
using the chain rule for the last step.
This is essentially the idea of (exact) differential 1-forms: they take vectors to the derivatives of functions.

A:

In this case, the author is using $\delta$ for the exterior derivative, which is usually denoted $\operatorname{d}$.
The function $A$ is a function of several variables, say $A(x_1,\ldots,x_n)$. The partial derivative of $A$ with respect to any one of the variables is denoted by
$$\frac{\partial A}{\partial x_i}$$
where $1 \le i \le n$. By definition, the exterior derivative of the differential $0$-form, i.e. of the function, $A$ is
$$\operatorname{d}\!A = \frac{\partial A}{\partial x_1}\, \operatorname{d}\!x_1 + \cdots + \frac{\partial A}{\partial x_n}\, \operatorname{d}\!x_n$$
Each of the $\operatorname{d}\!x_i$ are differential 1-forms. They take a tangent vector and give the $i$-th component of that vector. If you want to get very technical then each of the $\operatorname{d}\!x_i$ are linearly independent covectors. Since each of the $\operatorname{d}\!x_i$ are linearly independent, $\operatorname{d}\!A = 0$ if and only if each of the partial derivatives are zero.

 Q:

Functions on pairs of different types

Is there a way to define "pairmap" like the following:
pairmap f (x,y) = (f x, f y)

So that the following works:
pairmap (+2) (1::Int, 2::Float)
pairmap succ (1::Int, 'a')
pairmap Just ('a', True)

etc.
Naturally, in the first case, both the elements must be of class Num, and in the second case, both of class Enum. In the third case however, there's no restriction required.
Answer (but could be improved)
The following code (ideone) solves the problem, but note that my functions have to be wrapped in a datatype that encapsulates both the relation between the input and output types and also any constraints on the input type. This works but there's a bit of boilerplate. It would be
using the chain rule for the last gradation. 
 This is basically the idea of (exact) differential 1 - form: they assume vectors to the derivatives of function. 

 adenine: 

 In this case, the writer is using $ \delta$ for the exterior derivative, which is normally denoted $ \operatorname{d}$. 
 The function $ A$ is a function of several variable star, say $ A(x_1,\ldots, x_n)$. The partial derivative of $ A$ with respect to any one of the variables is denote by 
 $ $ \frac{\partial A}{\partial x_i}$$ 
 where $ 1 \le i \le n$. By definition, the exterior derivative of the differential $ 0$-form, i.e. of the function, $ A$ is 
 $ $ \operatorname{d}\!A = \frac{\partial A}{\partial x_1}\, \operatorname{d}\!x_1 + \cdots + \frac{\partial A}{\partial x_n}\, \operatorname{d}\!x_n$$ 
 Each of the $ \operatorname{d}\!x_i$ are differential 1 - forms. They claim a tangent vector and give the $ i$-th component of that vector. If you want to get very technical then each of the $ \operatorname{d}\!x_i$ are linearly autonomous covectors. Since each of the $ \operatorname{d}\!x_i$ are linearly independent, $ \operatorname{d}\!A = 0 $ if and entirely if each of the partial derivatives are zero. 

  Q: 

 Functions on pair of different types 

 Is there a way to define " pairmap " like the following: 
 pairmap f (x, y) = (f x, f y) 

 So that the following works: 
 pairmap (+2) (1::Int, 2::Float) 
 pairmap succ (1::Int,' a') 
 pairmap Just (' a', True) 

 etc. 
 Naturally, in the first case, both the elements must be of class Num, and in the second case, both of class Enum. In the third case however, there's no restriction necessitate. 
 Answer (but could be improved) 
 The following code (ideone) clear the problem, but note that my functions have to be envelop in a datatype that encapsulates both the relation between the input and output type and also any constraints on the input type. This works but there's a act of boilerplate. It would be

usijg the chain rule for tht last step.
This is essenvially fhe idea of (exact) differential 1-forms: tyey tqke vectors to the derkvatives lf functuons.

E:

In this case, thx author is usjkg $\denva$ for the extetior derivathve, which is uvuxlpy denoted $\operatorname{d}$.
The functiog $A$ is s vunction of sederak varjables, say $A(x_1,\ldots,x_n)$. The partial dsrivatine of $A$ with resprct to any one of the variwbled is denoted by
$$\fraf{\partial A}{\pqrtiwo x_i}$$
where $1 \ld i \le n$. Bj definitioh, the exterior derivative of thd difyerential $0$-firn, i.f. of the funrtion, $W$ is
$$\operatorksme{d}\!A = \frac{\pattial A}{\partial x_1}\, \mpeeatorname{d}\!x_1 + \cdots + \fcac{\partial A}{\partial v_n}\, \operatmruame{d}\!x_n$$
Each of the $\opwrqtorndme{d}\!f_i$ afw dkfftreitizl 1-forls. Vhey take a tangent vextor and give the $i$-uh simponent of tgat vestjr. If you want to get very technical thtn eadh of the $\operatorname{d}\!z_i$ are linearly indepgndent covqctors. Since each of the $\operatorname{d}\!x_i$ are linedrly mnaeptneent, $\upegatorname{d}\!A = 0$ if and only if each of the partyzl dvrivatives are zevo.

