2. Partial derivatives. Total differential
Definition of partial derivatives
Given a function f(x,y)
defined in a neighborhood of
,
the partial derivative of f
with respect to x at
is denoted by
and is defined by
provided
that the limit exists. Likewise, the partial derivative of f with
respect to y at
is denoted by
and is defined by
.
If g is a differentiable function of a single
variable, and h is a function of two (or more) variables with
defined, then
,
and similarly with y and any other variables.
Total differential
In the case of a function of a single variable the differential of the function y = f(x) is the quantity dy = f '(x) Δx .
This quantity is used to compute the approximate change in the value of f(x) due to a change Δx in x. As is shown in Fig. 2,
Fig 2.
Δy = CB = f(x + Δx) - f(x), while dy = CT = f '(x)Δx .
When Δx is small the approximation is close. Line AT represents the tangent to the curve at point A. In the case of a function of two variables the situation is analogous. Let us start at point A(x1, y1, z1) on the surface z = f(x, y) shown in Fig. 3 and let x and y change by small amounts Δx and Δy, respectively. The change produced in the value of the function z is
Δz = CB = f(x1 + Δx, y1 + Δy) - f(x1, y1) .
An
approximation to Δz is given by
.
When Δx and Δy are small the approximation is close. Point T lies in that plane tangent to the surface at point A.
The quantity
.
is called the total differential of the function z = f(x, y). Because is customary to denote increments Δx and Δy by dx and dy, the total differential of a function z = f(x, y) is defined as
.
Gradients and directional derivatives
Definition
1. Let z=f(x, y), the gradient of
f, denoted by
,
is defined by the vector
If
w= f(x, y, z), the gradient of f denoted by
,
is defined by
Note: the gradient of a function is a vector.
Definition
of directional derivative. If
w=f(x, y, z) and
(unit vector), then the directional derivative of w in the direction
of
is defined by
provided the limit exists.
Theorem
1. If z=f(x, y), and if
are
both continuous, and
is a unit vector in the direction of
then
(dot product of the gradient and the unit vector gives the directional derivative in the direction of a given vector )
Theorem
2. If w=f(x, y, z), and if
are
continuous, and
,
then
(dot product of the gradient and the unit vector gives the directional derivative in the direction of the given vector)
The vector Chain rule:
is a differentiable function of t, then
.
Definition
2.
is called the normal derivative because it is normal to the level
curve and it is denoted by
.
3. Differentiation of composite and implicit functions. Tangent and surface normal
The case of one independent variable
Theorem 3. If
is differentiable function in point
and
,
are differentiable functions of independent variable
,
then derivative of composite function
calculated by the formula:
.
The case of several independent variable
If arguments х and у of function z = f(х; у) are functions of two variables,
,
then
also is function of two variables u and v:
and
.
Differential of composite function
Differential of composite function z
= z(х;у),
where
,
can be obtained if in the formula of differential
do change
and
.
As a result, we obtain
.