Math Problem Statement
If u and v are functionally dependent, then their Jacobian value is...
Solution
The problem asks: "If and are functionally dependent, then their Jacobian value is ..." with the following answer options:
- (A) 0
- (B) 1
- (C) Non-zero
- (D) Greater than zero
Explanation:
- Functional dependence means that one of the variables ( or ) can be expressed as a function of the other. Mathematically, if , then they are dependent.
- The Jacobian determinant of two functions and with respect to variables and is given by:
- When and are functionally dependent, the Jacobian determinant is 0. This is because functional dependence implies that there is no independent variation between and .
Correct Answer:
(A) 0
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Related Questions:
- What is the Jacobian determinant and how is it calculated for multivariable functions?
- How can functional dependence between two variables be determined using partial derivatives?
- What is the geometric interpretation of a Jacobian determinant?
- How does functional independence relate to the Jacobian being non-zero?
- Can functional dependence be generalized to more than two variables, and how does it affect the Jacobian matrix?
Tip:
Always check if the rank of the Jacobian matrix matches the number of independent variables to determine functional dependence.
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Math Problem Analysis
Mathematical Concepts
Functional Dependence
Jacobian Determinant
Partial Derivatives
Formulas
J(u, v) = (∂u/∂x)(∂v/∂y) - (∂u/∂y)(∂v/∂x)
Theorems
Jacobian Determinant Theorem for Functional Dependence
Suitable Grade Level
Grades 11-12 or College Level