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MathsIsFun

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Registered: 2005-01-21

Posts: 7,713

Partial Differentiation Formulas

Partial Differentiation Formulas

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

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,385

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

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Partial Differentiation Formulas

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"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..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,385

Re: Partial Differentiation Formulas

Jacobians

If

are functions of 3 variables

then the Jacobian of the transformation from

to

is defined by the determinant

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

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)