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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Partial Differentiation Formulas

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

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**ganesh****Administrator**- Registered: 2005-06-28
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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 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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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

"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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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,989

**Jacobians**

If

are functions of 3 variablesthen 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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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

**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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