Math Problem Statement

If u and v are functionally dependent, then their Jacobian value is...

Solution

The problem asks: "If uu and vv 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:

  1. Functional dependence means that one of the variables (uu or vv) can be expressed as a function of the other. Mathematically, if u=f(v)u = f(v), then they are dependent.
  2. The Jacobian determinant of two functions u(x,y)u(x, y) and v(x,y)v(x, y) with respect to variables xx and yy is given by: J(u,v)=uxvyuyvx.J(u, v) = \frac{\partial u}{\partial x} \frac{\partial v}{\partial y} - \frac{\partial u}{\partial y} \frac{\partial v}{\partial x}.
  3. When uu and vv are functionally dependent, the Jacobian determinant J(u,v)J(u, v) is 0. This is because functional dependence implies that there is no independent variation between uu and vv.

Correct Answer:

(A) 0

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Related Questions:

  1. What is the Jacobian determinant and how is it calculated for multivariable functions?
  2. How can functional dependence between two variables be determined using partial derivatives?
  3. What is the geometric interpretation of a Jacobian determinant?
  4. How does functional independence relate to the Jacobian being non-zero?
  5. 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