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Partial Differentiation FormulasPartial Differentiation Formulas "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #2 20060416 01:00:11
Re: Partial Differentiation FormulasIf 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 20060416 01:09:06
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 20060425 00:38:05
Re: Partial Differentiation FormulasJacobians then the Jacobian of the transformation from to is defined by the determinant Character is who you are when no one is looking. #5 20060806 13:05:20
Re: Partial Differentiation FormulasDifferential of a Multivariable Function Differentiation of Composite Functions If z = f(x_{1}, x_{2}, ..., x_{n}), where x_{1} = f_{1}(r_{1}, r_{2}, ..., r_{p}), ..., x_{n} = f_{n}(r_{1}, r_{2}, ..., r_{p}), 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_{a} and x_{b}, 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 (20060806 13:13:30) 