Chapter 3: Many Variables; Section 1: Differentiation; page 7
Jacobian, Exercise 4
Question
The spherical coordinates
are related to the Cartesian by the relations
Calculate the Jacobian in terms of
!
Calculate the derivatives of the spherical coordinates, with respect to the Cartesian, in terms of
!
For a given function
, what are
?
Reminder
.... the determinant of the third order becomes
(3.1.7.26)
.... the exchange of all the rows with the corresponding columns, conserves the value of the determinant. For instance it follows that
(3.1.7.27)
The Jacobian for the change of a set of three variables
(3.1.7.28)
is defined as
(3.1.7.29)
where the right-hand side determinant is obtained with the aid of (3.1.7.27).
.... one has
(3.1.7.30)
The relations (3.1.7.29) and (3.1.7.30), can be extended to any number of variables.
The relations between the partial derivatives of one set of variables with respect to the other, like (3.1.7.9), are not so simple for more than two variable. For instance in the case of three variables the relations are of the type:
(3.1.7.31)
The right-hand sides of the equations (3.1.7.31) are similar to Jacobian, but since they contain only partial variables, they do not obey the relations corresponding to the full derivatives, namely
(3.1.7.32)
except for some coincidence. For that reason the symbol
is used instead of D.
Parts 1-3
Solution of question 1.
According to the definition (3.1.7.29), the required Jacobian should be
The application of the relation between the Cartesian and the polar coordinates yields
which after evaluation (3.1.7.26), becomes
Parts 4-13
Solution of question 2.
The derivatives of the spherical coordinates with respect to the Cartesian will be obtained, by the aid of the relations (3.1.7.31).
Parts 14-17
Solution of question 3.
For the differentiation of the function f, with respect to the Cartesian coordinates, the chain rule, and the results of question 2 are used.
Score
Note: If the score is not an integer, round it up to the next integer.
By parts.
Parts 1,2,3 are worth 1 point each.
Parts 4,5,6,7,8,9,10,11,12,13,14,15,16,17 are worth half a point each.
By questions.
Question 1 is worth 3 points.
Question 2 is worth 5 points.
Question 3 is worth 2 points.