Vorige: Introduction to physical_constants, Nach oben: ezunits [Inhalt][Index]
The dimensional quantity operator.
An expression 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 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
Returns the declared constant value of a symbol, or value of an expression with declared constant values substituted for symbols.
Constant values are declared by declare_constvalue.
Note that constant values as recognized by constvalue
are separate from values declared by numerval and
recognized by constantp.
The physical_units package declares constant values
for a number of physical constants.
load("ezunits") loads these functions.
Examples:
Constant value of a physical constant.
(%i1) load ("physical_constants")$
(%i2) constvalue (%G);
3
m
(%o2) 6.67428 ` -----
2
kg s
(%i3) get ('%G, 'description);
(%o3) Newtonian constant of gravitation
Declaring a new constant.
(%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
Returns the units of a dimensional quantity x, or returns 1 if x is nondimensional.
x may be a literal dimensional expression declare_units,
or an expression containing either or both of those.
declare_units declares that units(a) should return u,
where u is an expression.
load("ezunits") loads these functions.
Examples:
units applied to literal dimensional expressions.
(%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
units applied to symbols with declared units.
(%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
qty returns the nondimensional part of a dimensional quantity x,
or returns x if x is nondimensional.
x may be a literal dimensional expression
declare_qty declares that qty(a) should return x,
where x is a nondimensional quantity.
load("ezunits") loads these functions.
Examples:
qty applied to literal dimensional expressions.
(%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
qty applied to symbols with declared quantity.
(%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.
load("ezunits") loads this function.
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
declare_dimensions 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.
remove_dimensions reverts the effect of declare_dimensions.
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
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 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:
Vorige: Introduction to physical_constants, Nach oben: ezunits [Inhalt][Index]