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#1 2006-03-30 09:03:24
- MathsIsFun
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Partial Differentiation Formulas
Partial Differentiation Formulas
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#2 2006-04-16 01:00:11
- ganesh
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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
Character is who you are when no one is looking.
#3 2006-04-16 01:09:06
- Ricky
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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..."
#4 2006-04-25 00:38:05
- ganesh
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Re: Partial Differentiation Formulas
Jacobians
If
then the Jacobian of the transformation from
to
is defined by the determinant
Character is who you are when no one is looking.
#5 2006-08-06 13:05:20
- Zhylliolom
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Re: Partial Differentiation Formulas
Differential of a Multivariable Function
If z = f(x1, x2, ..., xn), then
Differentiation of Composite Functions
If z = f(x1, x2, ..., xn), where x1 = f1(r1, r2, ..., rp), ..., xn = fn(r1, r2, ..., rp), 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 xa and xb, 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 (2006-08-06 13:13:30)