calc bc cylinders and quadric surfaces

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Calculus

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29 Terms

1
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<p>Ellipsoid</p>

Ellipsoid

All squared, all positive, equals 1

<p>All squared, all positive, equals 1</p>
2
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<p>Hyperboloid of One Sheet</p>

Hyperboloid of One Sheet

All squared, ONE negative, equals 1 (on negative’s axis)

<p>All squared, ONE negative, equals 1 (on negative’s axis)</p>
3
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<p>Hyperboloid of Two Sheets</p>

Hyperboloid of Two Sheets

All squared, TWO negatives, equals 1

<p>All squared, TWO negatives, equals 1</p>
4
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<p>Elliptic Cone</p>

Elliptic Cone

All squared, ONE negative, equals zero

<p>All squared, ONE negative, equals zero</p>
5
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<p>Elliptic Paraboloid</p>

Elliptic Paraboloid

TWO squared, all positive, equals non-squared variable

<p>TWO squared, all positive, equals non-squared variable</p>
6
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<p>Hyperbolic Paraboloid</p>

Hyperbolic Paraboloid

TWO squared, ONE squared negative, equal to non-squared variable

7
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Rectangular/Cartesian coordinates

(x, y, z)

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Cylindrical coordinates

(r, θ, z)

r = radius/distance from origin (xy axis, 2D)

θ = angle

z = height

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Spherical coordinates

( p, θ, Φ)

p = distance from origin (3D)

θ = angle

Φ = angle from z-axis

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find x (cylindrical to rectangular)

r cosθ

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find y (cylindrical to rectangular)

r sinθ

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find z (cylindrical or rectangular)

z

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find r (rectangular to cylindrical)

sqrt(x² + y²)

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find θ (rectangular to cylindrical)

tan-1 (y/x)

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find x (spherical to rectangular)

p sinΦ cosTHETA

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find y (spherical to rectangular)

p sinΦ sinTHETA

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find z (spherical to rectangular)

p cos Φ

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find p (rectangular to spherical)

sqrt (x² + y² + z²)

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find Φ (rectangular to spherical)

cos-1 (z/ sqrt(x² + y² + z²) )

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find p (cylindrical to spherical)

r / sinTHETA

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find r (spherical to cylindrical)

p sinΦ

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find r (spherical to cylindrical)

p sinΦ

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find z (spherical to cylindrical)

p cosΦ

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what shape?

p = n

sphere

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what shape?

THETA = pi/n

plane orthogonal to xy plane going through origin

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what shape?

Φ = pi/n

elliptic cone (half)

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what shape?

Φ = +— pi/n

elliptic cone (full)

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what shape?

r = n

right circular cylinder (toilet paper roll)

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what shape?

z = n

plane parallel to xy plane, up n units