Parametric Curves, Quadric Surfaces, and Coordinate Transformations (Calc 3, Lecture 8)

Flashcards covering quadric surfaces (cone, hyperboloids, ellipsoid, paraboloids, cylinder), parametric tangents, and coordinate transformations with Jacobians (polar, cylindrical, spherical).

What is the standard equation of a circular cone with vertex at the origin along the z-axis?

x^2 + y^2 = z^2

Hyperboloid of one sheet: standard form (centered at the origin, axis along z)?

x^2/a^2 + y^2/b^2 - z^2/c^2 = 1

Hyperboloid of two sheets: standard form?

z^2/c^2 - x^2/a^2 - y^2/b^2 = 1

Ellipsoid: standard form?

x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

Elliptic paraboloid: standard equation?

z = x^2/a^2 + y^2/b^2

Hyperbolic paraboloid (saddle shape): standard equation?

z = x^2/a^2 - y^2/b^2

Cylinder (circular cylinder along z): standard equation?

x^2 + y^2 = r^2

Parametric curve tangents: tangent vector at t0

If r(t) = ⟨x(t), y(t), z(t)⟩, then the tangent vector at t0 is r′(t0).

Rectangular to polar coordinates (2D): transformation and Jacobian

x = r cos θ, y = r sin θ; J(rect → polar) = r.

Rectangular to cylindrical coordinates (3D): transformation and Jacobian

x = r cos θ, y = r sin θ, z = z; J(rect → cylindrical) = r.

Spherical coordinates transformation and Jacobian

x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ; J(rect → spherical) = ρ^2 sin φ.