1. **State the problem:** We want to find the Jacobian of a transformation scaled by the determinant given a vector and then evaluate an integral involving the variables. 2. **Jacobian and determinant:** The Jacobian matrix $J$ of a vector function $\mathbf{F}(x,y,z)$ is the matrix of all first-order partial derivatives: $$J = \begin{bmatrix} \frac{\partial F_1}{\partial x} & \frac{\partial F_1}{\partial y} & \frac{\partial F_1}{\partial z} \\ \frac{\partial F_2}{\partial x} & \frac{\partial F_2}{\partial y} & \frac{\partial F_2}{\partial z} \\ \frac{\partial F_3}{\partial x} & \frac{\partial F_3}{\partial y} & \frac{\partial F_3}{\partial z} \end{bmatrix}$$ The determinant of $J$, denoted $\det(J)$, scales volumes under the transformation. 3. **Given vector:** $$\mathbf{v} = [2,6,5]$$ 4. **Integral expression:** $$x^2 + 6xy + 6z$$ 5. **Interpretation:** Assuming the problem asks to compute the determinant of the Jacobian matrix for the transformation defined by the vector components and then scale the integral by this determinant. 6. **Jacobian matrix:** Since the vector is constant, the Jacobian matrix of $\mathbf{v}$ with respect to $(x,y,z)$ is: $$J = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ Because each component is constant, all partial derivatives are zero. 7. **Determinant:** $$\det(J) = 0$$ 8. **Scaling the integral:** Multiplying the integral expression by $\det(J)$: $$0 \times (x^2 + 6xy + 6z) = 0$$ **Final answer:** The determinant of the Jacobian matrix is zero, so scaling the integral by this determinant results in zero.