Complete Proof

Complete Proof
What's the proof for the chain rule in the partial form df= fxdx+fydy+fzdz?

Let's say f is a scalar function of three variables (x,y,z) and fx, ty and fz are the partial derivatives. I would like to see the proof for df= fxdx+fydy+fzdz

the formula

df = fxdx + fydy + fzdz

is the definition of the total differential for f. there's nothing to prove, it's a definition.

this definition is motivated by the chain rule for higher dimensions:

df/dt = (fx)(dx/dt) + (fy)(dy/dt) + (fz)(dz/dt)

actually, this represents somewhat of an abuse of notation, because what is usally meant by this is f is a real-valued function of 3 variables, each of which is parametrized by t. so we actually have a composition:

(fop)(t), where p(t) = (x(t),y(t),z(t)).

so it really shouldn't be df/dt, but rather, d/dt(fop).

now p'(t) = (x'(t),y'(t),z'(t)) = (dx/dt,dy/dt,dz/dt). while the derivative of f is

∇f = (fx,fy,fz), and in this context, the chain rule is:

(fop)'(t) = ∇f(p(t))•p'(t)), where we have a dot product (actually, this is a special example of a matrix product).

the proof of this isn't terribly hard, but requires a lot of definitions to be made regarding derivatives of functions of more than one variable, and space prevents me from giving a complete proof here.

a "semi-intuitive proof" is the following: f:R^3--->R can be approximated (at a given point) by a linear function L, where L(x,y,z) = ax + by + cz + d. and the parametrization p can also be approximated by some line: p(t) = (x0 + tu, y0 + tv, z0 + tw).

now (Lop)(t) = a(x0 + tu) + b(y0 + tv) + c(z0 + tw) + d

= (au + bv + cw)t + ax0 + by0 + cz0 + d

computing (Lop)'(t) directly, we have:

(Lop)'(t) = au + bv + cw

while ∇L is the constant vector (a,b,c), and p'(t) = (u,v,w), so

(Lop)'(t) = (a,b,c)•(u,v,w).

now the derivative of a function g:R^n-->R^m can be defined as the unique linear transformation λ such that:

lim h-->0 |g(x+h) - g(x) - λ(h)|/|h| = 0

in other words λ(x) + g(x) is the "best linear approximation to g" at x, so if λ is the derivative of g, and μ is the derivative of h, then it makes sense for the best linear approximation of goh to be the matrix product, λμ, and when λ = g, μ = h, this is certainly the case (we just showed that for L and p above).

it is sometimes easier to see this is true with an example:

let f(x,y,z) = 2x + xy^2 + 3z

let p(t) = (x,y,z) = (t, t^2, t^3)

∇f(x,y,z) = (2 + y^2, 2xy,3) and p'(t) = (1, 2t,3t^2)

so ∇f(p(t)) = (2 + t^4, 2t^3, 3) so ∇f(p(t))•(1,2t,3t^2)

= 2 + t^4 + (2t^3)(2t) + 3(3t^2) = 2 + 9t^2 + 5t^4

on the other hand, f(p(t)) = 2t + t^5 + 3t^3, which has the derivative:

(fop)"(t) = 2 + 9t^2 + 5t^4.

Scroll Down To Find Complete Proof On eBay Below:

