:: Partial Differentiation of Real Ternary Functions
:: by Takao Inou\'e , Bing Xie and Xiquan Liang
::
:: Copyright (c) 2009-2021 Association of Mizar Users

theorem Th1: :: PDIFF_4:1
( dom (proj (1,3)) = REAL 3 & rng (proj (1,3)) = REAL & ( for x, y, z being Real holds (proj (1,3)) . <*x,y,z*> = x ) )
proof end;

theorem Th2: :: PDIFF_4:2
( dom (proj (2,3)) = REAL 3 & rng (proj (2,3)) = REAL & ( for x, y, z being Real holds (proj (2,3)) . <*x,y,z*> = y ) )
proof end;

theorem Th3: :: PDIFF_4:3
( dom (proj (3,3)) = REAL 3 & rng (proj (3,3)) = REAL & ( for x, y, z being Real holds (proj (3,3)) . <*x,y,z*> = z ) )
proof end;

theorem Th4: :: PDIFF_4:4
for x, y, z being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x,y,z*> & f is_partial_differentiable_in u,1 holds
SVF1 (1,f,u) is_differentiable_in x by Th1;

theorem Th5: :: PDIFF_4:5
for x, y, z being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x,y,z*> & f is_partial_differentiable_in u,2 holds
SVF1 (2,f,u) is_differentiable_in y by Th2;

theorem Th6: :: PDIFF_4:6
for x, y, z being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x,y,z*> & f is_partial_differentiable_in u,3 holds
SVF1 (3,f,u) is_differentiable_in z by Th3;

theorem Th7: :: PDIFF_4:7
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (1,f,u) is_differentiable_in x0 ) iff f is_partial_differentiable_in u,1 )
proof end;

theorem Th8: :: PDIFF_4:8
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (2,f,u) is_differentiable_in y0 ) iff f is_partial_differentiable_in u,2 )
proof end;

theorem Th9: :: PDIFF_4:9
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 ) iff f is_partial_differentiable_in u,3 )
proof end;

theorem :: PDIFF_4:10
for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,1 holds
ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) )
proof end;

theorem :: PDIFF_4:11
for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 holds
ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) )
proof end;

theorem :: PDIFF_4:12
for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 holds
ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) )
proof end;

theorem Th13: :: PDIFF_4:13
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_partial_differentiable_in u,1 iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) )
proof end;

theorem Th14: :: PDIFF_4:14
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_partial_differentiable_in u,2 iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) )
proof end;

theorem Th15: :: PDIFF_4:15
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_partial_differentiable_in u,3 iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) )
proof end;

Lm1: for x0, y0, z0, r being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,1 holds
( r = diff ((SVF1 (1,f,u)),x0) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )

proof end;

Lm2: for x0, y0, z0, r being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 holds
( r = diff ((SVF1 (2,f,u)),y0) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )

proof end;

Lm3: for x0, y0, z0, r being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 holds
( r = diff ((SVF1 (3,f,u)),z0) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) ) )

proof end;

theorem Th16: :: PDIFF_4:16
for x0, y0, z0, r being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,1 holds
( r = partdiff (f,u,1) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )
proof end;

theorem Th17: :: PDIFF_4:17
for x0, y0, z0, r being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 holds
( r = partdiff (f,u,2) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )
proof end;

theorem Th18: :: PDIFF_4:18
for x0, y0, z0, r being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 holds
( r = partdiff (f,u,3) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) ) )
proof end;

theorem :: PDIFF_4:19
for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> holds
partdiff (f,u,1) = diff ((SVF1 (1,f,u)),x0) by Th1;

theorem :: PDIFF_4:20
for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> holds
partdiff (f,u,2) = diff ((SVF1 (2,f,u)),y0) by Th2;

theorem :: PDIFF_4:21
for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> holds
partdiff (f,u,3) = diff ((SVF1 (3,f,u)),z0) by Th3;