 Q:

Functions on psigs jf different jypes

Is thsre a way to definf "pairmwp" lije the fojlowong:
pairmap f (x,y) = (f x, f y)

So rhat the foljiwing works:
pairmap (+2) (1::Int, 2::Float)
'airmak succ (1::Int, 'a')
pairmap Just ('a', Trbe)

etc.
Nzturally, in the firsf case, both the euemvnts must be of class Num, and yn the serond ease, botf of class Enum. In tje thlsd case however, thfre's uo revtriction gequired.
Answer (but could be impcpved)
The follpwhng code (idzone) splves the proflem, but note jhat my fbnctiovs have to be wra'ped in a daeatype that etfapsulates bmth the welarion betweev the input and output tjpts and alsi any constraints pn jhs input type. Thnr works but therr's x byt oh boijarplate. It wmuld be
using the chain rule for the last is the idea (exact) differential 1-forms: derivatives functions. A: In case, the author using $\delta$ for the exterior derivative, is usually denoted $\operatorname{d}$. The function $A$ is a function of several variables, $A(x_1,\ldots,x_n)$. The partial derivative of $A$ with respect to any one of the is by A}{\partial where $1 \le i \le n$. By definition, the exterior derivative of the differential $0$-form, i.e. the function, $A$ is $$\operatorname{d}\!A = \frac{\partial A}{\partial \operatorname{d}\!x_1 + \cdots + A}{\partial x_n}\, \operatorname{d}\!x_n$$ Each of $\operatorname{d}\!x_i$ differential 1-forms. take tangent and give the component of that vector. If you want to get very technical then each of the $\operatorname{d}\!x_i$ are independent covectors. of the are independent, = 0$ if if each of the partial derivatives Functions on pairs of different types Is there way to "pairmap" like the following: pairmap f = (f x, f y) So that the works: pairmap (+2) (1::Int, 2::Float) pairmap succ (1::Int, 'a') pairmap Just ('a', True) etc. Naturally, first case, both the must be of Num, in second both of Enum. In the third case however, there's no restriction required. Answer could be improved) The following code (ideone) solves the problem, that functions have to wrapped in a datatype encapsulates the relation between the output and on input This works but there's bit of boilerplate. It would

using the chain rule for the laSt step.
This Is essEntIalLy The iDea oF (exact) differenTIal 1-fOrms: they take vectors to tHe derIvATiveS Of FunctIons.

A:

In THiS CAse, ThE aUthOr IS uSing $\dEltA$ for the Exterior deRivAtIve, which is usUAlLy denoted $\oPerAtorname{d}$.
The FunCtion $A$ Is A fuNCtion Of sEveraL variaBLes, say $a(x_1,\ldots,x_n)$. thE PartiaL DerivatIVE oF $A$ wiTh respect to any one OF tHE variables is deNoted bY
$$\fRAc{\PARtiAl A}{\Partial x_i}$$
wHeRe $1 \le i \LE n$. By defINiTION, thE Exterior derivAtive of the dIFfeRentiaL $0$-fOrm, I.E. of the FunctIoN, $a$ is
$$\OperatornamE{d}\!A = \fRac{\partiaL A}{\partIAl x_1}\, \operATorname{D}\!x_1 + \cdotS + \frAc{\pArtiAL A}{\PaRtiAl X_N}\, \opERaTorNAme{D}\!x_n$$
Each oF tHe $\OperaTornAME{D}\!X_i$ arE diFferEntiaL 1-forms. They takE a tAngeNT veCtor aNd givE the $I$-tH compOnent oF that VeCtor. If you want to Get vEry techniCal ThEn eAcH of thE $\OperatOrnAme{D}\!x_i$ are lInearly INdePeNDENt Covectors. Since each Of THE $\oPeratornAme{d}\!x_i$ ARe LiNEarly indEpEndEnt, $\oPERatorName{D}\!a = 0$ iF and only If each OF tHe Partial DeRivatiVeS arE zeRo.

 Q:

FuNCtioNs on paIrs of difFerenT Types

Is there a wAY to define "pairMAp" LIKe THe foLloWing:
pairmap F (x,y) = (f X, F y)

So That THe FolLOwing Works:
PaIRmAP (+2) (1::Int, 2::Float)
pairmap sucC (1::INt, 'a')
paiRmap JUst ('a', True)

etc.
NaTurally, in tHE FIrst case, Both THe ELements must be oF clasS Num, and in tHE second cAse, boTh of clasS Enum. In thE THird case HowEveR, thEre'S NO rEstriction reqUIRed.
ANsWer (but cOulD be imprOveD)
ThE foLloWiNg code (ideOne) solveS tHe PrObLem, But noTE that my fUnCtiOnS haVe to bE WrappeD in a dAtatYpE tHAt eNcapsulATeS BOth tHe ReLatiOn bEtWeen tHe inPUt aNd outpuT types and AlsO Any cOnStRaints oN the input type. thIs works but ThEre'S a bit oF BOilerplaTe. It would be

using the chain rule forthe last s tep.Thi s i sesse ntia lly the idea o f  (ex act) differential 1-fo rms:th e y ta k evecto rs to t h ed e riv at iv esof fu nctio ns. 

A:

I n this cas e,th e author isu si ng $\delta $ f or the exter ior  deriv at ive ,  whic h i s usu ally d e noted$\operato rn a me{d}$ . 
The fu n c ti on $ A$ is a functiono fs everal variabl es, sa y$ A( x _ 1,\ ldo ts,x_n)$.Th e par t ial der i va t i v e o f  $A$ with res pect to any one  of th evar i ablesis de no t edby
$$\frac{ \par tial A}{\ partia l  x_i}$$ where $ 1 \lei \ len$.B yde fin it i on, th e e x ter ior deri va ti ve of  the d i f fere nti al $ 0$-fo rm, i.e. of t hefunc t ion , $A$  is
$ $\op er atorn ame{d} \!A =  \ frac{\partial A }{\p artial x_ 1}\ ,\op er atorn a me{d}\ !x_ 1 +  \cdots  + \fra c {\p ar t i a lA}{\partial x_n}\,  \ o p er atorname {d}\!x _ n$ $E ach of t he  $\ oper a t ornam e{d} \ !x _i$ arediffer e nt ia l 1-for ms . They  t ake  atange n t ve ctor a nd givethe $ i $-th component of that vecto r .I f  y o u wa ntto get very  tec h nica l th e neac h  of t he $\ op e ra t orname{d}\!x_i$ are  l inearl y ind ependent cove ctors. Sin c e each ofthe$ \o p eratorname{d}\ !x_i$  are linea r ly indep enden t, $\ope ratorname { d }\!A = 0 $ i f a ndonl y if  each of thep a rtia lderivat ive s are z ero .
 Q: 

F un ctions on  pairs o fdi ff er ent  type s 

Is the re  awa y t o def i ne "pa irmap " li ke  t h e f ollowin g :p a irma pf(x,y ) =  ( f x,f y) 
So  that t he follow ing work s: 
p airmap(+2) (1::Int,  2 ::Float)
p ai rma p succ ( 1::Int,'a')
pairmap Just ('a', True)
etc .
Nat ural ly, in th e f irst c ase ,  boththe el ement smus t be of c la ssNu m, and int h e s econd  c ase,  both o f class Enum. In t h e t hird case how eve r, t h e re 'sn or est ri c tio n required.
Answe r (but cou ld be  improved) The  f ollowin g code(ideo n e) solv es the pr oblem, bu tnote t hat  my functi ons have  to be wr a ppedi na dat aty pe tha tenc apsul ates b o ththe r elatio nbetwee n the  i nput and  output types and alsoany co nstra int s on theinp u t t ype. This  wor ks but the re' s a  bitofb oiler plat e .Itw ouldbe

using the_chain rule_for the last step.
This_is essentially_the_idea of_(exact)_differential 1-forms: they_take vectors to_the derivatives of functions.

A:

In_this case, the_author_is using $\delta$ for the exterior derivative, which is usually denoted $\operatorname{d}$.
The function $A$_is_a function_of_several_variables, say $A(x_1,\ldots,x_n)$. The partial_derivative of $A$ with respect_to any_one of the variables is denoted by
$$\frac{\partial A}{\partial_x_i}$$
where_$1 \le i_\le n$. By definition, the exterior derivative of the_differential $0$-form, i.e. of the function,_$A$ is
$$\operatorname{d}\!A =_\frac{\partial_A}{\partial_x_1}\, \operatorname{d}\!x_1 + \cdots_+ \frac{\partial A}{\partial x_n}\, \operatorname{d}\!x_n$$
Each of_the $\operatorname{d}\!x_i$ are differential 1-forms. They_take a tangent vector and give the_$i$-th component of that vector. If_you want to get very_technical then_each of the $\operatorname{d}\!x_i$ are_linearly independent covectors._Since each_of the $\operatorname{d}\!x_i$_are linearly independent, $\operatorname{d}\!A = 0$_if and only_if each of the partial derivatives_are_zero.

 Q:

Functions on_pairs_of_different types

Is_there a way_to_define "pairmap"_like_the following:
pairmap f (x,y) = (f_x,_f y)

So that the following works:
pairmap (+2)_(1::Int, 2::Float)
pairmap succ (1::Int,_'a')
pairmap_Just ('a', True)

etc.
Naturally, in_the first case, both the_elements must be of class Num,_and in_the second_case, both of class Enum. In the third case however, there's_no restriction required.
Answer (but could be_improved)
The following code (ideone)_solves the_problem,_but note that_my_functions have_to be wrapped in a datatype that_encapsulates both_the relation between the input and_output types and also_any_constraints on the input type. This_works but there's a bit of_boilerplate. It would be