2007 S Complete Proof Set P07 14 Coins w COA

2007 S Complete Proof Set P07 14 Coins w COA

$29.00

1985 United States 5 Coin Complete Proof Set OGP

1985 United States 5 Coin Complete Proof Set OGP

$5.59

2006 United States 10 Coin Complete Proof Set with Box and COA

2006 United States 10 Coin Complete Proof Set with Box and COA

$7.99

1971 s USProof set Genuine complete and original as issued by US Mint

1971 s USProof set Genuine complete and original as issued by US Mint

$4.25

2010 US Mint PROOF SET Complete 14 coin set with box  coa

2010 US Mint PROOF Set Complete 14 Coin Set with box coa

$26.98

1959 US Proof Set Complete As Issued By The US Mint Sealed Unopened Envelope

1959 US Proof Set Complete As Issued By The US Mint Sealed Unopened Envelope

$32.50

1987 United States 5 Coin Complete Proof Set

1987 United States 5 Coin Complete Proof Set

$5.89

Complete Lot of 60 1968 to 1998 US Coin PROOF SETS AND MINT SETS with packaging

Complete Lot of 60 1968 to 1998 US Coin Proof SETS AND MINT SETS with packaging

$375.00

1990 S US Complete Proof set Nice Set  Free shipping

1990 S US Complete Proof set Nice Set Free Shipping

$7.95

1982 s USProof set Genuine complete and original as issued by US Mint

1982 s USProof set Genuine complete and original as issued by US Mint

$3.25

1999 s US Proof Set Complete and Original in BLUE mint paper box

1999 s US Proof Set Complete and Original in BLUE mint paper box

$6.49

1969 s USProof set Genuine complete and original as issued by US Mint

1969 s USProof set Genuine complete and original as issued by US Mint

$5.95

1968 s USProof set Genuine complete and original as issued by US Mint

1968 s USProof set Genuine complete and original as issued by US Mint

$5.99

1996 OLYMPIC PRESTIGE Proof Set US Mint Complete  Original With Box RARE

1996 OLYMPIC Prestige Proof Set US Mint Complete Original With Box RARE

$224.98

2009 Proof Set Clad complete

2009 Proof Set Clad complete

$9.99

2015 SILVER PROOF SET COMPLETE 14 COINS BOX  CERTIFICATE FROM US MINT SHOWN

2015 Silver Proof SET COMPLETE 14 COINS BOX CERTIFICATE FROM US MINT SHOWN

$65.99

1993 s USProof set Genuine complete and original as issued by US Mint

1993 s USProof set Genuine complete and original as issued by US Mint

$4.05

COMPLETE SET SILVER 90 PROOF SETS 109 TOTAL COINS 1999 2008 W COAs

Complete Set SILVER 90 Proof Sets 109 TOTAL COINS 1999 2008 W COAs

$257.21

2002 s US Proof Set Complete and Original in BLUE mint paper box

2002 s US Proof Set Complete and Original in BLUE mint paper box

$6.25

1998 s US Proof Set Complete and Original

1998 s US Proof Set Complete and Original

$7.75

1970 s USProof set Genuine complete and original as issued by US Mint

1970 s USProof set Genuine complete and original as issued by US Mint

$10.95

2005 United States Complete 11 Coin Proof Set

2005 United States Complete 11 Coin Proof Set

$8.39

2005 s US Proof Set Complete and Original in BLUE mint paper box

2005 s US Proof Set Complete and Original in BLUE mint paper box

$5.99

1985 s USProof set Genuine complete and original as issued by US Mint

1985 s USProof set Genuine complete and original as issued by US Mint

$3.25

2000 s US Proof Set Complete and Original in BLUE mint paper box

2000 s US Proof Set Complete and Original in BLUE mint paper box

$5.25

2012 S 14 COIN PROOF SET COMPLETE ORIGINAL

2012 S 14 COIN PROOF SET COMPLETE ORIGINAL

$65.00

