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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