Question

Find the jacobian of the transformation. x = 4u/v, y = 6v/w, z = 7w/u

Answer 1

$\dfrac{\partial(x,y,z)}{\partial(u,v,w)}=\begin{bmatrix}x_u&x_v&x_w\\y_u&y_v&y_w\\z_u&z_v&z_w\end{bmatrix}=\begin{bmatrix}\dfrac4v&-\dfrac{4u}{v^2}&0\\\\0&\dfrac6w&-\dfrac{6v}{w^2}\\\\-\dfrac{7w}{u^2}&0&\dfrac7u\end{bmatrix}$

Answer 2

Answer:

The Jacobian of the transformation J(u, v, w) is equal to 0.

General Formulas and Concepts:
Pre-Calculus

Matrices

2x2 Matrix Determinant:
$\displaystyle \left| \begin{array}{ccc} a & b \\ c & d \end{array} \right| = ad - bc$

3x3 Matrix Determinant:
$\displaystyle \left| \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right| = a \left| \begin{array}{ccc} e & f \\ h & i \end{array} \right| - b \left| \begin{array}{ccc} d & f \\ g & i \end{array} \right| + c \left| \begin{array}{ccc} d & e \\ g & h \end{array} \right|$

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
$\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:
$\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Derivative Rule [Basic Power Rule]:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Multivariable Calculus

Partial Derivatives

Integration

Jacobian Substitution:
$\displaystyle J(u, v, w) = \frac{\partial (x, y, z)}{\partial (u, v, w)} = \left| \begin{array}{ccc} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \\ \frac{\partial x}{\partial w} & \frac{\partial x}{\partial w} & \frac{\partial x}{\partial w} \end{array} \right|$

Step-by-step explanation:

Step 1: Define

Identify.

$\displaystyle \left \{ \begin{array}{ccc} x = \frac{4u}{v} \\ y = \frac{6v}{w} \\ z = \frac{7w}{u} \end{array}$

Step 2: Find Jacobian Pt. 1

1. Find  $\displaystyle \frac{\partial x}{\partial u}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial x}{\partial u} = \frac{4}{v}$
2. Find  $\displaystyle \frac{\partial y}{\partial u}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial y}{\partial u} = 0$
3. Find  $\displaystyle \frac{\partial z}{\partial u}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial z}{\partial u} = \frac{-7w}{u^2}$
4. Find  $\displaystyle \frac{\partial x}{\partial v}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial x}{\partial v} = \frac{-4u}{v^2}$
5. Find  $\displaystyle \frac{\partial y}{\partial v}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial y}{\partial v} = \frac{6}{w}$
6. Find  $\displaystyle \frac{\partial z}{\partial v}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial z}{\partial v} = 0$
7. Find  $\displaystyle \frac{\partial x}{\partial w}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial x}{\partial w} = 0$
8. Find  $\displaystyle \frac{\partial y}{\partial w}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial y}{\partial w} = \frac{-6v}{w^2}$
9. Find  $\displaystyle \frac{\partial z}{\partial w}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial z}{\partial w} = \frac{7}{u}$

Step 3: Find Jacobian Pt. 2

1. Substitute in partial derivative values [Jacobian Substitution]:
   $\displaystyle J(u, v, w) = \left| \begin{array}{ccc} \frac{4}{v} & \frac{-4u}{v^2} & 0 \\ 0 & \frac{6}{w} & \frac{-6v}{w^2} \\ \frac{-7w}{u^2} & 0 & \frac{7}{u} \end{array} \right|$
2. [Jacobian] Simplify [3x3 Matrix Determinant]:
   $\displaystyle J(u, v, w) = \frac{4}{v} \left| \begin{array}{ccc} \frac{6}{w} & \frac{-6v}{w^2} \\ 0 & \frac{7}{u} \end{array} \right| + \frac{4u}{v^2} \left| \begin{array}{ccc} 0 & \frac{-6v}{w^2} \\ \frac{-7w}{u^2} & \frac{7}{u} \end{array} \right| + 0 \left| \begin{array}{ccc} 0 & \frac{6}{w} \\ \frac{-7w}{u^2} & 0 \end{array} \right|$
3. [Jacobian] Simplify:
   $\displaystyle J(u, v, w) = \frac{4}{v} \left| \begin{array}{ccc} \frac{6}{w} & \frac{-6v}{w^2} \\ 0 & \frac{7}{u} \end{array} \right| + \frac{4u}{v^2} \left| \begin{array}{ccc} 0 & \frac{-6v}{w^2} \\ \frac{-7w}{u^2} & \frac{7}{u} \end{array} \right|$
4. [Jacobian] Simplify [2x2 Matrix Determinant]:
   $\displaystyle J(u, v, w) = \frac{4}{v} \bigg[ \frac{6}{w} \bigg( \frac{7}{u} \bigg) - \frac{-6v}{w^2} (0) \bigg] + \frac{4u}{v^2} \bigg[ 0 \bigg( \frac{7}{u} \bigg) + \frac{6v}{w^2} \bigg( \frac{-7w}{u^2} \bigg) \bigg]$
5. [Jacobian] Simplify:
   $\displaystyle J(u, v, w) = 0$

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Learn more about Jacobian substitution: https://brainly.com/question/9381576
Learn more about multivariable calculus: https://brainly.com/question/17203772

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Topic: Multivariable Calculus

Unit: Triple Integral Applications