COMPLETE SET SILVER 90 PROOF SETS 109 TOTAL COINS 1999 2008 W COAs

COMPLETE Set Silver 90 PROOF SETS 109 TOTAL COINS 1999 2008 W COAs

$306.02

2004 s US Proof Set Complete and Original in BLUE mint paper box

2004 s US Proof Set Complete and Original in BLUE mint paper box

$9.99

2003 s US Proof Set Complete and Original in BLUE mint paper box

2003 s US Proof Set Complete and Original in BLUE mint paper box

$5.99

2010 S 14 COIN PROOF SET COMPLETE ORIGINAL

2010 S 14 COIN PROOF SET COMPLETE ORIGINAL

$8.00

1968 1998 COMPLETE RUN OF 31 US PROOF SETS 7077

1968 1998 COMPLETE RUN OF 31 US PROOF SETS 7077

$179.00

2015 CLAD PROOF SET COMPLETE 14 COINS BOX  CERTIFICATE FROM US MINT SHOWN

2015 Clad Proof SET COMPLETE 14 COINS BOX CERTIFICATE FROM US MINT SHOWN

$37.99

1983 s USProof set Genuine complete and original as issued by US Mint

1983 s USProof set Genuine complete and original as issued by US Mint

$3.75

2003 S US MINT SILVER PROOF SET ORIGINAL BOX  COA COMPLETE 10 COIN SET

2003 S US MINT SILVER PROOF Set Original BOX Coa Complete 10 COIN SET

$28.00

1979 S US Complete Proof set Susan B Anthony Nice Set Free Shipping

1979 S US Complete Proof set Susan B Anthony Nice Set Free Shipping

$10.50

TEN 10 Complete 2000 Silver US Proof Sets 10 Coin Sets 100 Coins Total

TEN 10 Complete 2000 Silver US Proof Sets 10 Coin Sets 100 Coins Total

$50.00

COMPLETE SACAGAWEA PROOF SET 2000 THRU 2016 ALL IN CAPSILS

COMPLETE SACAGAWEA PROOF SET 2000 THRU 2016 ALL IN CAPSILS

$60.00

2008 United States Mint ANNUAL 14 Coin Proof Set Original Box and COA Complete

2008 United States Mint ANNUAL 14 Coin Proof Set Original Box and COA Complete

$32.95

2011 US Mint Clad Proof Set Complete 14 Coin Set with Box  COA Ships FREE

2011 US Mint Clad Proof Set Complete 14 Coin Set with Box COA Ships FREE

$26.50

1987 s USProof set Genuine complete and original as issued by US Mint

1987 s USProof set Genuine complete and original as issued by US Mint

$3.09

COMPLETE US PRESIDENTIAL DOLLAR MINT PROOF SET 2007 to 2016 IN BOXES

COMPLETE US PRESIDENTIAL Dollar Mint PROOF SET 2007 to 2016 IN BOXES

$180.00

2002 United States 10 Coin Complete Proof Set with Box and COA

2002 United States 10 Coin Complete Proof Set with Box and COA

$8.99

2012 US MINT SILVER PROOF SET Complete Set W Box  COA A2617

2012 US MINT SILVER PROOF SET Complete Set W Box COA A2617

$152.50

2004 US MINT SILVER PROOF SET Complete Set W Box  COA A2615

2004 US MINT SILVER PROOF SET Complete Set W Box COA A2615

$20.50

1953 Complete Proof Set Franklin Half PCGS PR66 Roosevelt Dime Looks Cameo

1953 Complete Proof Set Franklin Half PCGS PR66 Roosevelt Dime Looks Cameo

$170.00

2000 Complete 10 Coin United States Silver Proof Set w Box and COA

2000 Complete 10 Coin United States Silver Proof Set w Box and COA

$30.99

2009 S 18 COIN CLAD PROOF SET COMPLETE ORIGINAL

2009 S 18 COIN CLAD PROOF SET COMPLETE ORIGINAL

$0.99

1973 s USProof set Genuine complete and original as issued by US Mint

1973 s USProof set Genuine complete and original as issued by US Mint

$6.99

1999 United States 9 Coin Complete Proof Set with Box COA

1999 United States 9 Coin Complete Proof Set with Box COA

$8.99

2006 US MINT SILVER PROOF SET Complete Set W Box  COA A2616

2006 US MINT SILVER PROOF SET Complete Set W Box COA A2616

$7.00

COMPLETE SET SILVER 90 PROOF SETS 109 TOTAL COINS 1999 2008 W COAs

COMPLETE SET SILVER 90 PROOF SETS 109 TOTAL COINS 1999 2008 W COAs

$302.00

