applied mechmatreianics of solids (a.f
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7/29/2019 Applied Mechmatreianics of Solids (a.F
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9/13/13 Applied Mechanics of Solids (A.F. Bower) Appendix A: Vectors and Matrices
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1. Objectives and Applications >1.1 Defining a Problem >
1.1.1 Deciding what to calculate
1.1.2 Defining geometry
1.1.3 Defining loading
1.1.4 Choosing physics
1.1.5 Defining material behavior
1.1.6 A representative problem
1.1.7 Choosing a method of
analysis
2. Governing Equations >
2.1 Deformation measures
>
2.1.1 Displacement and Velocity
2.1.2 Deformation gradient
2.1.3 Deformation gradient from two
deformations
2.1.4 Jacobian of deformation
gradient
2.1.5 Lagrange strain
2.1.6 Eulerian strain
2.1.7 Infinitesimal Strain
2.1.8 Engineering Shear Strain
2.1.9 Volumetric and Deviatoric strain2.1.10 Infinitesimal rotation
2.1.11 Principal strains
2.1.12 Cauchy-Green deformation
tensors
2.1.13 Rotation tensor, Stretch
tensors
2.1.14 Principal stretches
2.1.15 Generalized strain measures
2.1.16 Velocity gradient
2.1.17 Stretch rate and spin
2.1.18 Infinitesimal strain/rotation rate
2.1.19 Other deformation rates
2.1.20 Strain equations of
compatibility
2.2 Internal forces >
2.2.1 Surface traction/body force
2.2.2 Internal tractions
2.2.3 Cauchy stress
2.2.4 Kirchhoff, Nominal, Material
stress
2.2.5 Stress for infinitesimal motions
2.2.6 Principal stresses2.2.7 Hydrostatic, Deviatoric, Von
Mises stress
2.2.8 Stresses at a boundary
2.3 Equations of motion >
2.3.1 Linear momentum balance
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2.3.2 Angular momentum
balance
2.3.3 Equations using other
stresses
2.4 Work and Virtual
Work >
2.4.1 Work done by Cauchy
stress
2.4.2 Work done by other
stresses2.4.3 Work for infinitesimal
motions
2.4.4 Principle of virtual work
2.4.5 Virtual work with other
stresses
2.4.6 Virtual work for small
strains
3. Constitutive Equations >
3.1 General requirements
3.2 Linear elasticity >
3.2.1 Isotropic elastic behavior3.2.2 Isotropic stress-strain laws
3.2.3 Plane stress & strain
3.2.4 Isotropic material data
3.2.5 Lame, Shear, & Bulk modulus
3.2.6 Interpreting elastic constants
3.2.7 Strain energy density (isotropic)
3.2.8 Anisotropic stress-strain laws
3.2.9 Interpreting anisotropic
constants
3.2.10 Anisotropic strain energy
density
3.2.11 Basis change formulas
3.2.12 Effect of material symmetry
3.2.13 Orthotropic materials
3.2.14 Transversely isotropic
materials
3.2.15 Transversely isotropic data
3.2.16 Cubic materials
3.2.17 Cubic material data
3.3 Hypoelasticity
3.4 Elasticity w/ large rotations
3.5 Hyperelasticity >
3.5.1 Deformation measures3.5.2 Stress measures
3.5.3 Strain energy density
3.5.4 Incompressible materials
3.5.5 Energy density functions
3.5.6 Calibrating material
models
3.5.7 Representative
properties
3.6 Viscoelasticity >
3.6.1 Polymer behavior
3.6.2 General constitutive
equations
3.6.3 Spring-damper
approximations
3.6.4 Prony series
3.6.5 Calibrating constitutive
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aws
3.6.6 Calibrating material models
3.6.7 Representative properties
3.7 Rate independent plasticity
>
3.7.1 Plastic metal behavior
3.7.2 Elastic/plastic strain
decomposition
3.7.3 Yield criteria
3.7.4 Graphical yield surfaces3.7.5 Hardening laws
3.7.6 Plastic flow law
3.7.7 Unloading condition
3.7.8 Summary of stress-strain
relations
3.7.9 Representative properties
3.7.10 Principle of max. plastic
resistance
3.7.11 Drucker's postulate
3.7.12 Microscopic perspectives
3.8 Viscoplasticity >
3.8.1 Creep behavior
3.8.2 High strain rate behavior
3.8.3 Constitutive equations
3.8.4 Representative creep properties
3.8.5 Representative high rate
properties
3.9 Large strain plasticity >
3.9.1 Deformation measures
3.9.2 Stress measures
3.9.3 Elastic stress-strain
relations
3.5.4 Plastic stress-strainrelations
3.10 Large strain viscoelasticity
>
3.10.1 Deformation measures
3.10.2 Stress measures
3.10.3 Stress-strain energy
relations
3.10.4 Strain relaxation
3.10.5 Representative properties
3.11 Critical state soils >
3.11.1 Soil behavior3.11.2 Constitutive laws (Cam-
clay)
3.11.3 Response to 2D loading
3.11.4 Representative properties
3.12 Crystal plasticity >
3.12.1 Basic crystallography
3.12.2 Features of crystal
plasticity
3.12.3 Deformation measures
3.12.4 Stress measures
3.12.5 Elastic stress-strain
relations
3.12.6 Plastic stress-strain
relations
3.12.7 Representative properties
3.13 Surfaces and interfaces >
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. .
3.13.2 Contact and friction
4. Solutions to simple problems >
4.1 Axial/Spherical linear elasticity
>
4.1.1 Elastic governing equations
4.1.2 Spherically symmetric
equations
4.1.3 General spherical solution
4.1.4 Pressurized sphere4.1.5 Gravitating sphere
4.1.6 Heated spherical shell
4.1.7 Axially symmetric equations
4.1.8 General axisymmetric solution
4.1.9 Pressurized cylinder
4.1.10 Spinning circular disk
4.1.11 Interference fit
4.2 Axial/Spherical elastoplasticity
>
4.2.1 Plastic governing equations
4.2.2 Spherically symmetric
equations
4.2.3 Pressurized sphere
4.2.4 Cyclically pressurized sphere
4.2.5 Axisymmetric equations
4.2.6 Pressurized cylinder
4.3 Spherical hyperelasticity >
4.3.1 Governing equations
4.3.2 Spherically symmetric
equations
4.3.3 Pressurized sphere
4.4 1D elastodynamics >
4.4.1 Surface subjected to pressure4.4.2 Surface under tangential
loading
4.4.3 1-D bar
4.4.4 Plane waves
4.4.5 Wave speeds in isotropic
solid
4.4.6 Reflection at a surface
4.4.7 Reflection at an interface
4.4.8 Plate impact experiment
5. Solutions for elastic solids >
5.1 General Principles >
5.1.1 Governing equations
5.1.2 Navier equation
5.1.3 Superposition &
linearity
5.1.4 Uniqueness & existence
5.1.5 Saint-Venants principle
5.2 2D Airy function solutions >
5.2.1 Airy solution in rectangular
coords
5.2.2 Demonstration of Airy solution
5.2.3 Airy solution in polar coords
5.2.4 End loaded cantilever5.2.5 Line load perpendicular to
surface
5.2.6 Line load parallel to surface
4.4.7 Pressure on a surface
4.4.8 Uniform ressure on a stri
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4.4.8 Stress near a crack tip
5.3 2D Complex variable
solutions >
5.3.1 Complex variable solution
5.3.2 Demonstration of CV
solution
5.3.3 Line force
5.3.4 Edge dislocation
5.3.5 Circular hole in infinite
solid
5.3.6 Slit crack
5.3.7 Bimaterial interface crack
5.3.8 Rigid flat punch on a
surface
5.3.9 Parabolic punch on a
surface
5.3.10 General line contact
4.3.11 Frictional sliding contact
4.3.12 Dislocation near a surface
5.4 3D static problems >
5.4.1 Papkovich-Neuber potentials5.4.2 Demonstration of PN
potentials
5.4.3 Point force in infinite solid
5.4.4 Point force normal to surface
5.4.5 Point force tangent to surface
5.4.6 Eshelby inclusion problem
5.4.7 Inclusion in an elastic solid
5.4.8 Spherical cavity in infinite
solid
5.4.9 Flat cylindrical punch on
surface5.4.10 Contact between spheres
4.4.11 Relations for general
contacts
4.4.12 P-d relations for
axisymmetric contact
5.5 2D Anisotropic elasticity >
5.5.1 Governing equations
5.5.2 Stroh solution
5.5.3 Demonstration of Stroh solution
5.5.4 Stroh matrices for cubic
materials
5.5.5 Degenerate materials
5.5.6 Fundamental elasticity matrix
5.5.7 Orthogonality of Stroh matrices
5.5.8 Barnett/Lothe & Impedance
tensors
5.5.9 Properties of matrices
5.5.10 Basis change formulas
5.5.11 Barnett-Lothe integrals
5.5.12 Uniform stress state
5.5.13 Line load/dislocation in infinite
solid
5.5.14 Line load/dislocation near asurface
5.6 Dynamic problems >
5.6.1 Love potentials
5.6.2 Pressurized spherical
cavity
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5.6.3 Rayleigh waves
5.6.4 Love waves
5.6.5 Elastic waves in
waveguides
5.7 Energy methods >
5.7.1 Definition of potential energy
5.7.2 Minimum energy theorem
5.7.3 Simple example of energy
minimization
5.7.4 Variational approach to beam
theory
5.7.5 Estimating stiffness
5.8 Reciprocal theorem >
5.8.1 Statement and proof of
theorem
5.8.2 Simple example
5.8.3 Boundary-internal value
relations
5.8.4 3D dislocation loops
5.9 Energetics of dislocations >
5.9.1 Potential energy of isolatedloop
5.9.2 Nonsingular dislocation
theory
5.9.3 Dislocation in bounded solid
5.9.4 Energy of interacting loops
5.9.5 Peach-Koehler formula
5.10 Rayleigh Ritz method >
5.10.1 Mode shapes, nat. frequencies,
Rayleigh's principle
5.10.2 Natural frequency of a beam
6. Solutions for plastic solids >
6.1 Slip-line fields >
6.1.1 Interpreting slip-line fields
6.1.2 Derivation of slip-line fields
6.1.3 Examples of solutions
6.2 Bounding theorems
>
6.2.1 Definition of plastic dissipation
6.2.2 Principle of min plastic
dissipation
6.2.3 Upper bound collapse theorem
6.2.4 Lower bound collapse theorem
6.2.5 Examples of bounding theorems6.2.6 Lower bound shakedown
theorem
6.2.7 Examples of lower bound
shakedown theorem
6.2.8 Upper bound shakedown
theorem
6.2.9 Examples of upper bound
shakedown theorem
7. Introduction to FEA >
7.1 Guide to FEA >
7.1.1 FE mesh7.1.2 Nodes and elements
7.1.3 Special elements
7.1.4 Material behavior
7.1.5 Boundary conditions
7.1.6 Constraints
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7.1.7 Contacting surface/interfaces
7.1.8 Initial conditions/external fields
7.1.9 Soln procedures / time
increments
7.1.10 Output
7.1.11 Units in FEA calculations
7.1.12 Using dimensional analysis
7.1.13 Scaling governing equations
7.1.14 Remarks on dimensional
analysis7.2 Simple FEA program
>
7.2.1 FE mesh and connectivity
7.2.2 Global displacement vector
7.2.3 Interpolation functions
7.2.4 Element strains & energy density
7.2.5 Element stiffness matrix
7.2.6 Global stiffness matrix
7.2.7 Boundary loading
7.2.8 Global force vector
7.2.9 Minimizing potential energy7.2.10 Eliminating prescribed
displacements
7.2.11 Solution
7.2.12 Post processing
7.2.13 Example code
8. Theory & Implementation of
FEA >
8.1 Static linear elasticity >
8.1.1 Review of virtual work
8.1.2 Weak form of governing
equns
8.1.3 Interpolating displacements
8.1.4 Finite element equations
8.1.5 Simple 1D implementation
8.1.6 Summary of 1D procedure
8.1.7 Example 1D code
8.1.8 Extension to 2D/3D
8.1.9 2D interpolation functions
8.1.10 3D interpolation functions
8.1.11 Volume integrals
8.1.12 2D/3D integration schemes
8.1.13 Summary of element
matrices8.1.14 Sample 2D/3D code
8.2 Dynamic elasticity >
8.2.1 Governing equations
8.2.2 Weak form of governing eqns
8.2.3 Finite element equations
8.2.4 Newmark time integration
8.2.5 Simple 1D implementation
8.2.6 Example 1D code
8.2.7 Lumped mass matrices
8.2.8 Example 2D/3D code
8.2.9 Modal time integration
8.2.10 Natural frequencies/mode
shapes
8.2.11 Example 1D modal dynamic
code
8.2.12 Example 2D/3D modal
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ynamc co e
8.3 Hypoelasticity >
8.3.1 Governing equations
8.3.2 Weak form of governing eqns
8.3.3 Finite element equations
8.3.4 Newton-Raphson iteration
8.3.5 Tangent moduli for hypoelastic
solid
8.3.6 Summary of Newton-Raphson
method8.3.7 Convergence problems
8.3.8 Variations on Newton-Raphson
8.3.9 Example code
8.4 Hyperelasticity >
8.4.1 Governing equations
8.4.2 Weak form of governing
eqns
8.4.3 Finite element equations
8.4.4 Newton-Raphson iteration
8.4.5 Neo-Hookean tangent
moduli
8.4.6 Evaluating boundary
integrals
8.4.7 Convergence problems
8.4.8 Example code
8.5 Viscoplasticity >
8.5.1 Governing equations
8.5.2 Weak form of governing eqns
8.5.3 Finite element equations
8.5.4 Integrating the stress-strain
law
8.5.5 Material tangent
8.5.6 Newton-Raphson solution8.5.7 Example code
8.6 Advanced elements >
8.6.1 Shear locking/incompatible
modes
8.6.2 Volumetric locking/Reduced
integration
8.6.3 Incompressible materials/Hybrid
elements
9. Modeling Material Failure >
9.1 Mechanisms of failure >
9.1.1 Monotonic loading9.1.2 Cyclic loading
9.2 Stress/strain based criteria >
9.2.1 Stress based criteria
9.2.2 Probabilistic methods
9.2.3 Static fatigue criterion
9.2.4 Models of crushing
failure
9.2.5 Ductile failure criteria
9.2.6 Strain localization
9.2.7 High cycle fatigue
9.2.8 Low cycle fatigue
9.2.9 Variable amplitude
loading
9.3 Elastic fracture mechanics >
9.3.1 Crack tip fields
9.3.2 Linear elastic fracture
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9.3.3 Calculating stress intensities
9.3.4 Using FEA
9.3.5 Measuring toughness
9.3.6 Values of fracture toughness
9.3.7 Stable tearing
9.3.8 Mixed mode fracture
9.3.9 Static fatigue
9.3.10 Cyclic fatigue
9.3.11 Finding cracks9.4 Energy methods in fracture >
9.4.1 Definition of energy release
rate
9.4.2 Energy based fracture
criterion
9.4.3 G-K relations
9.4.4 G-compliance relation
9.4.5 Calculating K with
compliance
9.4.6 Integral expression for G
9.4.7 The J integral
9.4.8 Calculating K using J
9.5 Plastic fracture mechanics >
9.5.1 Dugdale-Barenblatt model
9.5.2 HRR crack tip fields
9.5.3 J based fracture mechanics
9.6 Interface fracture mechanics
>
9.6.1 Interface crack tip fields
9.6.2 Interface fracture
mechanics
9.6.3 Stress intensity factors
9.6.4 Crack path selection10. Rods, Beams, Plates & Shells
>
10.1 Dyadic notation
10.2 Deformable rods - general
>
10.2.1 Characterizing the x-section
10.2.2 Coordinate systems
10.2.3 Kinematic relations
10.2.4 Displacement, velocity and
acceleration
10.2.5 Deformation gradient
10.2.6 Strain measures
10.2.7 Kinematics of bent rods
10.2.8 Internal forces and moments
10.2.9 Equations of motion
10.2.10 Constitutive equations
10.2.11 Strain energy density
10.3 String / beam theory >
10.3.1 Stretched string
10.3.2 Straight beam (small
deflections)
10.3.3 Axially loaded beam
10.4 Solutions for rods >10.4.1 Vibration of a straight beam
10.4.2 Buckling under gravitational
loading
10.4.3 Post buckled shape of a rod
10.4.4 Rod bent into a helix
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10.4.5 Helical spring
10.5 Shells - general >
10.5.1 Coordinate systems
10.5.2 Using non-orthogonal
bases
10.5.3 Deformation measures
10.5.4 Displacement and
velocity
10.5.5 Deformation gradient
10.5.6 Other strain measures
10.5.7 Internal forces and
moments
10.5.8 Equations of motion
10.5.9 Constitutive relations
10.5.10 Strain energy
10.6 Plates and membranes >
10.6.1 Flat plates (small strain)
10.6.2 Flat plates with in-plane
loading
10.6.3 Plates with large
displacements10.6.4 Membranes
10.6.5 Membranes in polar
coordinates
10.7 Solutions for shells >
10.7.1 Circular plate bent by pressure
10.7.2 Vibrating circular membrane
10.7.3 Natural frequency of
rectangular plate
10.7.4 Thin film on a substrate
(Stoney eqs)
10.7.5 Buckling of heated plate10.7.6 Cylindrical shell under axial
load
10.7.7 Twisted open walled cylinder
10.7.8 Gravity loaded spherical shell
A: Vectors & Matrices
B: Intro to tensors
C: Index Notation
D: Using polar coordinates
E: Misc derivations
Problems
1. Objectives and Applications >1.1 Defining a Problem
2. Governing Equations >
2.1 Deformation measures
2.2 Internal forces
2.3 Equations of motion
2.4 Work and Virtual
Work
3. Constitutive Equations >
3.1 General requirements
3.2 Linear elasticity
3.3 Hypoelasticity3.4 Elasticity w/ large rotations
3.5 Hyperelasticity
3.6 Viscoelasticity
3.7 Rate independent plasticity
3.8 Viscoplasticity
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3.9 Large strain plasticity
3.10 Large strain viscoelasticity
3.11 Critical state soils
3.12 Crystal plasticity
3.13 Surfaces and interfaces
4. Solutions to simple problems >
4.1 Axial/Spherical linear elasticity
4.2 Axial/Spherical elastoplasticity
4.3 Spherical hyperelasticity
4.4 1D elastodynamics
5. Solutions for elastic solids >
5.1 General Principles
5.2 2D Airy function solutions
5.3 2D Complex variable
solutions
5.4 3D static problems
5.5 2D Anisotropic elasticity
5.6 Dynamic problems
5.7 Energy methods
5.8 Reciprocal theorem
5.9 Energetics of dislocations5.10 Rayleigh Ritz method
6. Solutions for plastic solids >
6.1 Slip-line fields
6.2 Bounding theorems
7. Introduction to FEA >
7.1 Guide to FEA
7.2 Simple FEA program
8. Theory & Implementation of
FEA >
8.1 Static linear elasticity
8.2 Dynamic elasticity
8.3 Hypoelasticity
8.4 Hyperelasticity
8.5 Viscoplasticity
8.6 Advanced elements
9. Modeling Material Failure >
9.1 Mechanisms of failure
9.2 Stress/strain based criteria
9.3 Elastic fracture mechanics
9.4 Energy methods in fracture
9.5 Plastic fracture mechanics
9.6 Interface fracture mechanics
10. Rods, Beams, Plates & Shells>
10.1 Dyadic notation
10.2 Deformable rods - general
10.3 String / beam theory
10.4 Solutions for rods
10.5 Shells - general
10.6 Plates and membranes
10.7 Solutions for shells
A: Vectors & Matrices
B: Intro to tensors
C: Index NotationD: Using polar coordinates
E: Misc derivations
FEA codes
Maple
Matlab
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Appendix A
Review of Vectors and Matrices
A.1. VECTORS
A.1.1 Definition
For the purposes of this text, a vector is an object which has magnitude and direction. Examples include forces,
electric fields, and the normal to a surface. A vector is often represented pictorially as an arrow and
symbolically by an underlined letter or using bold type . Its magnitude is denoted or . There are two
special cases of vectors: the unit vector has ; and the null vector has .
