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#1 2006-03-29 10:03:24

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,618

Partial Differentiation Formulas

Partial Differentiation Formulas


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#2 2006-04-15 03:00:11

ganesh
Moderator
Registered: 2005-06-28
Posts: 20,935

Re: Partial Differentiation Formulas

If f is a function of two variables, its partial derivatives fx and fy are also function of two variables; their partial derivatives (fx)x, (fx)y, (fy)x, and (fy)y are second order partial derivatives. If z=f(x,y), then


Homogenous function :- A function f(x,y) of two independent variables x and y is said to be homogenous in x and y of degree n if

for any positive quantity t where t is independent of x and y.
For example,

Therefore, f(x,y) is a homogenous function of degree 2 in x and y.

Euler's theorem on homogenous functions
If f is a homogenous function of degree n in x and y, then


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#3 2006-04-15 03:09:06

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Partial Differentiation Formulas


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2006-04-24 02:38:05

ganesh
Moderator
Registered: 2005-06-28
Posts: 20,935

Re: Partial Differentiation Formulas

Jacobians

If

are functions of 3 variables

then the Jacobian of the transformation from
to

is defined by the determinant


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#5 2006-08-05 15:05:20

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: Partial Differentiation Formulas

Differential of a Multivariable Function

If z = f(x[sub]1[/sub], x[sub]2[/sub], ..., x[sub]n[/sub]), then

Differentiation of Composite Functions

If z = f(x[sub]1[/sub], x[sub]2[/sub], ..., x[sub]n[/sub]), where x[sub]1[/sub] = f[sub]1[/sub](r[sub]1[/sub], r[sub]2[/sub], ..., r[sub]p[/sub]), ..., x[sub]n[/sub] = f[sub]n[/sub](r[sub]1[/sub], r[sub]2[/sub], ..., r[sub]p[/sub]), then

where k = 1, 2, ..., p.

Implicit Functions

For the implicit equation F[x, y, z(x, y)] = 0, we have

and

Surface Area

The area of a surface z = f(x, y) is given by

Theorems on Jacobians

If x and y are functions of u and v and u and v are functions of r and s, then

For 2 equations in n > 2 variables to be possibly solved for the variables x[sub]a[/sub] and x[sub]b[/sub], it is necessary and sufficient that

This may be extended to m equations in n > m variables.

If u = f(x, y) and v = g(x, y), then a necessary and sufficient condition that a functional relation of the form Φ(x, y) = 0 exists between u and v is that

This may be extended to n functions of n variables.

Partial Derivatives with Jacobians

Given the equations F(x, y, u, v) = 0 and G(x, y, u, v) = 0, we have

This process may be extended to functions of more variables.

Differentiation Under the Integral Sign

If

then

Last edited by Zhylliolom (2006-08-05 15:13:30)

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