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

]]>If

are functions of 3 variablesthen the Jacobian of the transformation from

to

is defined by the determinant]]>

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