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Calculus/Cal II. Vectors
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Question
Describe the region of R^3 represented by the inequality
x^2+y^2+z^2>2z
The equation should be transformed into what I can understand.
The eqatuion x² + y² + z² = 2z needs to be looked at to see
where the boundary is. To do this, subtract 2z from both sides.
This gives x² + y² + z² - 2z = 0.
Also, we need to add 1 to both sides to complete the square in z.
This would be x² + y² + z² - 2z + 1 = 1.
Now it can be rewritten as x² + y² + (z-1)² = 1.
Aha! Now I see what the boundary is.
It is a sphere, centered at (0,0,1) with radius 1.
What we really have is then x² + y² + (z-1)² > 1,
so the answer is all points outside the sphere
defined by the equation x² + y² + (z-1)² = 1.
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