Please disable adblock to view this page.

← Go home

Jacobians, Errors and Approximations, Maxima and Minima

December 8, 2016
Published By : Pratik Kataria
Categorised in:

Denotation: $J = {δ(u,v)}/{δ(x,y)} = |\table u_x,u_y;v_x,v_y|$

Chain rule of jacobians:

x,y -> u,v -> r,s
${δ(x,y)}/{δ(u,v)} . {δ(u,v)}/{δ(r,s)} = {δ(x,y)}/{δ(r,s)}$

J. J’ = 1

Jacobian of implicit functions:
$u_1, u_2, u_3$ be implicit functions of variables $x_1, x_2, x_3$ connected by $f_1, f_2, f_3$ such that
$f_1(u_1,u_2,u_3,x_1,x_2,x_3)=0$
$f_2(u_1,u_2,u_3,x_1,x_2,x_3)=0$
$f_3(u_1,u_2,u_3,x_1,x_2,x_3)=0$

${δ(u_1,u_2,u_3)}/{δ(x_1,x_2,x_3)} = (-1)^3 {{δ(f_1,f_2,f_3)}/{δ(x_1,x_2,x_3)}}/{{δ(f_1,f_2,f_3)}/{δ(u_1,u_2,u_3)}}$

General:
${δ(u_1,u_2,u_3,…,u_n)}/{δ(x_1,x_2,x_3,…,x_n)} = (-1)^n {{δ(f_1,f_2,f_3,…,f_n)}/{δ(x_1,x_2,x_3,…,x_n)}}/{{δ(f_1,f_2,f_3,…,f_n)}/{δ(u_1,u_2,u_3,…,u_n)}}$

Four variables u,v,x,y related by implicit functions:
$f_1(u,v,x,y)=0$ & $f_2(u,v,x,y) = 0$

${δu}/{δx} = – {{δ(f_1,f_2)}/{δ(x,y)}}/{{δ(f_1,f_2)}/{δ(u,v)}}$

${δv}/{δx} = – {{δ(f_1,f_2)}/{δ(u,x)}}/{{δ(f_1,f_2)}/{δ(u,v)}}$

${δu}/{δy} = – {{δ(f_1,f_2)}/{δ(y,v)}}/{{δ(f_1,f_2)}/{δ(u,v)}}$

${δv}/{δy} = – {{δ(f_1,f_2)}/{δ(u,y)}}/{{δ(f_1,f_2)}/{δ(u,v)}}$


Functional Dependence
$J = {δ(f_1,f_2)}/{δ(x,y)} = {δ(u,v)}/{δ(x,y)} = 0$

${δ(f_1,f_2)}/{δ(x,y)} = 0, {δ(f_1,f_2)}/{δ(y,z)} = 0, {δ(f_1,f_2)}/{δ(z,x)} = 0$

If z = f(x,y) we have: $dz = {δz}/{δx} dx + {δz}/{δy} dy$
dx, dy, dz are actual errors in x, y and z, respectively

${dx}/{x}, {dy}/y, {dz}/{z}$ are relative errors in x, y and z, respectively.

${100 dx}/{x} , {100 dy}/y, {100 dz}/z$ are known as percentage errors in x, y and z, respectively.

Maxima and Minima of Function z=f(x,y):
Find and equal ${δf}/{δx} = 0$ and ${δf}/{δy} = 0$
Calculate: $r = {δ^2 f}/{δx^2}$ , $s = {δ^2 f}/{δx δy}$ , $t = {δ^2 f}/{δy^2}$

$rt – s^2 > 0$ & r<0 at $(a_1, b_1)$, f(x,y) is a maximum at $(a_1,b_1)$ & $f(a_1,b_1)$ is maximum value
$rt – s^2 > 0 $ & r>0 at $(a_1, b_1)$, f(x,y) is a minimum at $(a_1,b_1)$ & $f(a_1,b_1)$ is minimum value
$rt – s^2 < 0$ at $(a_1, b_1)$, f(x,y) is neither maximum nor minimum at $(a_1,b_1)$ i.e. $f(a_1,b_1)$ is not an extreme value. Such a point is called saddle point.

$rt – s^2 = 0$ at $(a_1, b_1)$, case is undecided

Lagrange’s Method
Let u = f(x,y,z)
ϕ(x,y,z) = 0

Construct: F = u + λϕ (λ is non zero constant)

From equations:
${δF}/{δx} = 0$ , ${δF}/{δy} = 0$ , ${δF}/{δz} = 0$

Eliminate x, y, z and λ… Get equation in terms of u.