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
pred f is_partial_differentiable1_on D means :: PDIFF_4:def 1
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_partial_differentiable_in u,1 ) );
pred f is_partial_differentiable2_on D means :: PDIFF_4:def 2
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_partial_differentiable_in u,2 ) );
pred f is_partial_differentiable3_on D means :: PDIFF_4:def 3
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_partial_differentiable_in u,3 ) );
end;

:: deftheorem defines is_partial_differentiable1_on PDIFF_4:def 1 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_partial_differentiable1_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_partial_differentiable_in u,1 ) ) );

:: deftheorem defines is_partial_differentiable2_on PDIFF_4:def 2 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_partial_differentiable2_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_partial_differentiable_in u,2 ) ) );

:: deftheorem defines is_partial_differentiable3_on PDIFF_4:def 3 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_partial_differentiable3_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_partial_differentiable_in u,3 ) ) );

theorem :: PDIFF_4:22
for D being set
for f being PartFunc of (REAL 3),REAL st f is_partial_differentiable1_on D holds
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f is_partial_differentiable_in u,1 ) )
proof end;

theorem :: PDIFF_4:23
for D being set
for f being PartFunc of (REAL 3),REAL st f is_partial_differentiable2_on D holds
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f is_partial_differentiable_in u,2 ) )
proof end;

theorem :: PDIFF_4:24
for D being set
for f being PartFunc of (REAL 3),REAL st f is_partial_differentiable3_on D holds
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f is_partial_differentiable_in u,3 ) )
proof end;

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_partial_differentiable1_on D ;
func f partial1| D -> PartFunc of (REAL 3),REAL means :: PDIFF_4:def 4
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = partdiff (f,u,1) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = partdiff (f,u,1) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = partdiff (f,u,1) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = partdiff (f,u,1) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines partial1| PDIFF_4:def 4 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_partial_differentiable1_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f partial1| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = partdiff (f,u,1) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_partial_differentiable2_on D ;
func f partial2| D -> PartFunc of (REAL 3),REAL means :: PDIFF_4:def 5
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = partdiff (f,u,2) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = partdiff (f,u,2) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = partdiff (f,u,2) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = partdiff (f,u,2) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines partial2| PDIFF_4:def 5 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_partial_differentiable2_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f partial2| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = partdiff (f,u,2) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_partial_differentiable3_on D ;
func f partial3| D -> PartFunc of (REAL 3),REAL means :: PDIFF_4:def 6
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = partdiff (f,u,3) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = partdiff (f,u,3) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = partdiff (f,u,3) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = partdiff (f,u,3) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines partial3| PDIFF_4:def 6 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_partial_differentiable3_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f `partial3| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = partdiff (f,u,3) ) ) );

theorem :: PDIFF_4:25
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (1,3)) . u0 st f is_partial_differentiable_in u0,1 & N c= dom (SVF1 (1,f,u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c)) is convergent & partdiff (f,u0,1) = lim ((h ") (#) (((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c))) )
proof end;

theorem :: PDIFF_4:26
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (2,3)) . u0 st f is_partial_differentiable_in u0,2 & N c= dom (SVF1 (2,f,u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (2,f,u0)) /* (h + c)) - ((SVF1 (2,f,u0)) /* c)) is convergent & partdiff (f,u0,2) = lim ((h ") (#) (((SVF1 (2,f,u0)) /* (h + c)) - ((SVF1 (2,f,u0)) /* c))) )
proof end;

theorem :: PDIFF_4:27
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (3,3)) . u0 st f is_partial_differentiable_in u0,3 & N c= dom (SVF1 (3,f,u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (3,f,u0)) /* (h + c)) - ((SVF1 (3,f,u0)) /* c)) is convergent & partdiff (f,u0,3) = lim ((h ") (#) (((SVF1 (3,f,u0)) /* (h + c)) - ((SVF1 (3,f,u0)) /* c))) )
proof end;

theorem :: PDIFF_4:28
for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_partial_differentiable_in u0,1 & f2 is_partial_differentiable_in u0,1 holds
f1 (#) f2 is_partial_differentiable_in u0,1
proof end;

theorem :: PDIFF_4:29
for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_partial_differentiable_in u0,2 & f2 is_partial_differentiable_in u0,2 holds
f1 (#) f2 is_partial_differentiable_in u0,2
proof end;

