# Question 20 Analysis Let us analyze and explain each step thoroughly, as required. --- ## Step 1: Understanding the Jacobian The Jacobian matrix (and its determinant) arises in multivariable calculus. For a function that maps ℝⁿ to ℝⁿ, the Jacobian is the matrix of all first-order partial derivatives. It is especially significant when transforming coordinates or variables in multidimensional spaces. --- ## Step 2: Evaluating Each Option - **Option A: Solving differential equations** - While the Jacobian can appear in systems of differential equations, its primary role is not directly in solving them, but rather in analyzing stability or transformations. - **Option B: Changing variables in multiple integrals** - In multiple integrals, when changing variables (for example, from Cartesian to polar coordinates), the Jacobian determinant accounts for how areas or volumes change under the transformation. This is a primary and direct application. - **Option C: Matrix inversion** - The Jacobian itself is a matrix, but it is not specifically used for inverting matrices. - **Option D: Finding eigenvalues** - The process of finding eigenvalues is unrelated to the Jacobian matrix or its determinant. --- ## Step 3: Identifying the Correct Use The most direct and correct use of the Jacobian is in the change of variables in multiple integrals. The formula, \[ \iint_{R} f(x, y) \, dx \, dy = \iint_{S} f(g(u, v), h(u, v)) \left| \frac{\partial(x, y)}{\partial(u, v)} \right| du \, dv \] demonstrates how the Jacobian determinant is essential for correct transformation between coordinate systems. --- ## Final Answer **The Jacobian is used in:** **B) Changing variables in multiple integrals**