Summary of Multivariable Calculus MVC Lecture 10 for MATH2310 Engineering Calculus

MATH2310 – MVC Lecture 10


Surface integrals:
The relationship between surface integrals and surface area is analogous to the
relationship between line integrals and arc length.
For a parametric surface, S, where S is the graph of a function of 2 variables and has
vector equation r : R2 → R3 given by:

r ( u , v )= ⟨ x (u , v )+ y ( u , v )+ z (u , v) ⟩

we can write:

ru =
∂r
∂u
= ⟨
∂x ∂ y ∂ z
, ,
∂ u ∂ u ∂u
∂r
, rv= =
∂v
, ⟩
∂x ∂ y ∂z
,
∂v ∂v ∂ v ⟨ ⟩
For S,
❑ ❑

∬ f ( x , y , z ) dS=∬ f ( r ( u , v ) )|ru ×r v| dA
S D


The surface area of S is given by:
❑ ❑
SA ( S )=∬ 1 dS=∬|r u ×r v| dA
S D




Application of line and surface integrals:
An example of an application of line integrals is determining the centre of mass of an
object given the density and an equation defining the mass with two variables.
Similarly, the centre of mass of an object defined using three variables can be
determined using a surface integral.



Specialising to graphs of functions:
Any surface S, given by z=g ( x , y ) can be regarded as a parametrised surface:

r ( u , v )= ⟨ u , v , g (u , v ) ⟩

with:


⟨
r u = 1,0 ,
∂g
∂u ⟩ ⟨
,r v = 0 , 1,
∂g
∂v ⟩