“Stegosaurus”

Its array of plates and spikes has been the subject of much speculation. The spikes were

most likely used for defense, while the plates have been proposed as having displayed

thermoregulatory (heat control) functions.

Marsh's 1896 illustration of Stegosaurus. Note the twelve dorsal plates and eight tail spikes; Stegosaurus actually had

17 plates and just four spikes

A dynamometer or "dyno" for short, is a device for measuring force or power. For

example, the power produced by an engine, motor or other rotating prime mover can be

calculated by simultaneously measuring torque and rotational speed (rpm).

Mass is a concept used in the physical sciences to explain a number of observable

behaviors, and in everyday usage, it is common to identify mass with those resulting

behaviors. In particular, mass is commonly identified with weight. But according to our

modern scientific understanding, the weight of an object results from the interaction of its

mass with a gravitational field, so while mass is part of the explanation of weight, it is not

the complete explanation.

A thermodynamic potential is a scalar potential function used to represent the

thermodynamic state of a system. One main thermodynamic potential which has a

physical interpretation is the internal energy, U. It is the energy of configuration of a

given system of conservative forces (that is why it is a potential) and only has meaning

with respect to a defined set of references (or data).

In physics, thermodynamics (from the Greek θερμ-<θερμότης, therme, meaning "heat"

[1]

and δυναμις, dynamis, meaning "power") is the study of the conversion of energy into

work and heat and its relation to macroscopic variables such as temperature and pressure.

Its underpinnings, based upon statistical predictions of the collective motion of particles

from their microscopic behavior, is the field of statistical thermodynamics, a branch of

statistical mechanics.

four laws are:

Zeroth law of thermodynamics, about thermal equilibrium:

If two thermodynamic systems are separately in thermal equilibrium with a third,

they are also in thermal equilibrium with each other.

If we grant that all systems are (trivially) in thermal equilibrium with themselves,

the Zeroth law implies that thermal equilibrium is an equivalence relation on the

set of thermodynamic systems. This law is tacitly assumed in every measurement

of temperature. Thus, if we want to know if two bodies are at the same

temperature, it is not necessary to bring them into contact and to watch whether

their observable properties change with time.

[13]

First law of thermodynamics, about the conservation of energy:

The change in the internal energy of a closed thermodynamic system is equal to

the sum of the amount of heat energy supplied to the system and the work done on

the system.

Second law of thermodynamics, about entropy:

The total entropy of any isolated thermodynamic system tends to increase over

time, approaching a maximum value.

Third law of thermodynamics, about the absolute zero of temperature:

As a system asymptotically approaches absolute zero of temperature all processes

virtually cease and the entropy of the system asymptotically approaches a

minimum value; also stated as: "the entropy of all systems and of all states of a

system is zero at absolute zero" or equivalently "it is impossible to reach the

absolute zero of temperature by any finite number of processes".

The following has sometimes been called the "Fourth Law of Thermodynamics", about

the transfer of heat energy between systems.

Onsager reciprocal relations:

In connected thermodynamic systems which are in equilibrium neither for

pressure nor temperature, heat flow between is caused by forces proportional with

unit of pressure difference, and equal to the proportional density flow caused per

unit of temperature difference.

In the history of thermodynamics, Reflections on the Motive Power of Fire and on

Machines Fitted to Develop that Power (French title: Réflexions sur la puissance motrice

du feu et sur les machines propres à développer cette puissance) is an 1824, 65-page

book by French physicist Sadi Carnot on a generalized theory of heat engines and is

considered the founding work in the science of thermodynamics.

[1]

In it is found the

preliminary outline of the second law of thermodynamics, namely that motive power is

due to the fall of caloric (heat) from a hot to cold body.

In thermodynamics, a thermodynamic system is said to be in thermodynamic

equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical

equilibrium.

In thermodynamics, the Onsager reciprocal relations express the equality of certain

relations between flows and forces in thermodynamic systems out of equilibrium, but

where a notion of local equilibrium exists.

In physics and thermodynamics, an equation of state is a relation between state

variables.

[1]

More specifically, an equation of state is a thermodynamic equation

describing the state of matter under a given set of physical conditions. It is a constitutive

equation which provides a mathematical relationship between two or more state functions

associated with the matter, such as its temperature, pressure, volume, or internal energy.

