Chapter 3: Many Variables; Section 1: Differentiation; page 4
Change of Variables, Exercise 3
Question
The Cartesian coordinate system (x', y') is obtained by rotating the system (x, y) by an angle β about the origin. For a given function
Calculate
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. It is not necessary, and will be usually omitted, except for a few particular cases. Since the first derivatives are also functions of two variables, the chain rules (3.1.4.4) can be reused, in order to obtain second order derivatives, and so on, up to any order of differentiation.
....The rotation of coordinates, about the origin, by an angle Δφ ...
(3.1.4.11)
yielding this time a different result, in comparison with (3.1.4.9)
(3.1.4.12)
and finally
(3.1.4.13)
Parts 1-3
Solution of question 1
By substituting Δφ with β in (3.1.4.13) we obtain
Finally
Parts 4-9
Solution of question 2
The second derivatives will be calculated, independently from the exercise 2. The necessary derivatives of the variables (x, y) with respect to the transformed (x', y') are given by (3.1.4.12):
The results of parts 1 and 4 are used for the second differentiation.
In the same way we calculate
After adding the results of parts 6 and 8, we obtain
Score
Notice.
If the total score is at least 6 points, one have to add one additional point.
Question 2 can be also solved, by using the results of exercise 2.
By parts.
Each one of the 9 parts is worth 1 point.
By questions.
Question 1 is worth 3 points.
Question 2 is worth 6 points.