Chapter 3: Many Variables; Section 1: Differentiation; page 7
Jacobian, Exercise 1
Question
The change of variables
is defined by
Obtain the Jacobian in terms of u and v !
Calculate the partial derivatives of u and of v with respect to
x and to y !
For a given function
, express the derivatives
in terms of u and v !
Reminder
(3.1.7.6)
that allows one to calculate
from the known derivatives of (3.1.7.1) . The expression in brackets of (3.1.7.6) is called the Jacobian and will be denoted here as
(3.1.7.7)
Notice that according to the definition (3.1.7.7):
(3.1.7.10)
It means, that the exchange of two variables, inverts the sign of the Jacobian.
By use of (3.1.7.9), and just by exchanging x with y, and u with v, one obtains the following four relations:
(3.1.7.11)
(3.1.7.15)
The relation (3.1.7.15) states, that in the case of inversion, the Jacobian behaves like the full derivative, namely
.
Parts 1-3
Solution of question 1.
The Jacobian is defined by (3.1.7.7), and the derivatives of the Cartesian coordinates, with respect to u and v , should be first calculated, according to the given relations.
From (3.1.7.7) and the part 2, we obtain for the Jacobian
Parts 4-8
Solution of question 2.
For the calculation of the derivatives for this question, we have to use (3.1.7.10), together with (3.1.7.11), applied on the results from parts 2 and 3.
Parts 9-10
Solution of question 3.
From the chain rule, we have
where the results of question 2 were used.
Similarly
Score
Note 1: The way that the Jacobian was simplified in part 3, is not unique, and another way of simplification is also acceptable. As a matter of fact, the Jacobian can be left, without any simplification.
Note 2: In the case of an exam, it would be wise to check yourself, by calculating the Jacobian , directly from the results of question 2.
By parts.
Any one of the 10 parts is worth 1 point.
By questions.
Question 1 is worth 3 points.
Question 2 is worth 5 points.
Question 3 is worth 2 points.