Previous: Introduction to physical_constants, Up: ezunits [Contents][Index]
The dimensional quantity operator.
An expression \(a ` b\) represents a dimensional quantity,
with a
indicating a nondimensional quantity and b
indicating the dimensional units.
A symbol can be used as a unit without declaring it as such;
unit symbols need not have any special properties.
The quantity and unit of an expression \(a ` b\) can
be extracted by the qty
and units
functions, respectively.
Arithmetic operations on dimensional quantities are carried out by conventional rules for such operations.
y
is nondimensional.
ezunits
does not require that units in a sum have the same dimensions;
such terms are not added together, and no error is reported.
load ("ezunits")
enables this operator.
Examples:
SI (Systeme Internationale) units.
(%i1) load ("ezunits")$ (%i2) foo : 10 ` m; (%o2) 10 ` m (%i3) qty (foo); (%o3) 10 (%i4) units (foo); (%o4) m (%i5) dimensions (foo); (%o5) length
"Customary" units.
(%i1) load ("ezunits")$ (%i2) bar : x ` acre; (%o2) x ` acre (%i3) dimensions (bar); 2 (%o3) length (%i4) fundamental_units (bar); 2 (%o4) m
Units ad hoc.
(%i1) load ("ezunits")$ (%i2) baz : 3 ` sheep + 8 ` goat + 1 ` horse; (%o2) 8 ` goat + 3 ` sheep + 1 ` horse (%i3) subst ([sheep = 3*goat, horse = 10*goat], baz); (%o3) 27 ` goat (%i4) baz2 : 1000`gallon/fortnight; gallon (%o4) 1000 ` --------- fortnight (%i5) subst (fortnight = 14*day, baz2); 500 gallon (%o5) --- ` ------ 7 day
Arithmetic operations on dimensional quantities.
(%i1) load ("ezunits")$ (%i2) 100 ` kg + 200 ` kg; (%o2) 300 ` kg (%i3) 100 ` m^3 - 100 ` m^3; 3 (%o3) 0 ` m (%i4) (10 ` kg) * (17 ` m/s^2); kg m (%o4) 170 ` ---- 2 s (%i5) (x ` m) / (y ` s); x m (%o5) - ` - y s (%i6) (a ` m)^2; 2 2 (%o6) a ` m
The unit conversion operator.
An expression a ` b `` c converts from unit b
to unit c
.
ezunits
has built-in conversions for SI base units,
SI derived units, and some non-SI units.
Unit conversions not already known to ezunits
can be declared.
The unit conversions known to ezunits
are specified by the
global variable known_unit_conversions
,
which comprises built-in and user-defined conversions.
Conversions for products, quotients, and powers of units are
derived from the set of known unit conversions.
There is no preferred system for display of units;
input units are not converted to other units
unless conversion is explicitly indicated.
ezunits
does not attempt to simplify units by prefixes
(milli-, centi-, deci-, etc)
unless such conversion is explicitly indicated.
load ("ezunits")
enables this operator.
Examples:
The set of known unit conversions.
(%i1) load ("ezunits")$ (%i2) display2d : false$ (%i3) known_unit_conversions; (%o3) {acre = 4840*yard^2,Btu = 1055*J,cfm = feet^3/minute, cm = m/100,day = 86400*s,feet = 381*m/1250,ft = feet, g = kg/1000,gallon = 757*l/200,GHz = 1000000000*Hz, GOhm = 1000000000*Ohm,GPa = 1000000000*Pa, GWb = 1000000000*Wb,Gg = 1000000*kg,Gm = 1000000000*m, Gmol = 1000000*mol,Gs = 1000000000*s,ha = hectare, hectare = 100*m^2,hour = 3600*s,Hz = 1/s,inch = feet/12, km = 1000*m,kmol = 1000*mol,ks = 1000*s,l = liter, lbf = pound_force,lbm = pound_mass,liter = m^3/1000, metric_ton = Mg,mg = kg/1000000,MHz = 1000000*Hz, microgram = kg/1000000000,micrometer = m/1000000, micron = micrometer,microsecond = s/1000000, mile = 5280*feet,minute = 60*s,mm = m/1000, mmol = mol/1000,month = 2629800*s,MOhm = 1000000*Ohm, MPa = 1000000*Pa,ms = s/1000,MWb = 1000000*Wb, Mg = 1000*kg,Mm = 1000000*m,Mmol = 1000000000*mol, Ms = 1000000*s,ns = s/1000000000,ounce = pound_mass/16, oz = ounce,Ohm = s*J/C^2, pound_force = 32*ft*pound_mass/s^2, pound_mass = 200*kg/441,psi = pound_force/inch^2, Pa = N/m^2,week = 604800*s,Wb = J/A,yard = 3*feet, year = 31557600*s,C = s*A,F = C^2/J,GA = 1000000000*A, GC = 1000000000*C,GF = 1000000000*F,GH = 1000000000*H, GJ = 1000000000*J,GK = 1000000000*K,GN = 1000000000*N, GS = 1000000000*S,GT = 1000000000*T,GV = 1000000000*V, GW = 1000000000*W,H = J/A^2,J = m*N,kA = 1000*A, kC = 1000*C,kF = 1000*F,kH = 1000*H,kHz = 1000*Hz, kJ = 1000*J,kK = 1000*K,kN = 1000*N,kOhm = 1000*Ohm, kPa = 1000*Pa,kS = 1000*S,kT = 1000*T,kV = 1000*V, kW = 1000*W,kWb = 1000*Wb,mA = A/1000,mC = C/1000, mF = F/1000,mH = H/1000,mHz = Hz/1000,mJ = J/1000, mK = K/1000,mN = N/1000,mOhm = Ohm/1000,mPa = Pa/1000, mS = S/1000,mT = T/1000,mV = V/1000,mW = W/1000, mWb = Wb/1000,MA = 1000000*A,MC = 1000000*C, MF = 1000000*F,MH = 1000000*H,MJ = 1000000*J, MK = 1000000*K,MN = 1000000*N,MS = 1000000*S, MT = 1000000*T,MV = 1000000*V,MW = 1000000*W, N = kg*m/s^2,R = 5*K/9,S = 1/Ohm,T = J/(m^2*A),V = J/C, W = J/s}
Elementary unit conversions.
(%i1) load ("ezunits")$ (%i2) 1 ` ft `` m; Computing conversions to base units; may take a moment. 381 (%o2) ---- ` m 1250 (%i3) %, numer; (%o3) 0.3048 ` m (%i4) 1 ` kg `` lbm; 441 (%o4) --- ` lbm 200 (%i5) %, numer; (%o5) 2.205 ` lbm (%i6) 1 ` W `` Btu/hour; 720 Btu (%o6) --- ` ---- 211 hour (%i7) %, numer; Btu (%o7) 3.412322274881517 ` ---- hour (%i8) 100 ` degC `` degF; (%o8) 212 ` degF (%i9) -40 ` degF `` degC; (%o9) (- 40) ` degC (%i10) 1 ` acre*ft `` m^3; 60228605349 3 (%o10) ----------- ` m 48828125 (%i11) %, numer; 3 (%o11) 1233.48183754752 ` m
Coercing quantities in feet and meters to one or the other.
(%i1) load ("ezunits")$ (%i2) 100 ` m + 100 ` ft; (%o2) 100 ` m + 100 ` ft (%i3) (100 ` m + 100 ` ft) `` ft; Computing conversions to base units; may take a moment. 163100 (%o3) ------ ` ft 381 (%i4) %, numer; (%o4) 428.0839895013123 ` ft (%i5) (100 ` m + 100 ` ft) `` m; 3262 (%o5) ---- ` m 25 (%i6) %, numer; (%o6) 130.48 ` m
Dimensional analysis to find fundamental dimensions and fundamental units.