theorem :: PDIFF_4:30
for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_partial_differentiable_in u0,3 & f2 is_partial_differentiable_in u0,3 holds
f1 (#) f2 is_partial_differentiable_in u0,3
proof end;

theorem :: PDIFF_4:31
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_partial_differentiable_in u0,1 holds
SVF1 (1,f,u0) is_continuous_in (proj (1,3)) . u0 by FDIFF_1:24;

theorem :: PDIFF_4:32
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_partial_differentiable_in u0,2 holds
SVF1 (2,f,u0) is_continuous_in (proj (2,3)) . u0 by FDIFF_1:24;

theorem :: PDIFF_4:33
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_partial_differentiable_in u0,3 holds
SVF1 (3,f,u0) is_continuous_in (proj (3,3)) . u0 by FDIFF_1:24;

Lm4: for x1, x2, y1, y2, z1, z2 being Real holds |[x1,y1,z1]| + |[x2,y2,z2]| = |[(x1 + x2),(y1 + y2),(z1 + z2)]|
proof end;

definition
let f be PartFunc of (REAL 3),REAL;
let p be Element of REAL 3;
func grad (f,p) -> Element of REAL 3 equals :: PDIFF_4:def 7
(((partdiff (f,p,1)) * <e1>) + ((partdiff (f,p,2)) * <e2>)) + ((partdiff (f,p,3)) * <e3>);
coherence
(((partdiff (f,p,1)) * <e1>) + ((partdiff (f,p,2)) * <e2>)) + ((partdiff (f,p,3)) * <e3>) is Element of REAL 3
;
end;

:: deftheorem defines grad PDIFF_4:def 7 :
for f being PartFunc of (REAL 3),REAL
for p being Element of REAL 3 holds grad (f,p) = (((partdiff (f,p,1)) * <e1>) + ((partdiff (f,p,2)) * <e2>)) + ((partdiff (f,p,3)) * <e3>);

reconsider jj = 1 as Real ;

theorem Th34: :: PDIFF_4:34
for p being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds grad (f,p) = |[(partdiff (f,p,1)),(partdiff (f,p,2)),(partdiff (f,p,3))]|
proof end;

theorem Th35: :: PDIFF_4:35
for p being Element of REAL 3
for f, g being PartFunc of (REAL 3),REAL st f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 & f is_partial_differentiable_in p,3 & g is_partial_differentiable_in p,1 & g is_partial_differentiable_in p,2 & g is_partial_differentiable_in p,3 holds
proof end;

theorem Th36: :: PDIFF_4:36
for p being Element of REAL 3
for f, g being PartFunc of (REAL 3),REAL st f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 & f is_partial_differentiable_in p,3 & g is_partial_differentiable_in p,1 & g is_partial_differentiable_in p,2 & g is_partial_differentiable_in p,3 holds
proof end;

theorem Th37: :: PDIFF_4:37
for r being Real
for p being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 & f is_partial_differentiable_in p,3 holds
proof end;

theorem :: PDIFF_4:38
for s, t being Real
for p being Element of REAL 3
for f, g being PartFunc of (REAL 3),REAL st f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 & f is_partial_differentiable_in p,3 & g is_partial_differentiable_in p,1 & g is_partial_differentiable_in p,2 & g is_partial_differentiable_in p,3 holds
grad (((s (#) f) + (t (#) g)),p) = (s * (grad (f,p))) + (t * (grad (g,p)))
proof end;

theorem :: PDIFF_4:39
for s, t being Real
for p being Element of REAL 3
for f, g being PartFunc of (REAL 3),REAL st f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 & f is_partial_differentiable_in p,3 & g is_partial_differentiable_in p,1 & g is_partial_differentiable_in p,2 & g is_partial_differentiable_in p,3 holds
grad (((s (#) f) - (t (#) g)),p) = (s * (grad (f,p))) - (t * (grad (g,p)))
proof end;

theorem :: PDIFF_4:40
for p being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is total & f is constant holds
proof end;

definition
let a be Element of REAL 3;
func unitvector a -> Element of REAL 3 equals :: PDIFF_4:def 8
|[((a . 1) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2)))),((a . 2) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2)))),((a . 3) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2))))]|;
coherence
|[((a . 1) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2)))),((a . 2) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2)))),((a . 3) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2))))]| is Element of REAL 3
;
end;

