Multivariable Calculus: Curves, Surfaces, and 3D Shapes

Curve

r(t) =

Surface

r(u,v) =

Line Segment A → B

r(t) = (1−t)A + tB, 0 ≤ t ≤ 1

Circle (radius r)

r(t) =

Ellipse

r(t) =

Parabola

r(t) =

Helix

r(t) =

Plane

ax + by + cz = d → r(u,v) =

Graph z = f(x,y)

r(u,v) =

Vertical Surface x = g(y,z)

r(u,v) =

Cylinder x² + y² = r²

r(θ,z) =

Cone x² + y² = z²

r(r,θ) =

Sphere radius R

r(φ,θ) =

Ellipsoid

r(φ,θ) =

Elliptic Paraboloid z = x² + y²

r(r,θ) =

Hyperbolic Paraboloid

r(u,v) =

Hyperboloid (1 sheet)

r(u,v) =

Hyperboloid (2 sheets)

r(u,v) =

Torus

r(u,v) = <(R + r cos v) cos u, (R + r cos v) sin u, r sin v>

Normal Vector

n = r_u × r_v

Swap order

flips orientation

Outward normals for flux integrals