(%i1) load ("ezunits")$ (%i2) foo : 1 ` acre * ft; (%o2) 1 ` acre ft (%i3) dimensions (foo); 3 (%o3) length (%i4) fundamental_units (foo); 3 (%o4) m (%i5) foo `` m^3; Computing conversions to base units; may take a moment. 60228605349 3 (%o5) ----------- ` m 48828125 (%i6) %, numer; 3 (%o6) 1233.48183754752 ` m
Declared unit conversions.
(%i1) load ("ezunits")$ (%i2) declare_unit_conversion (MMBtu = 10^6*Btu, kW = 1000*W); (%o2) done (%i3) declare_unit_conversion (kWh = kW*hour, MWh = 1000*kWh, bell = 1800*s); (%o3) done (%i4) 1 ` kW*s `` MWh; Computing conversions to base units; may take a moment. 1 (%o4) ------- ` MWh 3600000 (%i5) 1 ` kW/m^2 `` MMBtu/bell/ft^2; 1306449 MMBtu (%o5) ---------- ` -------- 8242187500 2 bell ft
Shows the value and the units of one of the constants declared by package
physical_constants
, which includes a list of physical constants, or
of a new constant declared in package ezunits
(see
declare_constvalue
).
Note that constant values as recognized by constvalue
are separate from values declared by numerval
and
recognized by constantp
.
Example:
(%i1) load ("physical_constants")$ (%i2) constvalue (%G); 3 m (%o2) 6.67428 ` ----- 2 kg s (%i3) get ('%G, 'description); (%o3) Newtonian constant of gravitation
Declares the value of a constant to be used in package ezunits
. This
function should be loaded with load ("ezunits")
.
Example:
(%i1) load ("ezunits")$ (%i2) declare_constvalue (FOO, 100 ` lbm / acre); lbm (%o2) 100 ` ---- acre (%i3) FOO * (50 ` acre); (%o3) 50 FOO ` acre (%i4) constvalue (%); (%o4) 5000 ` lbm
Reverts the effect of declare_constvalue
. This function should be
loaded with load ("ezunits")
.
Returns the units of a dimensional quantity x, or returns 1 if x is nondimensional.
x may be a literal dimensional expression \(a ` b\),
a symbol with declared units via declare_units
,
or an expression containing either or both of those.
This function should be loaded with load ("ezunits")
.
Example:
(%i1) load ("ezunits")$ (%i2) foo : 100 ` kg; (%o2) 100 ` kg (%i3) bar : x ` m/s; m (%o3) x ` - s (%i4) units (foo); (%o4) kg (%i5) units (bar); m (%o5) - s (%i6) units (foo * bar); kg m (%o6) ---- s (%i7) units (foo / bar); kg s (%o7) ---- m (%i8) units (foo^2); 2 (%o8) kg
Declares that units
should return units u for a,
where u is an expression. This function should be loaded with
load ("ezunits")
.
Example:
(%i1) load ("ezunits")$ (%i2) units (aa); (%o2) 1 (%i3) declare_units (aa, J); (%o3) J (%i4) units (aa); (%o4) J (%i5) units (aa^2); 2 (%o5) J (%i6) foo : 100 ` kg; (%o6) 100 ` kg (%i7) units (aa * foo); (%o7) kg J
Returns the nondimensional part of a dimensional quantity x, or returns x if x is nondimensional. x may be a literal dimensional expression \(a ` b\), a symbol with declared quantity, or an expression containing either or both of those.
This function should be loaded with load ("ezunits")
.
Example:
(%i1) load ("ezunits")$ (%i2) foo : 100 ` kg; (%o2) 100 ` kg (%i3) qty (foo); (%o3) 100 (%i4) bar : v ` m/s; m (%o4) v ` - s (%i5) foo * bar; kg m (%o5) 100 v ` ---- s (%i6) qty (foo * bar); (%o6) 100 v
Declares that qty
should return x for symbol a, where
x is a nondimensional quantity. This function should be loaded
with load ("ezunits")
.
Example:
(%i1) load ("ezunits")$ (%i2) declare_qty (aa, xx); (%o2) xx (%i3) qty (aa); (%o3) xx (%i4) qty (aa^2); 2 (%o4) xx (%i5) foo : 100 ` kg; (%o5) 100 ` kg (%i6) qty (aa * foo); (%o6) 100 xx
Returns true
if x is a literal dimensional expression,
a symbol declared dimensional,
or an expression in which the main operator is declared dimensional.
unitp
returns false
otherwise.
load ("ezunits")
loads this function.
Examples:
unitp
applied to a literal dimensional expression.
(%i1) load ("ezunits")$ (%i2) unitp (100 ` kg); (%o2) true
unitp
applied to a symbol declared dimensional.
(%i1) load ("ezunits")$ (%i2) unitp (foo); (%o2) false (%i3) declare (foo, dimensional); (%o3) done (%i4) unitp (foo); (%o4) true
unitp
applied to an expression in which the main operator is declared dimensional.
(%i1) load ("ezunits")$ (%i2) unitp (bar (x, y, z)); (%o2) false (%i3) declare (bar, dimensional); (%o3) done (%i4) unitp (bar (x, y, z)); (%o4) true
Appends equations u = v, ... to the list of unit conversions known to the unit conversion operator ``. u and v are both multiplicative terms, in which any variables are units, or both literal dimensional expressions.
At present, it is necessary to express conversions such that the left-hand side of each equation is a simple unit (not a multiplicative expression) or a literal dimensional expression with the quantity equal to 1 and the unit being a simple unit. This limitation might be relaxed in future versions.
known_unit_conversions
is the list of known unit conversions.
This function should be loaded with load ("ezunits")
.
Examples:
Unit conversions expressed by equations of multiplicative terms.
(%i1) load ("ezunits")$ (%i2) declare_unit_conversion (nautical_mile = 1852 * m, fortnight = 14 * day); (%o2) done (%i3) 100 ` nautical_mile / fortnight `` m/s; Computing conversions to base units; may take a moment. 463 m (%o3) ---- ` - 3024 s
Unit conversions expressed by equations of literal dimensional expressions.
(%i1) load ("ezunits")$ (%i2) declare_unit_conversion (1 ` fluid_ounce = 2 ` tablespoon); (%o2) done (%i3) declare_unit_conversion (1 ` tablespoon = 3 ` teaspoon); (%o3) done (%i4) 15 ` fluid_ounce `` teaspoon; Computing conversions to base units; may take a moment. (%o4) 90 ` teaspoon
Declares a_1, ..., a_n to have dimensions d_1, ..., d_n, respectively.
Each a_k is a symbol or a list of symbols. If it is a list, then every symbol in a_k is declared to have dimension d_k.
load ("ezunits")
loads these functions.
Examples:
(%i1) load ("ezunits") $ (%i2) declare_dimensions ([x, y, z], length, [t, u], time); (%o2) done (%i3) dimensions (y^2/u); 2 length (%o3) ------- time (%i4) fundamental_units (y^2/u); 0 errors, 0 warnings 2 m (%o4) -- s
Reverts the effect of declare_dimensions
. This function should be
loaded with load ("ezunits")
.
declare_fundamental_dimensions
declares fundamental dimensions.
Symbols d_1, d_2, d_3, ... are appended to the list of
fundamental dimensions, if they are not already on the list.
remove_fundamental_dimensions
reverts the effect of declare_fundamental_dimensions
.
fundamental_dimensions
is the list of fundamental dimensions.
By default, the list comprises several physical dimensions.
load ("ezunits")
loads these functions.
Examples:
(%i1) load ("ezunits") $ (%i2) fundamental_dimensions; (%o2) [length, mass, time, current, temperature, quantity] (%i3) declare_fundamental_dimensions (money, cattle, happiness); (%o3) done (%i4) fundamental_dimensions; (%o4) [length, mass, time, current, temperature, quantity, money, cattle, happiness] (%i5) remove_fundamental_dimensions (cattle, happiness); (%o5) done (%i6) fundamental_dimensions; (%o6) [length, mass, time, current, temperature, quantity, money]
declare_fundamental_units
declares u_1, ..., u_n
to have dimensions d_1, ..., d_n, respectively.