Equations of state are useful in describing the properties of fluids, mixtures of fluids,

solids, and even the interior of stars.

Molecular attraction occurs when neutrally -charged molecules nevertheless experience the uneven

distribution of electrons over their structure due to London Forces induced by random variations of electron

density occurring in non -polar compounds, hydrogen bonding due to the production of what is essentially a

bare proton when it is bonded with a strongly electronegative element such as fluorine , oxygen or nitrogen,

and the dipole-dipole interaction between compounds that are polar, such as salt. The properties of liquids

and the expansion of water ice when it freezes are due to molecular attraction.

In physical chemistry , the van der Waals force (or van der Waals interaction ), named after Dutch scientist

Johannes Diderik van der Waals , is the attractive or repulsive force between molecules (or between parts of

the same molecule) other than those due to covalent bonds or to the electrostatic interaction of ions with

one another or with neutral molecules.

[1]

The term includes:

permanent dipole–permanent dipole forces

permanent dipole –induced dipole forces

instantaneous induced dipole -induced dipole (London dispersion forces).

It is also sometimes used loosely as a synonym for the totality of intermolecular forces. Van der Waals

forces are relatively weak compared to normal chemical bonds, but play a fundament al role in fields as

diverse as supramolecular chemistry , structural biology, polymer science, nanotechnology, surface science,

and condensed matter physics .

Van der Waals forces include attractions between atoms, molecules, and surface s. They differ from

covalent and ionic bonding in that they are caused by correlations in the fluctuating polarizations of nearby

particles (a consequence of quantum dynamics).

Intermolecular forces have four major contributions. In general an intermolecular potential has a repulsive

component (which prevents the collapse of molecules because as entities move closer to one another these

repulsions dominate). It also has an attractive component, which, in turn, consists of three distinct

contributions:

1. The electrostatic interactions between charges (in the case of molecular ions), dipoles (in the case

of molecules without inversion center), quadrupoles (all molecules with symmetry lower than

cubic), and in general between permanent multipoles. The electrostatic interaction is sometimes

called the Keesom interaction or Keesom force after Willem Hendrik Keesom .

2. The second source of attraction is induction (also known as polarization), which is the interaction

between a permanent multipole on one molecule with an induced multipole o n another. This

interaction is sometimes measured in debyes after Peter J.W. Debye.

3. The third attraction is us ually named after Fritz London who himself called it dispersion. This is

the only attraction experienced by non -polar atoms, but it is operative between any pair of

molecules, irrespective of their symmetry.

All intermolecular/van der Waals forces are anisotropic (except those between two noble gas atoms), which

means that they depend on the relative orientation of the molecules. The induction and dispersion

interactions are always attractive, irrespective of orientation, but the electrost atic interaction changes sign

upon rotation of the molecules. That is, the electrostatic force can be attractive or repulsive, depending on

the mutual orientation of the molecules.

In physics, chemistry , and biology, intermolecular forces are forces that act between stable molecules or

between functional groups of macromolecules. Intermolecular forces (the weakest of which are van der

Waals forces- also known as the London Dispersion Forces) include momentary attractions between

molecules, diatomic free elements, and individual atoms. They differ from covalent and ionic bonding in

that they are not stable, but are caused by momentary polarization of particles. Because electrons have no

fixed position in the structure of an atom or molecule, but rather are distributed in a probabilistic fashion

based on quantum probability, there is a positive chance that the electrons are not evenly distributed and

thus their electrical charges are not evenly distributed. See Schrödinger equation for the theories on wave

functions and descriptions of position and velocity of quantum particles.

Dipolar couplings in macromolecular structure determination.

Structures of invisible, excited protein states by relaxation dispersion NMR spectroscopy.

Variation of molecular alignment as orientational ambiguities in couplings

Biological Magnetic Resonance

Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three

dimensions by superposing a height ( ) axis. Unfortunately, there are a number of

different notations used for the other two coordinates. Either or is used to refer to the

radial coordinate and either or to the azimuthal coordinates. Arfken (1985), for

instance, uses , while Beyer (1987) uses . In this work, the notation is

used.

Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken

1985), are a system of curvilinear coordinates that are natural for describing positions on

a sphere or spheroid. Define to be the azimuthal angle in the -plane from the x-axis

with (denoted when referred to as the longitude), to be the polar angle (also

known as the zenith angle and colatitude, with where is the latitude) from the

positive z-axis with , and to be distance (radius) from a point to the origin. This

is the convention commonly used in mathematics.

The coordinates are the asymptotic angle of confocal hyperbolic cylinders symmetrical

about the x-axis. The coordinates are confocal elliptic cylinders centered on the origin.

A field is any set of elements that satisfies the field axioms for both addition and

multiplication and is a commutative division algebra. An archaic name for a field is

rational domain. The French term for a field is corps and the German word is Körper,

both meaning "body." A field with a finite number of members is known as a finite field

or Galois field.

Because the identity condition is generally required to be different for addition and

multiplication, every field must have at least two elements. Examples include the

complex numbers ( ), rational numbers ( ), and real numbers ( ), but not the integers (

), which form only a ring.

Function Space

is the collection of all real-valued continuous functions defined on some interval .

is the collection of all functions with continuous th derivatives. A function

space is a topological vector space whose "points" are functions.

Complex Plane

The plane of complex numbers spanned by the vectors 1 and , where is the imaginary

number. Every complex number corresponds to a unique point in the complex plane. The

line in the plane with is the real line. The complex plane is sometimes called the

Argand plane or Gauss plane, and a plot of complex numbers in the plane is sometimes

called an Argand diagram.

Cartesian Plane

The Euclidean plane parametrized by coordinates, so that each point is located based on

its position with respect to two perpendicular lines, called coordinate axes. They are two

copies of the real line, and the zero point lies at their intersection, called the origin. The

coordinate axes are usually called the x-axis and y-axis, depicted above. Point is

associated with the coordinates corresponding to its orthogonal projections onto the

-axis and the -axis respectively.

Cartesian coordinates are rectilinear two-dimensional or three-dimensional coordinates

(and therefore a special case of curvilinear coordinates) which are also called rectangular

coordinates. The three axes of three-dimensional Cartesian coordinates, conventionally

denoted the x-, y-, and z-axes (a notation due to Descartes) are chosen to be linear and

mutually perpendicular. In three dimensions, the coordinates , , and may lie anywhere

in the interval . In René Descartes' original treatise (1637), which introduced the

use of coordinates for describing plane curves, the axes were omitted, and only positive

values of the - and the -coordinates were considered, since they were defined as

distances between points. For an ellipse this meant that, instead of the full picture which

we would plot nowadays, Descartes drew only the upper half.

Analytic Geometry

The study of the geometry of figures by algebraic representation and manipulation of

equations describing their positions, configurations, and separations. Analytic geometry

is also called coordinate geometry since the objects are described as -tuples of points

(where in the plane and 3 in space) in some coordinate system.

A tensor is an object which includes and extends the notion of scalar, vector, and matrix.

The term is slightly ambiguous, and this often leads to misunderstandings; roughly

speaking, there are different default meanings in mathematics and physics. In the

mathematical fields of multilinear algebra and differential geometry, a tensor is first of all

an element of a tensor product of vector spaces. In physics and engineering, the same

term often means what a mathematician would call a tensor field: an association of a

different mathematical tensor with each point of a geometric space, varying continuously

with position. This difference of emphasis conceals the agreement there is on the

geometric nature of tensors, and in application of tensors there may be different types of

notation used, but for what are the same underlying calculations.

In mathematics, a tensor is a generalized 'quantity' or 'geometrical entity'. If necessary it

can be expressed as a multi-dimensional array relative to a choice of basis of the

particular space on which it is defined; just as a linear transformation can be represented

as a matrix of numbers with respect to given vector bases, so a tensor of any type can be

written as an organized collection of numbers. The intuition underlying the tensor

concept, on the other hand, is inherently geometrical: as an object in and of itself, a tensor

is independent of any chosen frame of reference , just as a geometric vector, generally

pictured as an arrow, is independent of the basis used to represent it.

