OpenMath Content Dictionary: dimensions1

Canonical URL:

http://www.openmath.org/cd/dimensions1.ocd

CD File:

dimensions1.ocd

CD as XML Encoded OpenMath:

dimensions1.omcd

Defines:

acceleration, area, charge, concentration, current, density, energy, force, length, mass, momentum, power, pressure, resistance, speed, temperature, time, velocity, voltage, volume

Date:

2003-08-20

Version:

4

Review Date:

2004-04-01

Status:

experimental

Uses CD:

norm1, arith1, relation1

This CD defines symbols which represent basic physical dimensions.

length

Description:

This symbol represents the length physical dimension.

Signatures:

sts

area

Description:

This symbol represents the area physical dimension.

Commented Mathematical property (CMP):

area = length*length

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="area"/>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMS cd="dimensions1" name="length"/>
      <OMS cd="dimensions1" name="length"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (area, times (length, length) )

$$\mathrm{area}=\mathrm{length}\times \mathrm{length}$$

Signatures:

sts

volume

Description:

This symbol represents the volume physical dimension.

Commented Mathematical property (CMP):

volume = length*length*length

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="volume"/>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMS cd="dimensions1" name="length"/>
      <OMS cd="dimensions1" name="length"/>
      <OMS cd="dimensions1" name="length"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (volume, times (length, length, length) )

$$\mathrm{volume}=\mathrm{length}\times \mathrm{length}\times \mathrm{length}$$

Signatures:

sts

speed

Description:

This symbol represents the speed physical dimension. It is the size of the derivative of distance with respect to time.

Commented Mathematical property (CMP):

speed = length/time

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="speed"/>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMS cd="dimensions1" name="length"/>
      <OMS cd="dimensions1" name="time"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (speed, divide (length, time) )

$$\mathrm{speed}=\frac{\mathrm{length}}{\mathrm{time}}$$

Signatures:

sts

velocity

Description:

This symbol represents the velocity physical dimension. It is the derivative of (vector) distance with respect to time.

Commented Mathematical property (CMP):

velocity = displacement/second

Commented Mathematical property (CMP):

speed is the Euclidean norm of velocity

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS name="eq" cd="relation1"/>&gt;
    <OMS name="speed" cd="dimensions1"/>&gt;
    <OMA>
      <OMS name="Euclidean_norm" cd="norm1"/>&gt;
      <OMS name="velocity" cd="dimensions1"/>&gt;
    </OMA>
  </OMA>
</OMOBJ>

eq (speed, Euclidean_norm (velocity) )

$$\mathrm{speed}=\mathrm{Euclidean\_norm}\left(\mathrm{velocity}\right)$$

Signatures:

sts

acceleration

Description:

This symbol represents the acceleration physical dimension. It is the second derivative of distance with respect to time.

Commented Mathematical property (CMP):

acceleration = length/(time^2)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="acceleration"/>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMS cd="dimensions1" name="length"/>
      <OMA>
        <OMS cd="arith1" name="divide"/>
        <OMS cd="dimensions1" name="time"/>
	<OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (acceleration, divide (length, divide (time, 2 ) ) )

$$\mathrm{acceleration}=\frac{\mathrm{length}}{\frac{\mathrm{time}}{2}}$$

Signatures:

sts

time

Description:

This symbol represents the time physical dimension. Note that the main units for time are defined in the units_time1 CD.

Signatures:

sts

mass

Description:

This symbol represents the mass physical dimension.

Signatures:

sts

force

Description:

This symbol represents the force physical dimension.

Commented Mathematical property (CMP):

force = mass*length/time^2

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="force"/>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMS cd="dimensions1" name="mass"/>
      <OMS cd="dimensions1" name="acceleration"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (force, times (mass, acceleration) )

$$\mathrm{force}=\mathrm{mass}\times \mathrm{acceleration}$$

Signatures:

sts

temperature

Description:

This symbol represents the temperature physical dimension.

Signatures:

sts

pressure

Description:

This symbol represents the pressure physical dimension.

Commented Mathematical property (CMP):

pressure = force/area

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="pressure"/>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMS cd="dimensions1" name="force"/>
      <OMS cd="dimensions1" name="area"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (pressure, divide (force, area) )

$$\mathrm{pressure}=\frac{\mathrm{force}}{\mathrm{area}}$$

Signatures:

sts

charge

Description:

This symbol represents the charge physical dimension.

Commented Mathematical property (CMP):

charge = current/voltage

Signatures:

sts

current

Description:

This symbol represents the current physical dimension.

Commented Mathematical property (CMP):

current = voltage*charge

Signatures:

sts

voltage

Description:

This symbol represents the voltage physical dimension.

Commented Mathematical property (CMP):

voltage = current/charge

Signatures:

sts

resistance

Description:

This symbol represents the resistance physical dimension, it is the resistance that an electrical circuit has to flow of charge.

Commented Mathematical property (CMP):

resistance = voltage/current

Signatures:

sts

density

Description:

This symbol represents the density physical dimension, it is the mass per unit volume.

Commented Mathematical property (CMP):

density = mass/volume

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="density"/>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMS cd="dimensions1" name="mass"/>
      <OMS cd="dimensions1" name="volume"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (density, divide (mass, volume) )

$$\mathrm{density}=\frac{\mathrm{mass}}{\mathrm{volume}}$$

Signatures:

sts

energy

Description:

This symbol represents the energy physical dimension.

Commented Mathematical property (CMP):

energy = mass*length^2/time^2

Commented Mathematical property (CMP):

energy = force*length

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="energy"/>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMS cd="dimensions1" name="force"/>
      <OMS cd="dimensions1" name="length"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (energy, times (force, length) )

$$\mathrm{energy}=\mathrm{force}\times \mathrm{length}$$

Signatures:

sts

concentration

Description:

This symbol represents the concentration physical dimension, it is the amount of a substance in a volume.

Commented Mathematical property (CMP):

concentration = mass/length^3

Signatures:

sts

momentum

Description:

This symbol represents the momentum physical dimension, it is mass times velocity.

Commented Mathematical property (CMP):

momentum = mass*velocity

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="momentun"/>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMS cd="dimensions1" name="mass"/>
      <OMS cd="dimensions1" name="velocity"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (momentun, times (mass, velocity) )

$$\mathrm{momentun}=\mathrm{mass}\times \mathrm{velocity}$$

Signatures:

sts

power

Description:

This symbol represents the power physical dimension, it is energy per time.

Commented Mathematical property (CMP):

power = energy/time

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMS cd="dimensions1" name="power"/>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMS cd="dimensions1" name="energy"/>
      <OMS cd="dimensions1" name="time"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (power, divide (energy, time) )

$$\mathrm{power}=\frac{\mathrm{energy}}{\mathrm{time}}$$

Signatures:

sts