All arguments must be symbols.
After calling declare_fundamental_units
,
dimensions(u_k)
returns d_k for each argument u_1, ..., u_n,
and fundamental_units(d_k)
returns u_k for each argument d_1, ..., d_n.
remove_fundamental_units
reverts the effect of declare_fundamental_units
.
load ("ezunits")
loads these functions.
Examples:
(%i1) load ("ezunits") $ (%i2) declare_fundamental_dimensions (money, cattle, happiness); (%o2) done (%i3) declare_fundamental_units (dollar, money, goat, cattle, smile, happiness); (%o3) [dollar, goat, smile] (%i4) dimensions (100 ` dollar/goat/km^2); money (%o4) -------------- 2 cattle length (%i5) dimensions (x ` smile/kg); happiness (%o5) --------- mass (%i6) fundamental_units (money*cattle/happiness); 0 errors, 0 warnings dollar goat (%o6) ----------- smile
dimensions
returns the dimensions of the dimensional quantity x
as an expression comprising products and powers of base dimensions.
dimensions_as_list
returns the dimensions of the dimensional quantity x
as a list, in which each element is an integer which indicates the power of the
corresponding base dimension in the dimensions of x.
load ("ezunits")
loads these functions.
Examples:
(%i1) load ("ezunits")$ (%i2) dimensions (1000 ` kg*m^2/s^3); 2 length mass (%o2) ------------ 3 time (%i3) declare_units (foo, acre*ft/hour); acre ft (%o3) ------- hour (%i4) dimensions (foo); 3 length (%o4) ------- time
(%i1) load ("ezunits")$ (%i2) fundamental_dimensions; (%o2) [length, mass, time, charge, temperature, quantity] (%i3) dimensions_as_list (1000 ` kg*m^2/s^3); (%o3) [2, 1, - 3, 0, 0, 0] (%i4) declare_units (foo, acre*ft/hour); acre ft (%o4) ------- hour (%i5) dimensions_as_list (foo); (%o5) [3, 0, - 1, 0, 0, 0]
fundamental_units(x)
returns the units
associated with the fundamental dimensions of x.
as determined by dimensions(x)
.
x may be a literal dimensional expression \(a ` b\),
a symbol with declared units via declare_units
,
or an expression containing either or both of those.
fundamental_units()
returns the list of all known fundamental units,
as declared by declare_fundamental_units
.
load ("ezunits")
loads this function.
Examples:
(%i1) load ("ezunits")$ (%i2) fundamental_units (); (%o2) [m, kg, s, A, K, mol] (%i3) fundamental_units (100 ` mile/hour); m (%o3) - s (%i4) declare_units (aa, g/foot^2); g (%o4) ----- 2 foot (%i5) fundamental_units (aa); kg (%o5) -- 2 m
Returns a basis for the dimensionless quantities which can be formed from a list L of dimensional quantities.
load ("ezunits")
loads this function.
Examples:
(%i1) load ("ezunits") $ (%i2) dimensionless ([x ` m, y ` m/s, z ` s]); 0 errors, 0 warnings 0 errors, 0 warnings y z (%o2) [---] x
Dimensionless quantities derived from fundamental physical quantities. Note that the first element on the list is proportional to the fine-structure constant.
(%i1) load ("ezunits") $ (%i2) load ("physical_constants") $ (%i3) dimensionless([%h_bar, %m_e, %m_P, %%e, %c, %e_0]); 0 errors, 0 warnings 0 errors, 0 warnings 2 %%e %m_e (%o3) [--------------, ----] %c %e_0 %h_bar %m_P
Finds exponents e_1, ..., e_n such that
dimension(expr) = dimension(v_1^e_1 ... v_n^e_n)
.
load ("ezunits")
loads this function.
Examples:
Previous: Introduction to physical_constants, Up: ezunits [Contents][Index]