conic sections -...

1
Math 21a: Multivariable calculus Fall 2019 Conic sections Ellipse Parabola Hyperbola Specials x 2 + y 2 =1 y = x 2 x 2 - y 2 =1 x 2 = y 2 Quadrics Ellipsoid Paraboloid Hyperboloid Specials x 2 + y 2 + z 2 =1 z = x 2 + y 2 z = x 2 - y 2 x 2 + y 2 - z 2 =1 x 2 + y 2 - z 2 = -1 x 2 + y 2 =1 x 2 + y 2 = z 2 Advise You will have to know the quadrics in the exam. Here are some pointers: There is no need to memorize the quadrics. You can derive them: look at the traces (put one of the variables to zero) to see what conic section you get. The name usually reveals what the surface is: elliptic paraboloids contain ellipses and parabola, hyperbolic paraboloids contain hyperbola and parabola as traces. The paraboloids can be written as graphs z = f (x, y). This is not possible for ellipsoids or hyperboloids. The special surfaces are non-generic but useful and important. The cone or cylinder appear a lot in applications. Make sure to recognize the surfaces also if the variables are turned, scaled shifted or signs switched. 2x 2 - (y - 5) 2 - 4z 2 = -1 for example is a one-sheeted hyperboloid.

Upload: others

Post on 12-Feb-2020

3 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Conic sections - people.math.harvard.edupeople.math.harvard.edu/~knill/teaching/math21a2019/04-quadrics.pdfEllipsoid Paraboloid Hyperboloid Specials x2 +y 2+z = 1 z = x2 +y 2 z = x

Math 21a: Multivariable calculus Fall 2019

Conic sections

Ellipse Parabola Hyperbola Specials

x2 + y2 = 1 y = x2 x2 − y2 = 1 x2 = y2

Quadrics

Ellipsoid Paraboloid Hyperboloid Specials

x2 + y2 + z2 = 1z = x2 + y2

z = x2 − y2x2 + y2 − z2 = 1x2 + y2 − z2 = −1

x2 + y2 = 1x2 + y2 = z2

Advise

You will have to know the quadrics in the exam. Here are some pointers:

• There is no need to memorize the quadrics. You can derive them: look at the traces (putone of the variables to zero) to see what conic section you get.

• The name usually reveals what the surface is: elliptic paraboloids contain ellipses andparabola, hyperbolic paraboloids contain hyperbola and parabola as traces.

• The paraboloids can be written as graphs z = f(x, y). This is not possible for ellipsoids orhyperboloids.

• The special surfaces are non-generic but useful and important. The cone or cylinder appeara lot in applications.

• Make sure to recognize the surfaces also if the variables are turned, scaled shifted or signsswitched. 2x2 − (y − 5)2 − 4z2 = −1 for example is a one-sheeted hyperboloid.

1