Math Problem Statement
Compute the Jacobian for the transformation x = 8u + 5v, y = u - 2v.
Solution
Let's compute the Jacobian for the given transformation.
The transformation is defined as:
The Jacobian matrix is the matrix of partial derivatives of with respect to , which is:
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}$$ ### Step 1: Compute the partial derivatives - $$\frac{\partial x}{\partial u} = \frac{\partial}{\partial u}(8u + 5v) = 8$$ - $$\frac{\partial x}{\partial v} = \frac{\partial}{\partial v}(8u + 5v) = 5$$ - $$\frac{\partial y}{\partial u} = \frac{\partial}{\partial u}(u - 2v) = 1$$ - $$\frac{\partial y}{\partial v} = \frac{\partial}{\partial v}(u - 2v) = -2$$ ### Step 2: Form the Jacobian matrix $$J(u, v) = \begin{bmatrix} 8 & 5 \\ 1 & -2 \end{bmatrix}$$ ### Step 3: Compute the determinant of the Jacobian The determinant of the Jacobian matrix is: $$\det(J(u, v)) = (8)(-2) - (5)(1) = -16 - 5 = -21$$ Thus, the Jacobian determinant is: $$\det(J(u, v)) = -21$$ This confirms that the Jacobian is not always positive, as noted in the problem. Would you like more details or have any other questions? Here are 5 related questions to consider: 1. What is the significance of the Jacobian determinant in transformations? 2. How does the sign of the Jacobian determinant affect the orientation of the transformation? 3. Could the Jacobian determinant be zero? What would that imply about the transformation? 4. How would you interpret a negative Jacobian determinant geometrically? 5. How can you apply the Jacobian matrix in higher dimensions, such as in 3D transformations? **Tip:** The Jacobian determinant helps measure how a transformation affects area (in 2D) or volume (in higher dimensions).Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Multivariable Calculus
Transformation
Formulas
Jacobian matrix
Determinant of a 2x2 matrix
Theorems
Jacobian Determinant
Suitable Grade Level
Grades 11-12 and Undergraduate