Find the Jacobian 𝜕(𝑥,𝑦)𝜕(𝑢,𝑣) for the transformation 𝑢 = 3𝑥 + 2𝑦; 𝑣 = 𝑥 +4𝑦 . Further, find the image under this transformation of the triangularregion in the 𝑥𝑦- plane bounded by the 𝑥- axis, 𝑦- axis, and the line 𝑥 +𝑦 = 1. Sketch the transformed region in the 𝑢𝑣- plane.

To find the Jacobian 𝜕(𝑥,𝑦)𝜕(𝑢,𝑣) for the given transformation 𝑢 = 3𝑥 + 2𝑦 and 𝑣 = 𝑥 + 4𝑦, we need to compute the partial derivatives.

Step 1: Compute the partial derivative of 𝑢 with respect to 𝑥. 𝜕𝑢/𝜕𝑥 = 3

Step 2: Compute the partial derivative of 𝑢 with respect to 𝑦. 𝜕𝑢/𝜕𝑦 = 2

Step 3: Compute the partial derivative of 𝑣 with respect to 𝑥. 𝜕𝑣/𝜕𝑥 = 1

Step 4: Compute the partial derivative of 𝑣 with respect to 𝑦. 𝜕𝑣/𝜕𝑦 = 4

Therefore, the Jacobian matrix is: 𝐽 = [𝜕(𝑢,𝑣)/𝜕(𝑥,𝑦)] = [3 2; 1 4]

To find the image under this transformation of the triangular region in the 𝑥𝑦-plane bounded by the 𝑥-axis, 𝑦-axis, and the line 𝑥 + 𝑦 = 1, we substitute the coordinates of the vertices of the triangle into the transformation equations.

The vertices of the triangle are: A: (0, 0) B: (1, 0) C: (0, 1)

Substituting these coordinates into the transformation equations, we get: A': (𝑢, 𝑣) = (3(0) + 2(0), 0 + 4(0)) = (0, 0) B': (𝑢, 𝑣) = (3(1) + 2(0), 1 + 4(0)) = (3, 1) C': (𝑢, 𝑣) = (3(0) + 2(1), 0 + 4(1)) = (2, 4)

The image of the triangular region in the 𝑢𝑣-plane is a triangle with vertices A' (0, 0), B' (3, 1), and C' (2, 4).