Find the Jacobian ∂(x, y, z) ∂(u, v, w) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w), then the Jacobian of x, y, and z with respect to u, v, and w is ∂(x, y, z) ∂(u, v, w) = ∂x ∂u ∂x ∂v ∂x ∂w ∂y ∂u ∂y ∂v ∂y ∂w ∂z ∂u ∂z ∂v ∂z ∂w . x = 1 6 (u + v), y = 1 6 (u − v), z = 6uvw

The Jacobian ∂(x, y, z) ∂(u, v, w) for the indicated change of variables

= -3072uv

Step-by-step explanation:

Step :-(i)

Given  x = 1 6 (u + v)  …(i)

  Differentiating equation (i) partially with respective to 'u'

  Differentiating equation (i) partially with respective to 'v'

  Differentiating equation (i)  partially with respective to 'w'

Given  y = 1 6 (u − v) …(ii)

  Differentiating equation (ii) partially with respective to 'u'

 Differentiating equation (ii) partially with respective to 'v'

Differentiating equation (ii)  partially with respective to 'w'

Given   z = 6uvw   ..(iii)

Differentiating equation (iii) partially with respective to 'u'

Differentiating equation (iii) partially with respective to 'v'

Differentiating equation (iii) partially with respective to 'w'

Step :-(ii)

The Jacobian ∂(x, y, z)/ ∂(u, v, w) =

   Determinant       16(-16×6uv-0)-16(16×6uv)+0(0) = - 1536uv-1536uv

                                                                                 = -3072uv

Final answer:-

The Jacobian ∂(x, y, z)/ ∂(u, v, w) = -3072uv

your answer is the last one

step-by-step explanation:

divide the length of a'c' by the length of ac to get a scale factor of two

& then reflect vertically (over the y-axis)

so your answer is dilation by a scale factor of 2 followed by reflection over the y-axis

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step-by-step explanation: