Question

alenahelenash · Accepted Answer

To find the Jacobian of the transformation, we need to compute the partial derivatives of the new variables

x

and

y

with respect to the original variables

u

and

v

, respectively.
Let's start by finding the partial derivatives:

\frac{\partial x}{\partial u}

and

\frac{\partial x}{\partial v}

for

x = u^{2} + 4 v^{2}

Taking the partial derivative of

x

with respect to

u

while treating

v

as a constant:

\frac{\partial x}{\partial u} = \frac{\partial}{\partial u} (u^{2} + 4 v^{2}) = 2 u

Next, taking the partial derivative of

x

with respect to

v

while treating

u

as a constant:

\frac{\partial x}{\partial v} = \frac{\partial}{\partial v} (u^{2} + 4 v^{2}) = 8 v

Now, let's find the partial derivatives for

y = 3 u v

:

\frac{\partial y}{\partial u}

and

\frac{\partial y}{\partial v}

for

y = 3 u v

Taking the partial derivative of

y

with respect to

u

while treating

v

as a constant:

\frac{\partial y}{\partial u} = \frac{\partial}{\partial u} (3 u v) = 3 v

Next, taking the partial derivative of

y

with respect to

v

while treating

u

as a constant:

\frac{\partial y}{\partial v} = \frac{\partial}{\partial v} (3 u v) = 3 u

Now we can construct the Jacobian matrix by combining these partial derivatives:

J = [\begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{matrix}] = [\begin{matrix} 2 u & 8 v \\ 3 v & 3 u \end{matrix}]

Thus, the Jacobian matrix for the given transformation is:

J = [\begin{matrix} 2 u & 8 v \\ 3 v & 3 u \end{matrix}]

Answered question