.334-338 .

WEEK 7

Fri, 28 Sep 01:
Lecture 16:
Q&A on topics of the previous lecture.
Cauchy stress tensor (cont'd): Motivation for stress decomposition and stress invariants: 3-D metal plasticity, experimental observations (incompressible plastic flow, thought experiment), von Mises yield condition, relation to yield stresses in pure shear and pure tension; Mohr circles: Normal stress, tangential stress, maximum normal stress and maximum shear stress, domain of allowable stress states.
Reading assignment: Malvern, Sec 3.4-3.5 ; 6.5, pp.334-338 .
HW6: Please download.

Wed, 26 Sep 01:
Lecture 15:
Q&A, component form of the transpose of a tensor; motivation by matrix decomposition.
Cauchy stress tensor (cont'd): Analytical method to compute principal stresses based on stress deviator, relation between principal stresses and principal deviatoric stresses, de Moivre formula, binomial theorem, trigonometric relations, transformation of variable.
Reading assignment: Malvern, Sec 3.1-3.3 .

Mon, 24 Sep 01: HW4 due.
Lecture 14:
Return HW reports; comment on a HW problem.
Cauchy stress tensor (cont'd): Mean stress, identity tensor, decomposition of stress tensor into spherical part and deviatoric part, property of stress deviator, eigenvalue problems of stress deviator and its relation to the eigenvalue problem of the stress tensor, analytical method to compute principal stresses based on stress deviator.
Reading assignment: Malvern, Sec 3.1-3.3 .

WEEK 6

Fri, 21 Sep 01:
Lecture 13:
Cauchy stress tensor (cont'd): Q&A, review on the methods of derivation of the relation between traction, normal and stress tensor, mean-value theorem, cases of fluids and solids; principal stresses, principal directions, eigenvalue problem, stress invariants.
Reading assignment: Malvern, Sec 3.1-3.3 .
HW5: Please download.

Wed, 19 Sep 01:
Lecture 12:
Cauchy stress tensor (cont'd): Traction vectors on mutually-orthogonal planes, Cauchy tetrahedron, tensor product, contraction (from the left and from the right) between a tensor of order 1 and a tensor of order 2 (explanation using matrices), traction vector on an arbitrarily-inclined plane in terms of stress tensor, component form and tensor form, proof using statics and dynamics concepts (mean-value theorem, balance of linear momentum, volume of tetrahedron and relation among areas of tetrahedron, limiting process).
Reading assignment: Malvern, Sec 3.1-3.3 .

Did you know: " Cauchy's concept [of stresses] has the simplicity of genius. Its deep and thorough originality is fully outlined only against the background of the century of achievement by the brilliant geometers who preceded, treating the special kinds and case of deformable bodies by complicated and sometimes incorrect ways without ever hitting upon this basic idea, which immediately became and has remained the foundation of the mechanics of gross bodies. Nothing is harder to surmount than a corpus of true but too special knowledge; to reforge the tradition of his forebears is the greatest originality a man can have." From C. Truesdell, ``The Creation and Unfolding of the Concept of Stress'', Essays in the History of Mechanics, Berlin: Springer-Verlag, 1968, pp. 184-238.
More on Cauchy ( 1, 2, 3, 4 ).

Mon, 17 Sep 01: HW3 due.
Lecture 11:
Q&A regarding previous lectures or anything related to continuum mechanics.
General curvilinear coordinates, tangent vectors, covectors, basis, covariant and contravariant components of a vector. Remark on the connection between transformation of coordinates, diagonalization of matrix representation of general symmetric tensors, and invariants; application to matrix representation of stress and strain tensors and general tensors of second order, relationship to Cauchy stress tensor (principal stresses, principal directions, eigenvalue problems, invariants), notation (Truesdell and Noll's Nonlinear Field Theories of Mechanics), traction vector on an arbitrarily-inclined plane, Cauchy tetrahedron.
Reading assignment: Malvern, Sec 3.1-3.3 ; 2.5 .

