Calc III Exam 3

To find line of intersection of two planes

parametrize both equations with parameters that fulfill both

vector equation for a straight line

Unit Tangent Vector

Unit Normal Vector

Plane formed by Unit Tangent and Unit Normal Vectors

Osculating Plane

Arc Length Parametric

∫(a,b)|r’(t)|dt

Curvature

With scalar line integrals, what is ds =

ds = |r’(t)| dt = √(dx)²+(dy)²+(dz)² dt

How to parametrize a normal surface with given equation

get one variable in terms of two others

How to parametrize a surface given three points

• r(u,v) = point_1 + (vector_to_point_2)u + (vector_to_point_3)v

• combine

• x = i hat

• y = j hat

• z = k hat
  

How to parametrize the surface of a sphere

x=ρsinφcosθ
y=ρsinφsinθ

z=ρcosφ

How to parametrize “circular” shapes

usually use:

x=rcosθ

y=rsinθ

Surface area parametric

*****Jacobian is taken care of!!****

Scalar Surface Integral (Fundamental Plane)

just a double integral once you input the one variable’s constant value

Scalar Surface Integral (General)

Surface Area as surface integral = ?

surface integral of 1 (still include magnitude of cross product etc.)