Explanation Given information : The equation of quadratic surface, x 2 + y 2 + z 2 = 1 . Graph : The graph of the quadratic surface x 2 + y 2 + z 2 = 1 can be sketched using the traces, For the plane, z = 0 , the trace of the surfaces is, x 2 + y 2 = 1 which is a circle in a two dimensional. If we take any plane z = k , the trace of the surface is, x 2 + y 2 + k 2 = 1 x 2 + y 2 = 1 − k 2 Which is again a circle for k 2 ≤ 1 since, right hand side of the equation is a constant. Again for any arbitrary plane y = k , the trace of the surface is, x 2 + k 2 + z 2 = 1 x 2 + z 2 = 1 − k 2 Which is again a circle for k 2 ≤ 1 since, right hand side of the equation is a constant. Again for any arbitrary plane x = k , the trace of the surface is, k 2 + y 2 + z 2 = 1 y 2 + z 2 = 1 − k 2 Which is again a circle for k 2 ≤ 1 since, right hand side of the equation is a constant. Since, all traces are circles, the quadratic surface will be a sphere and graph is provided below in a 3 dimensional plane, Interpretation : The general equations of all quadratic surfaces along with the description of their traces is provided below, The standard equation of the sphere is x 2 + y 2 + z 2 = r 2 , where r is the radius of the sphere. The standard equation of ellipsoid is x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 , where a , b and c are constants. Ellipse shaped traces are found in coordinate planes. The standard equation of elliptic paraboloidis z = x 2 a 2 + y 2 b 2 , where a and b are constants. Parabola shaped traces are found in x z and y z coordinate planes. Trace is just a point in x y coordinate plane

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 9.7, Problem 29PS

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To graph: The quadratic surface x2+y2+z2=1 .

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Chapter 9.7, Problem 28PS

Chapter 9.7, Problem 30PS

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Chapter 9 Solutions

Calculus

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To graph : The quadratic surface x 2 + y 2 + z 2 = 1 .

Solution Summary: The author illustrates how the quadratic surface can be sketched using traces.

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To graph: The quadratic surface x2+y2+z2=1 .

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Chapter 9 Solutions