Determine the volume of the rectangular prism.
2 answers:
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Answer:
The Jacobian ∂(x, y, z) ∂(u, v, w) for the indicated change of variables
= -3072uv
Step-by-step explanation:
<u>Step :-(i)</u>
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'
<u>Step :-(ii)</u>
The Jacobian ∂(x, y, z)/ ∂(u, v, w) =
Determinant 16(-16×6uv-0)-16(16×6uv)+0(0) = - 1536uv-1536uv
= -3072uv
<u>Final answer</u>:-
The Jacobian ∂(x, y, z)/ ∂(u, v, w) = -3072uv
<h3>Answers:</h3>
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Explanation:
The scale factor is 2, which means we double each coordinate of each point. The general rule is with k = 2. So we can say the more specific dilation rule is
Something like W(-2,2) moves to W ' (-4, 4) after multiplying each coordinate by 2. Do the same for the other points as well.
The given preimage points
- V = (-2, -1)
- W = (-2, 2)
- X = (0, 2)
- Y = (3, -1)
will dilate to the corresponding image points
- V ' = (-4, -2)
- W ' = (-4, 4)
- X ' = (0, 4)
- Y ' = (6, -2)
as shown below. This causes the image to be larger compared to the preimage (since the scale factor is larger than 1). Any given point is twice as far from the origin as compared to before, which in turn means the distance between any two points is twice as much.