A.1.2 Vector Operations
Addition
Let and be vectors. Then is also a vector. The vector may be shown
diagramatically by placing arrows representing and head to tail, as shown in the
figure.
Multiplication
1. Multiplication by a scalar. Let be a vector, and a scalar. Then is a vector. The
direction of is parallel to and its magnitude is given by .
Note that you can form a unit vectorn which is parallel to a by setting .
2. Dot Product (also called the scalar product). Let a and b be two vectors. The
dot product ofa and b is a scalar denoted by , and is defined by
,
where is the angle subtended by a and b.Note that , and
. If and then if and only if ; i.e. a and b are
perpendicular.
3. Cross Product (also called the vector product). Let a and b be two
vectors. The cross product ofa and b is a vector denoted by .
The direction ofc is perpendicular to a and b, and is chosen so that (a,b,c)
form a right handed triad, Fig. 3. The magnitude ofc is given by
Note that and .
Some useful vector identities
A.1.3Cartesian components of vectors
Let be three mutually perpendicular unit vectors which form a right handed triad, Fig. 4. Then
are said to form and orthonormal basis. The vectors satisfy
We ma ex ress an vectora as a suitable combination of the unit vectors , and . For exam le, we ma
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write
where are scalars, called the components ofa in the basis . The components ofa have
a simple physical interpretation. For example, if we evaluate the dot product we find that
in view of the properties of the three vectors , and . Recall that
Then, noting that , we have
Thus, represents the projected length of the vectora in the direction of , as
illustrated in the figure. Similarly, and may be shown to represent the
projection of in the directions and , respectively.
The advantage of representing vectors in a Cartesian basis is that vector addition
and multiplication can be expressed as simple operations on the components of
the vectors. For example, let a, b and c be vectors, with components , and ,
respectively. Then, it is straightforward to show that
A.1.4 Change of basis
Let a be a vector, and let be a Cartesian basis. Suppose that the components of a in the basis
are known to be . Now, suppose that we wish to compute the components of a in a
second Cartesian basis, . This means we wish to find components , such that
To do so, note that
This transformation is conveniently written as a matrix operation
,
where is a matrix consisting of the components ofa in the basis , is a matrix consisting ofthe components ofa in the basis , and is a `rotation matrix as follows
Note that the elements of have a simple physical interpretation. For example, ,
where is the angle between the and axes. Similarly where
is the angle between the and axes. In practice, we usually know the angles between the axes
that make up the two bases, so it is simplest to assemble the elements of by putting the cosines of the known
angles in the appropriate places.
Index notation provides another convenient way to write this transformation:
You dont need to know index notation in detail to understand this all you need to know is that
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The same approach may be used to find an expression for in terms of . If you work through the details, you
will find that
Comparing this result with the formula for in terms of , we see that
where the superscript Tdenotes the transpose (rows and columns interchanged). The transformation matrix
is therefore orthogonal, and satisfies
where [I] is the identity matrix.
A.1.5Use ful vector operations
Calculating areas
The area of a triangle bounded by vectors a, band b-a is
The area of the parallelogram shown in the picture is 2A.
Calculating angles
The angle between two vectors a and b is
Calculating the normal to a surface .
If two vectors a and b can be found which are known to lie in the surface, then the unit normal to the
surface is
If the surface is specified by a parametric equation of the form , where s and t are two
parameters and r is the position vector of a point on the surface, then two vectors which lie in the plane
may be computed from
Calculating Volumes
The volume of the parallelopiped defined by three vectors a, b, c is
The volume of the tetrahedron shown outlined in red is V/6.
A.2. VECTOR FIELDS AND VECTOR CALCULUS
A.2.1. Scalar field.
Let be a Cartesian basis with origin O in three dimensional space. Let
denote the position vector of a point in space. A scalar field is a scalar valued function of position in space. A
scalar field is a function of the components of the position vector, and so may be expressed as . The
value of at a particular point in space must be independent of the choice of basis vectors. A scalar field may
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.
