This cell plots the image G(Rect), where the function G maps (u,v) to (x,y) and Rect = [0,2] x [0, 3].
The image is a triangle with vertices (0,0), (0,3), and (2,3)

G[u_, v_] := {u v/3, v}; (* (x,y) = G(u,v), so x = uv/3, y = v *)ParametricPlot[G[u,v], {u,0, 2}, {v, 0,3}, Mesh ->Automatic, FrameLabel->{x,y}, RotateLabel->False]

Note:  The Mesh -> Automatic prints 15 internal curves in each direction.  We can change that.  Let’s start by plotting the rectangle R = [0, 2] x [0,3], with squares of side length 1/2.  Each of these squares has area 1/4.

rect = ParametricPlot[{u,v}, {u,0, 2}, {v, 0,3}, Mesh ->{3, 5}, FrameLabel->{u,v}, RotateLabel->False]

Now, plot G(Rect), for the same function G we defined at the beginning.

G[u_, v_] := {u v/3, v}; (* (x,y) = G(u,v), so x = uv/3, y = v *)GofRect = ParametricPlot[G[u,v], {u,0, 2}, {v, 0,3}, Mesh ->{3, 5}, FrameLabel->{x,y}, RotateLabel->False]

The next plot shows Rect = [0,2] x [0,3] on the left, and G(Rect) on the right,
for G(u,v) = (u v/3, v).
The Jacobian of G is Jac(G)(u, v) = v/3. This means that the area of the rectangles on the right are v/3 times the area of the rectangles on the left.

GraphicsRow[{rect, GofRect}]

The variables don’t need to be called u and v. This shows the image of the rectangle 1 < r < 5 and 0 < θ < π, also called [1, 5] x [0, π], in polar coordinates.  In this case Jac(G)(r, θ) = r.  This is why dA = r dr dθ

G[r_, θ_] := {r Cos[θ], r Sin[θ]}; (* This is polar coordinates, x = r cos(θ), y = r sin(θ) *)ParametricPlot[G[r,θ], {r, 1, 5}, {θ, 0, π}, Mesh -> Automatic, FrameLabel->{x,y}, RotateLabel->False]