WEEK 5

Fri, 14 Sep 01:
Lecture 10:
National Day of Prayer and Remembrance: A minute of silence and prayer for the victims in the area of the collapsed buildings in New York, and for the quick recovery of this free society.
Q&A regarding previous lecture.
General tensors: Eigenvalue problems (cont'd): Real symmetric matrices, properties of eigenvectors, proof (cont'd). Proper/improper orthogonal transformation, 3x3 real symmetric matrices, invariants of second-order (real, symmetric) tensors.
Reading assignment: Malvern, Sections 3.1-3.3; Section 2.4, Parts 1-4; Appendix I.
HW4: Please download.

Wed, 12 Sep 01:
Lecture 9: A minute of silence and prayer for the victims in the area of the collapsed buildings in New York, and for the quick recovery of this free society.
Second-order tensors: Transformation of coordinates, diagonalization of matrix representation; Eigenvalue problems: general nxn matrices with complex coefficients, classification in terms of diagonalizability (simple, semi-simple, defective, Jordan canonical form); Real symmetric matrices, properties of eigenvalues and eigenvectors, proof.
Reading assignment: Malvern, Section 2.4, Parts 1-4; Appendix I, pp.569-572 (Euclidean vector space, linear independence, basis, scalar product, orthogonality; skip curvilinear coordinates).



Tue, 11 Sep 01: Day of infamy. New York Times: 12 Sep 01, 13 Sep 01, 17 Sep 01, 7 Oct 01.
Phantom Towers: The photo on the right was the creation of P. Myoda and J. LaVerdiere, based on the original photo by F.R. Conrad of the New York Times. It appeared as the cover of the New York Times Magazine on 23 Sep 01.
A MEMORIAL


Did you know: Einstein could be considered as the first person who ever worked in nanotechnology (or rather nanoscience), a hot topic these days, almost 100 years ago! In his doctoral dissertation, Einstein deduced the size of a sugar molecule to be a nanometer based on experimental data of the diffusion of sugar in water. (He published this result in 1905.)

Mon, 10 Sep 01: HW1  and HW2 due.
Lecture 8: Tensors: Meaning of quotation from Brillouin's book (revisited, second-order tensors), transformation of coordinates, orthogonality of transformation matrix, first-order and second-order tensors, matrix form of relation between new components and old components.
Reading assignment: Malvern, Section 2.4, Parts 1-4; Appendix I, pp.569-572 (Euclidean vector space, linear independence, basis, scalar product, orthogonality; skip curvilinear coordinates).

WEEK 4

Fri, 7 Sep 01:
Lecture 7: Epsilon-delta relation: Sketch of proof. Tensors: Meaning of quotation from Brillouin's book, tensors of order 1 (vectors), component form, change of basis, transformation of coordinates, orthogonal matrices.
Reading assignment: Malvern, Section 2.4, Parts 1-4; Appendix I, pp.569-572 (Euclidean vector space, linear independence, basis, scalar product, orthogonality; skip curvilinear coordinates).
HW3: Please download.

Wed, 5 Sep 01:
Lecture 6: Use of permutation symbol: Triple scalar-product; determinant, 3x3 matrix, permutation operator, generalization to nxn matrix; triple vector product, epsilon-delta identity.
Handout: Eigenvalue problem for general matrices (from the MS thesis of a previous student of mine).
Reading assignment: Malvern, Section 2.4, Parts 1-4; Appendix I, pp.569-572 (Euclidean vector space, linear independence, basis, scalar product, orthogonality; skip curvilinear coordinates).

Did you know: Woldemar Voigt not only contributed to pionneer the field of tensor analysis, as applied to crystal physics, and thus provided a rigorous thermodynamic foundation for the study of piezoelectric crystals (a highly popular topic, these days), but also contributed fundamental work (by anticipating the Lorentz transformations) that led to the special theory of relativity, which was first published in a paper by Einstein in 1905.

