Chapter 3: Many Variables; Section 1: Differentiation; page 3
Derivatives of Higher Order, Exercise 3
Question
The following function z represents the half of an ellipsoid
where a, b and c are positive constants.
Calculate the first and the second derivatives of z !
Obtain the second approximation
around the point (x, y) = (0, 0) !
Check your final result, by applying the balance of the physical dimensions !
Reminder
(3.1.3.1)
...the notation presented here, ....exhibits its physical dimensions. As an example let's look at ... :
(3.1.3.12)
where the square brackets mean the physical dimension (of what is inside the brackets).
For a function of two variables (3.1.3.1), the first approximation around the point (x, y)0 can be written as:
(3.1.3.15)
The second approximation includes an additional term of a squared expression containing symbolic operators. Its meaning could be understood by the following:
(3.1.3.16)
Parts 1-5
Solution of question 1
Parts 6-8
Solution of question 2
Parts 9-10
Solution of question 3
From the definition of the function it follows that
where the square brackets mean physical dimensions.
It follows that in the expression of the result of part 8, the content in the brackets is dimensionless, showing that the balance of the dimensions in the final result is kept.
Score
By parts.
All ten parts are worth one point each.
By questions.
Question 1 is worth 5 points.
Question 2 is worth 3 points.
Question 3 is worth 2 points.