Chapter 3: Many Variables; Section 1: Differentiation; page 4
Change of Variables, Exercise 4
Question
The function
undergoes the transformation
Use the expressions
in order to obtain
Notice: The first two questions of this exercise are simple, but the last two require a good understanding of the change of variables, and of the partial derivatives.
Reminder
... the change of the pair of variables (x,y) by a new pair (u,v), can be also expressed by a functional relation of the variables
(3.1.4.3)
The derivatives of the function with respect to the new variables can be obtained from the definition of partial derivatives, yielding the following chain rules:
(3.1.4.4)
The relations (3.1.4.4) contain partial derivatives, indexed by the variable that is kept constant during the differentiation. This notation is introduced just for clarity....
Parts 1-2
Solution of question 1
One has to calculate
which is not identical to
It can be calculated from the top expression of (3.1.4.4), by substituting u by x and v by z.
Parts 3-4
Solution of question 2
By applying the same substitution (of part 1) on the bottom expression of (3.1.4.4), we obtain
By definition
therefore
Parts 5-8
Solution of question 3
For this and the next question, the transformation is from (x, y) to (y, z), therefore one have to
substitute (u, v) by (y, z), in (3.1.4.4).
From the top expression of (3.1.4.4), after the substitution one obtains
By definition
The question is
The answer is in the relation
Since z is constant with respect to the differentiation, this relation behaves, as y being a function of the single variable x. Therefore
and finally
Parts 9-11
Solution of question 4
From the bottom expression of (3.1.4.4), and after the substitution of part 5, one obtains
By definition
The question is
The answer is again in the relation
Since y is constant with respect to the differentiation, this relation behaves as an implicit function
Therefore
and finally
Score
Notice.
If the total score is not an integer, add half a point.
By parts.
Parts 1,2,3,4,6,9 are worth ½ points each.
Parts 5,7,10 are worth 1 point each.
Parts 8,11 are worth 2 points each.
By questions.
Questions 1 and 2 are worth 1 point each.
Question 3 is worth 4½ points.
Question 4 is worth 3½ points.