Chapter 6: Surface Integrals Roberto S. Costas-Santos May 2013 - - PowerPoint PPT Presentation

Parameterized Surfaces

Chapter 6: Surface Integrals

Roberto S. Costas-Santos May 2013

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces

Outline

1

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

The Basics

A surface is an application c : I ⊆ R2 → Rn. S(u, v) = (x1(u, v), x2(u, v), . . . , xn(u, v)) u, v are the independent variables, and xi are the components

• f the surface.

If we have z = f (x, y), with f (x, y) good enough on an open set Ω ⊆ R2, then we can parameterize the defined surface by writing S(x, y) : (x, y, f (x, y)), (x, y) ∈ Ω.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

The Basics

A surface is an application c : I ⊆ R2 → Rn. S(u, v) = (x1(u, v), x2(u, v), . . . , xn(u, v)) u, v are the independent variables, and xi are the components

• f the surface.

If we have z = f (x, y), with f (x, y) good enough on an open set Ω ⊆ R2, then we can parameterize the defined surface by writing S(x, y) : (x, y, f (x, y)), (x, y) ∈ Ω.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

The Basics

A surface is an application c : I ⊆ R2 → Rn. S(u, v) = (x1(u, v), x2(u, v), . . . , xn(u, v)) u, v are the independent variables, and xi are the components

• f the surface.

If we have z = f (x, y), with f (x, y) good enough on an open set Ω ⊆ R2, then we can parameterize the defined surface by writing S(x, y) : (x, y, f (x, y)), (x, y) ∈ Ω.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

The Basics

A surface is an application c : I ⊆ R2 → Rn. S(u, v) = (x1(u, v), x2(u, v), . . . , xn(u, v)) u, v are the independent variables, and xi are the components

• f the surface.

If we have z = f (x, y), with f (x, y) good enough on an open set Ω ⊆ R2, then we can parameterize the defined surface by writing S(x, y) : (x, y, f (x, y)), (x, y) ∈ Ω.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

The tangent and normal vectors of a surface

Given a parameterized surface S(u, v) = (x(u, v), y(u, v), z(u, v)), (u, v) ∈ Ω, we define the vectors

• Su(u, v) =

∂x

∂u , ∂y ∂u , ∂z ∂u

• = ∂x

∂u i + ∂y ∂u j + ∂z ∂u k, and

• Sv(u, v) =

∂x

∂v , ∂y ∂v , ∂z ∂v

• = ∂x

∂v i + ∂y ∂v j + ∂z ∂v k.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

Normal and tangent vectors

Given a particular value of u and v, we have a particular point, namely P, on S. In fact Both vectors, Su and Sv, are in the tangent plane of the surface at the point P. And a normal vector

• f the surface at the point is the vectorial product

Su and Sv, i.e.

• N(u, v) = (

Su × Sv)(u, v) =

• i

j k

• Su(u, v)
• Sv(u, v)
• .

If at any point we can choose N so that it changes continuously on S, then we say that S is oriented. Orientation: if the normal vector at any point is outward then the surface is positive oriented. No every surface is oriented: a classical example is the M¨

• ebius
• strip. http://goo.gl/TrQh3

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

Area of a parameterised surface

Let S(u, v) = (x(u, v), y(u, v), z(u, v)) a parameterised surface in R3, with (u, v) ∈ Ω ⊆ R2. The area of such surface is defined by A(S) =

• Ω
• N(u, v) dudv.

If the parameterised surface is of the form: S(x, y) = (x, y, f (x, y)), with (x, y) ∈ D ⊆ R2, then Su(x, y) = (1, 0, fx), Sy(x, y) = (1, 0, fy), so A(S) =

• D
• 1 + f 2

x + f 2 y dxdy.

Here fx and fy are the respective partials derivatives with respect to x and y.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

Surface integral

As before to compute the surface integral of a function over it we integrate 1 over it. Now we will see how to integrate a scalar function over a surface. Let S be a parameterised surface given by S(u, v) with (u, v) ∈ ω. Given a continuous scalar function g(x, y, z) from R3 to R, we define the integral of the function g over S as

• S

g dS =

• Ω

g(S(u, v)) N(u, v) dudv. If S is the graph of the function z = f (x, y), with (x, y) ∈ D, and we want to integrate over S the function g(x, y, z), then we need to compute

• S

g dS =

• D

g(S(x, y, f (x, y)))

• 1 + f 2

x + f 2 y dxdy.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

Surface integral. Interpretation

Interpretation: let us imagine surface S as a very thin lamina of some material and the function g(x, y, z) as the mass (charge, or

• ther) superficial density of such lamina.

Then

• S g dS give us the total mass (charge, ...) of such lamina S

with superficial density g. Other possible applications are: To compute the mean of some physical magnitud of a surface, To compute the mass center, the inertia momentum of a surface, etc, with variable density.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals

Parameterized Surfaces The Basics The tangent and normal vectors of a surface Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field

Surface integral of vectorial field

Let F(x, y, z) = (F1(x, y, z), F2(x, y, z), F3(x, y, z)) a vectorial field defined on S. The integral of F on S, or flow of F through the surface S given S(u, v) is:

• S
• F d

S =

• Ω
• F(S(u, v))

N(u, v) dudv. Remark: We need to be aware about the sign of such integrals since the sign of such value depends on the orientation of the normal through the surface.

http://rscosan.com/docencia.html Chapter 6: Surface Integrals