#
OEF several variables functions
--- Introduction ---

This module actually contains 7 exercises on derivatives of
several variables functions.

### Linear approximation

Let
be the real function on
defined by
. Give the linear approximation of
at
. It it does not exist, answer by ** no **.

### Scalar Field 2D

Let a scalar field which represents at a point
of
given by
.
Calculate at the point (,) .
What is the equation of the level curve of constant
The domain where is is

### Directionnal derivatives

Let
be a function C^{1} in two variables with values in
, and
and
two vectors in
defined by

.

If you know the partial derivatives
and
of
in the two directions
and
at
, are you able to calculate the directionnal derivative of
at
in any other direction ?
Let
be the vector defined by w=(, ). Calculate the derivative of
in direction
if
with
.

You are right, it's not possible because the vectors
et
are colinear.

Is it possible that
, with
?

### Composition I, partial derivatives

Let
be a real function of two variables
and
on
and
the real function on
defined by
. Calculate the partial derivative of
with respect to
.
(x,y)=
(
,
) +
(
,
)

### Partial derivatives 1

Calculate the partial derivatives of the function
defined by

### Partial derivatives 2

Calculate
for the function
defined by
.

### Composition II Partial derivatives

Let
be a function of 2 variables
and
on
with values in
and
the function on
with values in
defined by
.

Calculate the second derivative of
with respect to
.

(x,y)=
(
,
) +
(
)

+
(
) +
(
)

+
(
)
(x,y)= (
)
(
) +
(
,
)

+ (
)
(
) + (
)
(
)

+
(
)
(x,y)=
(
) + (
)
(
)

+
(
,
) + (
)
(
)

+
(
)

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- Description: collection of exercises on several variables functions. interactive exercises, online calculators and plotters, mathematical recreation and games
- Keywords: interactive mathematics, interactive math, server side interactivity, analysis, partial derivative, scalar function, field