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 C1 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