:: deftheorem defines unitvector PDIFF_4:def 8 :
for a being Element of REAL 3 holds unitvector a = |[((a . 1) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2)))),((a . 2) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2)))),((a . 3) / (sqrt ((((a . 1) ^2) + ((a . 2) ^2)) + ((a . 3) ^2))))]|;

definition
let f be PartFunc of (REAL 3),REAL;
let p, a be Element of REAL 3;
func Directiondiff (f,p,a) -> Real equals :: PDIFF_4:def 9
(((partdiff (f,p,1)) * (() . 1)) + ((partdiff (f,p,2)) * (() . 2))) + ((partdiff (f,p,3)) * (() . 3));
coherence
(((partdiff (f,p,1)) * (() . 1)) + ((partdiff (f,p,2)) * (() . 2))) + ((partdiff (f,p,3)) * (() . 3)) is Real
;
end;

:: deftheorem defines Directiondiff PDIFF_4:def 9 :
for f being PartFunc of (REAL 3),REAL
for p, a being Element of REAL 3 holds Directiondiff (f,p,a) = (((partdiff (f,p,1)) * (() . 1)) + ((partdiff (f,p,2)) * (() . 2))) + ((partdiff (f,p,3)) * (() . 3));

theorem :: PDIFF_4:41
for x0, y0, z0 being Real
for p, a being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st a = <*x0,y0,z0*> holds
Directiondiff (f,p,a) = ((((partdiff (f,p,1)) * x0) / (sqrt (((x0 ^2) + (y0 ^2)) + (z0 ^2)))) + (((partdiff (f,p,2)) * y0) / (sqrt (((x0 ^2) + (y0 ^2)) + (z0 ^2))))) + (((partdiff (f,p,3)) * z0) / (sqrt (((x0 ^2) + (y0 ^2)) + (z0 ^2))))
proof end;

theorem :: PDIFF_4:42
for p, a being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds Directiondiff (f,p,a) = |((grad (f,p)),())|
proof end;

definition
let f1, f2, f3 be PartFunc of (REAL 3),REAL;
let p be Element of REAL 3;
func curl (f1,f2,f3,p) -> Element of REAL 3 equals :: PDIFF_4:def 10
((((partdiff (f3,p,2)) - (partdiff (f2,p,3))) * <e1>) + (((partdiff (f1,p,3)) - (partdiff (f3,p,1))) * <e2>)) + (((partdiff (f2,p,1)) - (partdiff (f1,p,2))) * <e3>);
coherence
((((partdiff (f3,p,2)) - (partdiff (f2,p,3))) * <e1>) + (((partdiff (f1,p,3)) - (partdiff (f3,p,1))) * <e2>)) + (((partdiff (f2,p,1)) - (partdiff (f1,p,2))) * <e3>) is Element of REAL 3
;
end;

:: deftheorem defines curl PDIFF_4:def 10 :
for f1, f2, f3 being PartFunc of (REAL 3),REAL
for p being Element of REAL 3 holds curl (f1,f2,f3,p) = ((((partdiff (f3,p,2)) - (partdiff (f2,p,3))) * <e1>) + (((partdiff (f1,p,3)) - (partdiff (f3,p,1))) * <e2>)) + (((partdiff (f2,p,1)) - (partdiff (f1,p,2))) * <e3>);

theorem :: PDIFF_4:43
for p being Element of REAL 3
for f1, f2, f3 being PartFunc of (REAL 3),REAL holds curl (f1,f2,f3,p) = |[((partdiff (f3,p,2)) - (partdiff (f2,p,3))),((partdiff (f1,p,3)) - (partdiff (f3,p,1))),((partdiff (f2,p,1)) - (partdiff (f1,p,2)))]|
proof end;