A.2.2.Vector field
Let be a Cartesian basis with origin O in three dimensional space. Let
denote the position vector of a point in space. A vector field is a vector valued function of position in space. A
vector field is a function of the components of the position vector, and so may be expressed as . The
vector may also be expressed as components in the basis
The magnitude and direction of at a particular point in space is independent of the choice of basis vectors. A
vector field may be a function of time (and possibly other parameters) as well as position in space.
A.2.3.Change of basis for scalar fields.
Let be a Cartesian basis with origin O in three
dimensional space. Express the position vector of a point relative to O
in as
and let be a scalar field.
Let be a second Cartesian basis, with origin P. Let
denote the position vector of P relative to O. Express the
position vector of a point relative to P in as
To find , use the following procedure. First, express p as components in the basis , using
the procedure outlined in Section 1.4:
where
or, using index notation
where the transformation matrix is defined in Sect 1.4.
Now, express c as components in , and note that
so that
A.2.4. Change of basis for vector fields .
Let be a Cartesian basis with origin O in three dimensional
space. Express the position vector of a point relative to O inas
and let be a vector field, with components
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, .
denote the position vector of P relative to O. Express the
position vector of a point relative to P in as
To express the vector field as components in and as a
function of the components ofp, use the following procedure. First,
express in terms of using the procedure outlined
for scalar fields in the preceding section
fork=1,2,3. Now, find the components of v in using the procedure outlined in Section 1.4.
Using index notation, the result is
A.2.5. Time derivatives of vectors
Let a(t)be a vector whose magnitude and direction vary with time, t. Suppose that is afixedbasis, i.e.
independent of time. We may express a(t) in terms of components in the basis as
.
The time derivativeofa is defined using the usual rules of calculus
,
or in component form as
The definition of the time derivative of a vector may be used to show the following rules
A.2.6. Using a rotating basis
It is often convenient to express position vectors as components in a basis which rotates with time. To writeequations of motion one must evaluate time derivatives of rotating vectors.
Let be a basis which rotates with instantaneous angular velocity . Then,
A.2.7. Gradient of a scalar field.
Let be a scalar field in three dimensional space. The gradient of is a vector field denoted by or
, and is defined so that
for every position r in space and for every vectora.
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Let be a Cartesian basis with origin O in three dimensional space. Let
denote the position vector of a point in space. Express as a function of the components ofr .
The gradient of in this basis is then given by
A.2.8. Gradient of a vector field
Let v be a vector field in three dimensional space. The gradient ofv is a tensor field denoted by or
, and is defined so that
for every position r in space and for every vectora.
Let be a Cartesian basis with origin O in three dimensional space. Let
denote the position vector of a point in space. Express v as a function of the components of r, so that
. The gradient of v in this basis is then given by
Alternatively, in index notation
A.2.9. Divergence of a vector field
Let v be a vector field in three dimensional space. The divergence ofv is a scalar field denoted by or
. Formally, it is defined as (the trace of a tensor is the sum of its diagonal terms).
Let be a Cartesian basis with origin O in three dimensional space. Let
denote the position vector of a point in space. Express v as a function of the components ofr: .
The divergence ofv is then
A.2.10. Curl of a vector field.
Let v be a vector field in three dimensional space. The curl of v is a vector field denoted by or .
It is best defined in terms of its components in a given basis, although its magnitude and direction are not
dependent on the choice of basis.
Let be a Cartesian basis with origin O in three dimensional space. Let
denote the position vector of a point in space. Express v as a function of the components ofr .
The curl of v in this basis is then given by
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Using index notation, this may be expressed as
A.2.11 The Divergence Theorem.
Let V be a closed region in three dimensional space, bounded by an
orientable surface S. Let n denote the unit vector normal to S, taken so
that n points out ofV. Let u be a vector field which is continuous and has
continuous first partial derivatives in some domain containing T. Then
alternatively, expressed in index notation
For a proof of this extremely useful theorem consult e.g. Kreyzig,Advanced Engineering Mathematics,Wiley,
New York, (1998).
