Describe the level surfaces of the following functions. f(x, y, z) = x 2 − y 2 − z 2 .
Answers
Answer:
The level curves verify
x2−y2+z2=A
x2−y2+z2=A
for some AA, or equivalently
y2−r2=−A
y2−r2=−A
where r2=x2+z2r2=x2+z2. This means that yy depends on xx and zz only through rr, which is the distance from the point (x,y,z)(x,y,z) to the YY axis. Therefore, the surface is rotationally symmetric about the YY axis.
There are three cases depending on the sign of AA. Nice plots of the three possibilities have been provided by Raffaele in their answer.
If AA is positive, let A=a2A=a2, and we have
r2a2−y2a2=1
r2a2−y2a2=1
which is the equation of a hyperbola with asymptotes r=±yr=±y and vertices at (r=±a,y=0)(r=±a,y=0). The surface is this hyperbola rotated about the YY axis.
If AA is 00, then we have the degenerate case y2=r2y2=r2, or y=±ry=±r, which is a cone along the YY axis.
If AA is negative, let A=−a2A=−a2, and we have
y2a2−r2a2=1
y2a2−r2a2=1
which is the equation of a hyperbola with asymptotes r=±yr=±y and vertices at (r=0,y=±a)(r=0,y=±a). This is the same hyperbola as in the positive AA case, but with the YY and RR axes swapped. The surface is this hyperbola rotated about the YY axis.
Answer:
f(x, y, z) = x2 − y2 − z2 a.) The level surfaces are a family of hyperboloids. ... The level surfaces are a family of parallel planes.