Many mathematical structures informally called 'tensors' are actually tensor fields—what

is given is a tensor-valued function defined on a geometric or topological space. This use

of the term is analogous to vector fields such as electromagnetic fields, with a vector

given at each point of a region. The whole advantage of tensors is that everything defined

in tensor terms can be tracked in a definite way under any change of coordinates.

Differential equations posed in terms of tensor quantities are basic to modern

mathematical physics, so that tensor fields are usually defined on differentiable

manifolds.

Structure tensors (or second-moment matrices) are matrix representations of partial

derivatives. In the field of image processing and computer vision, they are typically used

to represent gradients, edges or similar information. Structure tensors also provide a more

powerful description of local patterns when compared with the directional derivative

through its coherence measure. They have several other advantages that are detailed in

the structure tensor section of this tutorial. This article is written largely from the point of

view of image processing applications; however, it should also be useful to those from a

more general math background who want to learn more about this representation.

Tensor addition

Another desirable property of the structure tensor form is that the tensor addition equates

itself to the adding of the elliptical forms. For example, if the structure tensors for the

sphere case and step-edge case are added, the resulting structure tensor is an elongated

ellipsed along the direction of the step-edge case.

Vector Field

A vector field is a map that assigns each a vector . Several vector fields

are illustrated above. A vector field is uniquely specified by giving its divergence and

curl within a region and its normal component over the boundary, a result known as

Helmholtz's theorem (Arfken 1985, p. 79).

Scalar Field

A map which assigns each a scalar function .

Slope Field

Given an ordinary differential equation , the slope field for that differential

equation is the vector field that takes a point to a unit vector with slope . The

vectors in a slope field are usually drawn without arrowheads, indicating that they can be

followed in either direction. Using a visualization of a slope field, it is easy to graphically

trace out solution curves to initial value problems. For example, the illustration above

shows the slope field for the equation together with solution curves for

various initial values of .

Subfield

If a subset of the elements of a field satisfies the field axioms with the same operations

of , then is called a subfield of . In a finite field of field order , with a prime, there

exists a subfield of field order for every dividing .

A line passing through the origin (blue, thick) in R

3

is a linear subspace. It is the

intersection of two planes (green and yellow).

In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular

array of numbers, as shown at the right. Matrices consisting of only one column or row

are called vectors, while higher-dimensional, e.g. three-dimensional, arrays of numbers

are called tensors. Matrices can be added and subtracted entrywise, and multiplied

according to a rule corresponding to composition of linear transformations. These

operations satisfy the usual identities, except that matrix multiplication is not

commutative: the identity AB=BA can fail. One use of matrices is to represent linear

transformations, which are higher-dimensional analogs of linear functions of the form f(x)

= cx, where c is a constant. Matrices can also keep track of the coefficients in a system of

linear equations. For a square matrix, the determinant and inverse matrix (when it exists)

govern the behavior of solutions to the corresponding system of linear equations, and

eigenvalues and eigenvectors provide insight into the geometry of the associated linear

transformation.

In mathematics, curvature refers to any of a number of loosely related concepts in

different areas of geometry. Intuitively, curvature is the amount by which a geometric

object deviates from being flat, or straight in the case of a line, but this is defined in

different ways depending on the context. There is a key distinction between extrinsic

curvature, which is defined for objects embedded in another space (usually a Euclidean

space) in a way that relates to the radius of curvature of circles that touch the object, and

intrinsic curvature, which is defined at each point in a differential manifold. This article

deals primarily with the first concept.

The primordial example of extrinsic curvature is that of a circle, which has curvature

equal to the inverse of its radius everywhere. Smaller circles bend more sharply, and

hence have higher curvature. The curvature of a smooth curve is defined as the curvature

of its osculating circle at each point.

In a plane, this is a scalar quantity, but in three or more dimensions it is described by a

curvature vector that takes into account the direction of the bend as well as its sharpness.

The curvature of more complex objects (such as surfaces or even curved n-dimensional

spaces) is described by more complex objects from linear algebra, such as the general

Riemann curvature tensor.

In mathematics, the spherical harmonics are the angular portion of a set of solutions to

Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical

harmonics are a specific set of spherical harmonics which forms an orthogonal

system, first introduced by Pierre Simon de Laplace.