COMPLETE RUN US MINT PROOF SETS DATED 1961 TO 2012 INCLUDES 1965 1967 MINT SETS

COMPLETE RUN US MINT PROOF SETS DATED 1961 TO 2012 INCLUDES 1965 1967 MINT SETS

$699.95

1989 s USProof set Genuine complete and original as issued by US Mint

1989 s USProof set Genuine complete and original as issued by US Mint

$3.29

2004 Complete 11 Coin United States Silver Proof Set w Box and COA

2004 Complete 11 Coin United States Silver Proof Set w Box and COA

$30.99

2000 US MINT SILVER PROOF SET Complete Set W Box  COA A2613

2000 US MINT SILVER PROOF SET Complete Set W Box COA A2613

$15.50

1981 s USProof set Genuine complete and original as issued by US Mint

1981 s USProof set Genuine complete and original as issued by US Mint

$4.75

1972 s USProof set Genuine complete and original as issued by US Mint

1972 s USProof set Genuine complete and original as issued by US Mint

$3.35

1990 s USProof set Genuine complete and original as issued by US Mint

1990 s USProof set Genuine complete and original as issued by US Mint

$3.89

1991 PRESTIGE Proof Set US Mint Made Complete  Original With Box

1991 PRESTIGE Proof Set US Mint Made Complete Original With Box

$39.00

1963 SILVER PROOF SET COMPLETE

1963 SILVER PROOF SET COMPLETE

$21.00

1968 1998 FIRST 31 YEAR UNITED STATES PROOF SETS COMPLETE SET SAN FRANCISCO MINT

1968 1998 FIRST 31 YEAR UNITED STATES PROOF Sets Complete SET San Francisco MINT

$127.50

2002 US MINT SILVER PROOF SET Complete Set W Box  COA A2614

2002 US MINT SILVER PROOF SET Complete Set W Box COA A2614

$7.50

1992 s USProof set Genuine complete and original as issued by US Mint

1992 s USProof set Genuine complete and original as issued by US Mint

$3.25

2009 United States 18 Coin Complete Silver Proof Set w Box and COA

2009 United States 18 Coin Complete Silver Proof Set w Box and COA

$48.99

2008 s US 14 Coin Proof Set Complete and Original in BLUE mint paper box

2008 s US 14 Coin Proof Set Complete and Original in BLUE mint paper box

$29.00

COMPLETE RUN US MINT STATE QUARTER SILVER PROOF SETS DATED 2004 TO 2009

COMPLETE RUN US MINT STATE QUARTER SILVER PROOF SETS DATED 2004 TO 2009

$134.95

1973 to 1992 US Mint Proof Sets Complete Sets Lot Collection 20 Twenty Sets

1973 to 1992 US Mint Proof Sets Complete Sets Lot Collection 20 Twenty Sets

$81.00

1984 s USProof set Genuine complete and original as issued by US Mint

1984 s USProof set Genuine complete and original as issued by US Mint

$3.75

2010 s US 14 Coin Proof Set Complete and Original in BLUE mint paper box

2010 s US 14 Coin Proof Set Complete and Original in BLUE mint paper box

$28.50

1999 US MINT SILVER PROOF SET Complete Set W Box  COA A2612

1999 US MINT SILVER PROOF SET Complete Set W Box COA A2612

$41.00

1968 1998 FIRST 31 YEAR UNITED STATES PROOF SETS COMPLETE SET SAN FRANCISCO MINT

1968 1998 FIRST 31 YEAR UNITED STATES PROOF SETS COMPLETE SET SAN FRANCISCO MINT

$80.00

2008 S 14 COIN PROOF SET COMPLETE ORIGINAL

2008 S 14 COIN PROOF SET COMPLETE ORIGINAL

$16.25

1979 s USProof set Genuine complete and original as issued by US Mint

1979 s USProof set Genuine complete and original as issued by US Mint

$5.99

1976 s USProof set Genuine complete and original as issued by US Mint

1976 s USProof set Genuine complete and original as issued by US Mint

$7.95

Complete Run of Blue Box Proof Sets 1968 1972  1983 FREE US SHIP

Complete Run of Blue Box Proof Sets 1968 1972 1983 FREE US SHIP

$34.99

