Final answer:
To find the Jacobian of the transformation with given functions x and y in terms of r and θ, we compute the first-order partial derivatives and place them in a 2x2 determinant.
Step-by-step explanation:
The question asks to find the Jacobian of the transformation given by the equations:
x = -6e⁴ᵣ sin(3θ),
y = e⁻⁴ᵣ cos(3θ)
The Jacobian of a transformation is a matrix of all first-order partial derivatives, and it represents the transformation from one coordinate system to another, in this case from polar to Cartesian coordinates. For a function defined by two variables x and y that depend on two variables r and θ, the Jacobian matrix is given by:
|J| = \[ \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{vmatrix} \]
By computing the necessary partial derivatives, we can find:
\[ \frac{\partial x}{\partial r} = -24e⁴ᵣ sin(3θ) \]
\[ \frac{\partial x}{\partial θ} = -18e⁴ᵣ cos(3θ) \]
\[ \frac{\partial y}{\partial r} = -4e⁻⁴ᵣ cos(3θ) \]
\[ \frac{\partial y}{\partial θ} = -3e⁻⁴ᵣ sin(3θ) \]
Inserting these into the determinant formula, we get:
\[ |J| = \begin{vmatrix} -24e⁴ᵣ sin(3θ) & -18e⁴ᵣ cos(3θ) \\ -4e⁻⁴ᵣ cos(3θ) & -3e⁻⁴ᵣ sin(3θ) \end{vmatrix} \]
Calculating the determinant, the Jacobian determinant for the transformation |J| = ∂(x,y)/∂(r,θ) is obtained. This shows the rate of change of area in the Cartesian coordinate system corresponding to a unit of area in the polar coordinate system.