Let $f$ be a transformation from $\mathrm{R}^{3}$ to $\mathrm{R}^{3}$. Its Jacobian matrix is given below.
$$
J(f)=\left[\begin{array}{ccc}
-1 & 0 & -1 \
3 & 0 & 0 \
0 & \cos (y) & 1
\end{array}\right]
$$

Find the Jacobian determinant of $f$.
$$
|J(f)|=
$$
$\square$
How will $f$ expand or contract space around the point $(0, \pi, 0)$ ?
Choose 1 answer:
(A) Leave it the same
(B) Expand it finitely
(C) Contract it finitely
(D) Contract it infinitely

Question

Let $f$ be a transformation from $\mathrm{R}^{3}$ to $\mathrm{R}^{3}$. Its Jacobian matrix is given below.
$$
J(f)=\left[\begin{array}{ccc}
-1 & 0 & -1 \
3 & 0 & 0 \
0 & \cos (y) & 1
\end{array}\right]
$$

Find the Jacobian determinant of $f$.
$$
|J(f)|=
$$
$\square$
How will $f$ expand or contract space around the point $(0, \pi, 0)$ ?
Choose 1 answer:
(A) Leave it the same
(B) Expand it finitely
(C) Contract it finitely
(D) Contract it infinitely

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the determinant of the Jacobian matrix $J(f)$, we use the formula for the determinant of a 3x3 matrix. ‖ ‖ step 2 ⋮ The determinant of $J(f)$ is calculated as follows: $|J(f)| = -1 \cdot (0 \cdot 1 - 0 \cdot 0) - 0 \cdot (3 \cdot 1 - 0 \cdot 0) + (-1) \cdot (3 \cdot \cos(y) - 0 \cdot 0)$. ‖ ‖ step 3 ⋮ Simplifying the expression, we get $|J(f)| = -1 \cdot 0 - 0 + (-1) \cdot (3 \cdot \cos(y))$. ‖ ‖ step 4 ⋮ Substituting $y = \pi$, we have $\cos(\pi) = -1$, so $|J(f)| = -1 \cdot (3 \cdot -1) = 3$. ‖ ‖ step 5 ⋮ Since the determinant is positive and non-zero, the transformation $f$ will expand space around the point $(0, \pi, 0)$ finitely. ‖ ***B*** ∻Key Concept∻ ⚹Determinant of a Jacobian matrix⚹ ∻Explanation∻ ⚹The determinant of the Jacobian matrix at a point gives us information about the behavior of the transformation at that point. A positive determinant indicates an expansion, while a negative determinant indicates a contraction. The magnitude of the determinant suggests the extent of the expansion or contraction.⚹◊What is the formula for calculating the determinant of a 3x3 matrix?⍭ Generate me a similar question◊

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