1991 s USProof set Genuine complete and original as issued by US Mint

1991 s USProof set Genuine complete and original as issued by US Mint

$3.75

2016 United States Mint Clad Proof Set Complete with Mint Packaging and COA

2016 United States Mint Clad Proof Set Complete with Mint Packaging and COA

$9.99

2004 US Mint Silver Proof set complete w Certificate of Authenticity

2004 US Mint Silver Proof set complete w Certificate of Authenticity

$26.00

COMPLETE COLLECTION PROOF ROOSEVELT DIMES 1968 2007 40 GEM PROOF COINS

COMPLETE COLLECTION Proof Roosevelt DIMES 1968 2007 40 GEM Proof Coins

$15.50

1968 S US Mint Complete Proof set 40 Silver Five Coins Nice Set Free Shipping

1968 S US Mint Complete Proof set 40 Silver Five Coins Nice Set Free Shipping

$10.95

2003 US Mint Silver Proof set complete w Certificate of Authenticity

2003 US Mint Silver Proof set complete w Certificate of Authenticity

$26.77

2000 S US Complete Proof set Nice Set Free shipping

2000 S US Complete Proof set Nice Set Free shipping

$9.95

1988 s USProof set Genuine complete and original as issued by US Mint

1988 s USProof set Genuine complete and original as issued by US Mint

$3.75

1995 s USProof set Genuine complete and original as issued by US Mint

1995 s USProof set Genuine complete and original as issued by US Mint

$7.45

2005 US Mint Proof Westward Journey Nickel Series Coin Set Complete Box

2005 US Mint Proof Westward Journey Nickel Series Coin Set Complete Box

$4.50

COMPLETE 1999 2008 STATE QUARTER SET All BU PROOF SILVER PROOF 2 Dansco Albums

COMPLETE 1999 2008 STATE QUARTER SET All BU Proof Silver PROOF 2 Dansco Albums

$469.45

COMPLETE COLLECTION PROOF JEFFERSON NICKELS 1968 2007 42 GEM PROOF COINS

COMPLETE COLLECTION Proof Jefferson NICKELS 1968 2007 42 Gem Proof COINS

$15.00

2006 s US Proof Set Complete and Original in BLUE mint paper box

2006 s US Proof Set Complete and Original in BLUE mint paper box

$7.75

2003 S US MINT SILVER PROOF SET ORIGINAL BOX  COA COMPLETE 10 COIN SET

2003 S US MINT SILVER PROOF SET ORIGINAL BOX COA COMPLETE 10 COIN SET

$17.50

1992 SILVER US Mint PROOF SET Complete 5 coin set with box  coa

1992 SILVER US Mint PROOF SET Complete 5 coin set with box coa

$18.29

2015 US CLAD 14 Coin Proof Set Complete 3 Sets INV 276

2015 US CLAD 14 Coin Proof Set Complete 3 Sets INV 276

$41.95

1978 s USProof set Genuine complete and original as issued by US Mint

1978 s USProof set Genuine complete and original as issued by US Mint

$5.25

2006 US Mint Proof Westward Journey Nickel Series Coin Set Complete Box

2006 US Mint Proof Westward Journey Nickel Series Coin Set Complete Box

$5.95

2006 S United States Mint Proof Set Complete in Box with COA Mint Condition

2006 S United States Mint Proof Set Complete in Box with COA Mint Condition

$12.99

1999 2003 P D S STATE QUARTER SET DANSCO COMPLETE SET with 25 SILVER PROOFS

1999 2003 P D S STATE QUARTER Set Dansco COMPLETE SET with 25 SILVER PROOFS

$15.50

1986 S US Complete Proof set Nice Set  Free shipping

1986 S US Complete Proof set Nice Set Free shipping

$7.95

1986 s USProof set Genuine complete and original as issued by US Mint

1986 s USProof set Genuine complete and original as issued by US Mint

$4.75

2006 US MINT SILVER PROOF SET Complete w Original Box and COA

2006 US MINT SILVER PROOF SET Complete w Original Box and COA

$19.99



« Previous12

Leave a comment

You must be logged in to post a comment.