A.3. MATRICES
A.3.1 Definition
An matrix is a set of numbers, arranged in m rows and n columns
A square matrix has equal numbers of rows and columns
A diagonal matrix is a square matrix with elements such that for
The identity matrix is a diagonal matrix for which all diagonal elements
A symmetric matrix is a square matrix with elements such that
A skew symmetric matrix is a square matrix with elements such that
A.3.2 Matrix operations
Addition Let and be two matrices of order with elements and . Then
Multiplication by a scalar. Let be a matrix with elements , and let kbe a scalar. Then
Multiplication by a matrix. Let be a matrix of order with elements , and let be a matrix
of order with elements . The product is defined only ifn=p, and is an matrix
such that
Note that multiplication is distributive and associative, but not commutative, i.e.
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The multiplication of a vector by a matrix is a particularly important operation. Let b and c be two vectors with
n components, which we think of as matrices. Let be an matrix. Thus
Now,
i.e.
Transpose. Let be a matrix of order with elements . The transpose of is denoted .
If is an matrix such that , then , i.e.
Note that
Determinant The determinant is defined only for a square matrix. Let be a matrix with
components . The determinant of is denoted by or and is given by
Now, let be an matrix. Define the minors of as the determinant formed by omitting the ith
row and jth column of . For example, the minors and for a matrix are computed as
follows. Let
Then
Define the cofactors of as
Then, the determinant of the matrix is computed as follows
The result is the same whichever row i is chosen for the expansion. For the particular case of a matrix
The
determinant may also be evaluated by summing over rows, i.e.
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and as before the result is the same for each choice of column j. Finally, note that
Inversion. Let be an matrix. The inverse of is denoted by and is defined such that
The inverse of exists if and only if . A matrix which has no inverse is said to besingular. Theinverse of a matrix may be computed explicitly, by forming the cofactor matrix with components as
defined in the preceding section. Then
In practice, it is faster to compute the inverse of a matrix using methods such as Gaussian elimination.
Note that
For a diagonal matrix, the inverse is
For a matrix, the inverse is
Eigenvalues and eigenvectors. Let be an matrix, with coefficients . Consider the vector
equation
(1)
where x is a vector with n components, and is a scalar (which may be complex). The n nonzero vectors x
and corresponding scalars which satisfy this equation are the eigenvectors and eigenvalues of .
Formally, eighenvalues and eigenvectors may be computed as follows. Rearrange the preceding equation to
(2)
This has nontrivial solutions forx only if the determinant of the matrix vanishes. The equation
is an nth order polynomial which may be solved for . In general the polynomial will have n roots, which may
be complex. The eigenvectors may then be computed using equation (2). For example, a matrixgenerally has two eigenvectors, which satisfy
Solve the quadratic equation to see that
The two corresponding eigenvectors may be computed from (2), which shows that
so that, multiplying out the first row of the matrix (you can use the second row too, if you wish since we chose
to make the determinant of the matrix vanish, the two equations have the same solutions. In fact, if ,
you will need to do this, because the first equation will simply give 0=0 when trying to solve for one of the
ei envectors
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which are satisfied by any vector of the form
wherep and q are arbitrary real numbers.
It is often convenient to normalize eigenvectors so that they have unit length. For this purpose, choosep and q
so that . (For vectors of dimension n, the generalized dot product is defined such that
)
One may calculate explicit expressions for eigenvalues and eigenvectors for any matrix up to order , but
the results are so cumbersome that, except for the results, they are virtually useless. In practice,
numerical values may be computed using several iterative techniques. Packages like Mathematica, Maple or
Matlab make calculations like this easy.
The eigenvalues of a real symmetric matrix are always real, and its eigenvectors are orthogonal, i.e. the ith and
jth eigenvectors (with ) satisfy .
The eigenvalues of a skew symmetric matrix are pure imaginary.
Spectral and singular value decomposition. Let be a real symmetric matrix. Denote the n
(real) eigenvalues of by , and let be the corresponding normalized eigenvectors, such that
. Then, for any arbitrary vectorb,
Let be a diagonal matrix which contains the n eigenvalues of as elements of the diagonal, and let be
a matrix consisting of the n eigenvectors as columns, i.e.
Then
Note that this gives another (generally quite useless) way to invert
where is easy to compute since is diagonal.
Square root of a matrix. Let be a real symmetric matrix. Denote the singular value
decomposition of by as defined above. Suppose that denotes the square
root of , defined so that
One way to compute is through the singular value decomposition of
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