Question

Find the jacobian of the transformation. x = 6v + 6w2, y = 8w + 8u2, z = 2u + 2v2

Answer 1

Answer:

The Jacobian of the transformation J(u, v, w) is equal to 768uvw + 96.

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.

$\displaystyle \left \{ \begin{array}{ccc}x = 6v + 6w^2 \\ y = 8w + 8u^2 \\ z = 2u + 2v^2 \end{array}$

Step 2: Find Jacobian Pt. 1

1. Find  $\displaystyle \frac{\partial x}{\partial u}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial x}{\partial u} = 0$
2. Find  $\displaystyle \frac{\partial y}{\partial u}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial y}{\partial u} = 16u$
3. Find  $\displaystyle \frac{\partial z}{\partial u}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial z}{\partial u} = 2$
4. Find  $\displaystyle \frac{\partial x}{\partial v}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial x}{\partial v} = 6$
5. Find  $\displaystyle \frac{\partial y}{\partial v}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial y}{\partial v} = 0$
6. Find  $\displaystyle \frac{\partial z}{\partial v}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial z}{\partial v} = 4v$
7. Find  $\displaystyle \frac{\partial x}{\partial w}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial x}{\partial w} = 12w$
8. Find  $\displaystyle \frac{\partial y}{\partial w}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial y}{\partial w} = 8$
9. Find  $\displaystyle \frac{\partial z}{\partial w}$  [Derivative Properties and Rules]:
   $\displaystyle \frac{\partial z}{\partial w} = 0$

Step 3: Find Jacobian Pt. 2

1. Substitute in partial derivative values [Jacobian Substitution]:
   $\displaystyle J(u, v, w) = \left| \begin{array}{ccc} 0 & 16u & 2 \\ 6 & 0 & 4v \\ 12w & 8 & 0 \end{array} \right|$
2. [Jacobian] Simplify [3x3 Matrix Determinant]:
   $\displaystyle J(u, v, w) = 0 \left| \begin{array}{ccc} 0 & 4v \\ 8 & 0 \end{array} \right| - 16u \left| \begin{array}{ccc} 6 & 4v \\ 12w & 0 \end{array} \right| + 2 \left| \begin{array}{ccc} 6 & 0 \\ 12w & 8 \end{array} \right|$
3. [Jacobian] Simplify [2x2 Matrix Determinant]:
   $\displaystyle J(u, v, w) = 0 \bigg[ 0(0) - 4v(8) \bigg] - 16u \bigg[ 6(0) - 4v(12w) \bigg] + 2 \bigg[ 6(8) - 0(12w) \bigg]$
4. [Jacobian] Simplify:
   $\displaystyle J(u, v, w) = 768uvw + 96$

---

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

---

Topic: Multivariable Calculus

Unit: Triple Integral Applications