[1]

Spherical harmonics are important

in many theoretical and practical applications, particularly in the computation of atomic

orbital electron configurations, representation of gravitational fields, geoids, and the

magnetic fields of planetary bodies and stars, and characterization of the cosmic

microwave background radiation. In 3D computer graphics, spherical harmonics plays a

special role in a wide variety of topics including indirect lighting (ambient occlusion,

global illumination, precomputed radiance transfer, etc.) and in recognition of 3D shapes.

The term "potential theory" arises from the fact that, in 19th-century physics, the

fundamental forces of nature were believed to be derived from potentials which satisfied

Laplace's equation. Hence, potential theory was the study of functions which could serve

as potentials. Nowadays, we know that nature is more complicated: the equations which

describe forces are systems of non-linear partial differential equations such as the

Einstein equations and the Yang-Mills equations, and the Laplace equation is only valid

as a limiting case. Nevertheless, the term "potential theory" has remained as a convenient

term for describing the study of functions which satisfy the Laplace equation. It is also

still the case that the Laplace equation is used in applications in several areas of physics

like heat conduction and electrostatics.

Obviously, there is considerable overlap between potential theory and the theory of the

Laplace equation. To the extent that it is possible to draw a distinction between these two

fields, the difference is more one of emphasis than subject matter and rests on the

following distinction: potential theory focuses on the properties of the functions as

opposed to the properties of the equation. For example, a result about the singularities of

harmonic functions would be said to belong to potential theory whilst a result on how the

solution depends on the boundary data would be said to belong to the theory of the

Laplace equation. Of course, this is not a hard and fast distinction, and in practice there is

considerable overlap between the two fields, with methods and results from one being

used in the other.

Modern potential theory is also intimately connected with probability and the theory of

Markov chains. In the continuous case, this is closely related to the analytic theory. In the

finite state space case, this connection can be introduced by introducing an electric

network on the state space, with resistance between points inversely proportional to

transition probabilities and densities proportional to potentials. Even in the finite case, the

analogue I-K of the Laplacian in potential theory has its own maximum principal,

uniqueness principal, balance principal, and others.

The complex plane together with the point at infinity is known as the Riemann

sphere or extended complex plane and denoted or . However, the notation is also

used to denote the punctured plane .

In geometry and linear algebra, a rotation is a transformation in a plane or in space that

describes the motion of a rigid body around a fixed point. A rotation is different from a

translation, which has no fixed points, and from a reflection, which "flips" the bodies it is

transforming. A rotation and the above-mentioned transformations are isometries; they

leave the distance between any two points unchanged after the transformation.

Spin (physics)

Yaw, pitch, and roll, also known as Tait–Bryan angles, named after Peter Guthrie Tait

and George Bryan, are a specific kind of Euler angles very often used in aerospace

applications to define the relative orientation of a vehicle. The three angles specified in

this formulation are defined as the roll angle, pitch angle, and yaw angle.

The Penrose triangle, also called the tribar (Cerf), tri-bar (Ernst 1987), impossible tribar

(Pappas 1989, p. 13), or impossible triangle, is an impossible figure published by Penrose

and Penrose (1958). Penrose triangles appear prominently in the works of Escher, who

not only inspired creation of this object (Escher 1954, Penrose and Penrose 1958), but

also subsequently publicized it.

Kanizsa Triangle

An optical illusion, illustrated above, in which the eye perceives a white upright

equilateral triangle where none is actually drawn.

Hering Illusion

An optical illusion due to the physiologist Ewald Hering in 1861. The two horizontal

lines are both straight, but they look as if they were bowed outwards. The distortion is

produced by the lined pattern on the background, that simulates a perspective design, and

creates a false impression of depth.

Necker Cube

The necker cube is an illusion in which a two-dimensional drawing of an array of cubes

appears to simultaneously protrude from and intrude into the page.

The Necker cube is also a tiling that was

used in ancient times, including as a mosaic on the floor of one of the houses in Pompeii,

as illustrated in the photograph above (courtesy of S. Jaskulowski).

Computational Systems

Polish Space

The homeomorphic image of a so-called "complete separable" metric space. The

continuous image of a Polish space is called a Souslin set.