Jacobian Dependence A6308D

1. **State the problem:** We want to test if the pairs of functions $$Y_1, y_2$$ are functionally dependent using the Jacobian determinant. 2. **Recall the rule:** Two functions $$f(x,y)$$ and $$g(x,y)$$ are functionally dependent if the Jacobian determinant $$J = \frac{\partial(f,g)}{\partial(x,y)} = 0$$ everywhere in the domain. 3. **Part (a):** - Given $$Y_1 = 3xf + Xz$$ and $$y_2 = 9xt + 6xf (x^2 + 4) + x^2(x^2 + 8) + 12$$. - We need to identify variables with respect to which we take partial derivatives. Assuming variables are $$x$$ and $$f$$ (since $$Y_1$$ and $$y_2$$ depend on $$x$$ and $$f$$). 4. **Calculate partial derivatives for (a):** - $$\frac{\partial Y_1}{\partial x} = 3f + \frac{\partial (Xz)}{\partial x}$$ but $$Xz$$ is ambiguous; assuming $$Xz$$ is a constant or independent of $$x$$ and $$f$$, then $$\frac{\partial Y_1}{\partial x} = 3f$$. - $$\frac{\partial Y_1}{\partial f} = 3x$$. - For $$y_2$$, partial derivatives are complex due to undefined variables $$t$$ and $$z$$; assuming $$t$$ and $$z$$ are constants or independent of $$x$$ and $$f$$. - $$\frac{\partial y_2}{\partial x} = 9t + 6f (2x) + 2x(x^2 + 8) + x^2(2x) = 9t + 12fx + 2x^3 + 16x + 2x^3 = 9t + 12fx + 4x^3 + 16x$$. - $$\frac{\partial y_2}{\partial f} = 6x(x^2 + 4)$$. 5. **Jacobian determinant for (a):** $$J = \begin{vmatrix} \frac{\partial Y_1}{\partial x} & \frac{\partial Y_1}{\partial f} \\ \frac{\partial y_2}{\partial x} & \frac{\partial y_2}{\partial f} \end{vmatrix} = \begin{vmatrix} 3f & 3x \\ 9t + 12fx + 4x^3 + 16x & 6x(x^2 + 4) \end{vmatrix}$$ 6. **Calculate determinant:** $$J = 3f \cdot 6x(x^2 + 4) - 3x \cdot (9t + 12fx + 4x^3 + 16x)$$ $$= 18fx(x^2 + 4) - 3x(9t + 12fx + 4x^3 + 16x)$$ $$= 18fx^3 + 72fx - 27xt - 36fx^2 - 12x^4 - 48x^2$$ 7. **Simplify:** $$J = 18fx^3 + 72fx - 27xt - 36fx^2 - 12x^4 - 48x^2$$ 8. **Interpretation:** Since $$J$$ is not identically zero for all $$x, f, t$$, the functions are not functionally dependent. 9. **Part (b):** - Given $$Y_1 = 3xf + 2xf = 5xf$$ (simplified) and $$y_2 = Sx1 + 1$$ (ambiguous notation, assuming $$Sx1$$ means $$S \cdot x_1$$ or a constant times $$x_1$$). - Assuming variables are $$x$$ and $$f$$. 10. **Calculate partial derivatives for (b):** - $$\frac{\partial Y_1}{\partial x} = 5f$$. - $$\frac{\partial Y_1}{\partial f} = 5x$$. - $$\frac{\partial y_2}{\partial x} = \frac{\partial (Sx1 + 1)}{\partial x}$$; if $$Sx1$$ is constant or independent of $$x$$ and $$f$$, then $$0$$. - $$\frac{\partial y_2}{\partial f} = 0$$. 11. **Jacobian determinant for (b):** $$J = \begin{vmatrix} 5f & 5x \\ 0 & 0 \end{vmatrix} = 5f \cdot 0 - 5x \cdot 0 = 0$$ 12. **Interpretation:** Since $$J = 0$$ everywhere, the functions $$Y_1$$ and $$y_2$$ are functionally dependent. **Final answers:** - (a) Functions are not functionally dependent. - (b) Functions are functionally dependent.