16.6 Vector Calculus

for CALC III UIUC

Parametric Surface

The set of all points (x,y,z) in R^3 such that x = x(u, v), y=y(u, v), z=z(u, v)

Parametric Equations

x = x(u,v), y = y(u,v), z = z(u,v)
Such that:
the vector equation r(x,y) = x(u,v)i + y(u,v)j + z(u,v)k

Grid Curves

If v is kept constant by putting v=v_0, we get a curve C given r(u, v_0) that lies on S.

Surfaces of Revolution

a surface obtained by rotating a curve about an axis ( can be represented parametrically)

Tangent Planes

IF IS A SMOOTH SURFACE
1. find tangent vectors
2. find normal vector to tangent plane
3. plug in point to normal vector (replacing i,j,k with x-x_0, etc.)

Smooth Surface

the surface S is called smooth if the cross product of the tangent vectors is not 0 (ie. ru X rv!= 0)

Surface Area

Surface area of a smooth Surface S:

Surface Area of the Graph of a Function

For the special case of a surface S with equation z = f(x,y) where (x,y) ;ies in D and f has continuous partial derivatives: