Question

Find the image of the set S under the given transformation.

\begin{align} S &=\{(u,v)\mid 0u3,0v2\}; \\ x &=2u+3v \\ y &=u-v \\ \end{align}

Short answer

The image of the set s under given is the rectangle region.

Step-by-step solution

Define Jacobian matrix

The name "Jacobian" is frequently used to refer to both the Jacobian matrix and determinants, which are both defined for finite numbers of functions with the same number of variables. With respect to the variables, each row contains the first partial derivative of the same function.

Any type of Jacobian matrix can be used. It could be a square matrix (with the same number of rows and columns) or a rectangle matrix (the number of rows and columns are not equal).

 Step 2: Find the image of the set.

The substitution

\(\begin{array}{l}v = \frac{{x - 2y}}{5}\\u = \frac{{x + 2y}}{5}\end{array}\)

With

\(\begin{array}{*{20}{r}}{0 \le u \le 3{\rm{ and }}0 \le v \le 2}\\{0 \le x \le 25{\rm{ and }}0 \le y \le \frac{5}{4}}\end{array}\)

\(\begin{array}{l}0 \le \frac{{x + 3y}}{5} \le 3\\0 \le x + 3y \le 15\end{array}\)

\(\begin{array}{l}0 \le \frac{{x - 2y}}{5} \le 2\\0 \le x - 2y \le 10\end{array}\)

Step 3: Plot the graph.

Now plot the lines

\(\begin{array}{l}x + 3y = 0\\x + 3y = 15{\rm{ }}\\x - 2y = 0,\\x - 2y = 10\end{array}\)

Therefore, the image of the set s under given is the rectangle region.