Mon, 3 Sep 01: Labor Day. No class.

WEEK 3

Did you know: ``... certain chapters of physics, such as elasticity, already requires the introduction of quantities of a new nature, more complex than vectors: When one look for the forces or tensions in the interior of a deformed solid, one finds a set of 6 numbers, not separable from each other, that behaves as the 6 components of a certain new object. For a long time, physicists have since hesitated to give a name to this entity; but the study of crystal physics have revealed the existence of a large number of similar examples. The famous crystallographer and physicist [Woldemar] Voigt, the first to recognize the relationship between these diverse objects, insisted on their common characters, and baptised them "tensors" . This word clearly reminds one of the origin of tensors, since the first tensor indentified was the system of tensions in a deformed solid.''
L. Brillouin (NAS member, son and father, father), Les Tenseurs en Mecanique et en Elasticite, Dover, 1946, p.6.

"The first theoretician who discovered tensors, who identified them and gave them this name, was W. [Woldermar] Voigt, the famous crystallographer of Gottingen, in his remarkable treatise Lehrburch der Kristallphysik [Textbook on crystal physics](1910)."
L. Brillouin (NAS member, son and father, father), Les Tenseurs en Mecanique et en Elasticite, Dover, 1946, p.212.
(I translated the above paragraphs from the book written in French.)

Fri, 31 Aug 01:
Lecture 5: Finite rotations in 2-D and in 3-D; commutativity and noncommutativity, rotation matrices. Vector product, indicial notation, permutation tensor, triple scalar-product, determinant of nxn matrices, Laplace expansion.
Handout: Strang [1976], pp.154-167, properties of determinants.
Reading assignment: Malvern, Section 2.4, Parts 1-4; Appendix I, pp.569-572 (Euclidean vector space, linear independence, basis, scalar product, orthogonality; skip curvilinear coordinates).
HW2: Please download.

Did you see the infrared picture of Earth and Moon taken by the Mars Odyssey spacecraft on 23 Apr 01, from 3.5 million kilometers away (The word "Odyssey" is of course related to this course...).

Wed, 29 Aug 01:
Lecture 4: Generalization: Linear spaces, properties, generalization to matrices and functions; scalar product, general properties, norm, angle, examples (Kolmogorov & Fomin [1975]). Back to R^n: Scalar product, vector product, indicial notation. Tensors: Component form, matrix of components, dyadic notation, expansion.
Reading assignment: Malvern, Section 2.4, Parts 1-4; Appendix I, pp.569-572 (Euclidean vector space, linear independence, basis, scalar product, orthogonality; skip curvilinear coordinates).

Mon, 27 Aug 01:
Lecture 3: First and second-order tensors, tensor product, component form, motivation: matrix decomposition; scalar product, angle, property; orthonormal basis, Kronecker delta.
Handout: Kolmogorov & Fomin [1975], Linear spaces, pp.118-119, pp.138-139, pp.142-143.
Reading assignment: Malvern, Section 2.4, Parts 1-4.

WEEK 2

Fri, 24 Aug 01:
Lecture 2: Traction vector in continuum mechanics; Tensors, order, scalar functions (order 0), vectors (order 1), second-order tensors, examples, component form in different coordinate systems, matrix representation, indicial notation.
Reading assignment: Malvern, Section 2.1-2.3 (tensors); Appendix I, pp.569-572 (Euclidean vector space, linear independence, basis, scalar product, orthogonality; skip curvilinear coordinates).
HW1: Please download.
(Note that the HW number corresponds to the Week number; thus HW1 corresponds to Week 1.)

Wed, 22 Aug 01:
Lecture 1: Course organization, textbooks, cooperative learning, continuum, mass density, mathematical limiting process, matters at different length scales; Power of ten: From meter to nanometer (from Scientific American, Aug 2001).
Reading assignment: Malvern, Introduction, pp.1-6.

WEEK 1


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