Parameterized Surfaces Chapter 6: Surface Integrals Roberto S. Costas-Santos May 2013 http://rscosan.com/docencia.html Chapter 6: Surface Integrals
Parameterized Surfaces Outline Parameterized Surfaces 1 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
The Basics The tangent and normal vectors of a surface Parameterized Surfaces Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field The Basics A surface is an application c : I ⊆ R 2 → R n . S ( u , v ) = ( x 1 ( u , v ) , x 2 ( u , v ) , . . . , x n ( u , v )) u , v are the independent variables, and x i are the components of the surface. If we have z = f ( x , y ) , with f ( x , y ) good enough on an open set Ω ⊆ R 2 , 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
The Basics The tangent and normal vectors of a surface Parameterized Surfaces Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field The Basics A surface is an application c : I ⊆ R 2 → R n . S ( u , v ) = ( x 1 ( u , v ) , x 2 ( u , v ) , . . . , x n ( u , v )) u , v are the independent variables, and x i are the components of the surface. If we have z = f ( x , y ) , with f ( x , y ) good enough on an open set Ω ⊆ R 2 , 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
The Basics The tangent and normal vectors of a surface Parameterized Surfaces Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field The Basics A surface is an application c : I ⊆ R 2 → R n . S ( u , v ) = ( x 1 ( u , v ) , x 2 ( u , v ) , . . . , x n ( u , v )) u , v are the independent variables, and x i are the components of the surface. If we have z = f ( x , y ) , with f ( x , y ) good enough on an open set Ω ⊆ R 2 , 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
The Basics The tangent and normal vectors of a surface Parameterized Surfaces Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field The Basics A surface is an application c : I ⊆ R 2 → R n . S ( u , v ) = ( x 1 ( u , v ) , x 2 ( u , v ) , . . . , x n ( u , v )) u , v are the independent variables, and x i are the components of the surface. If we have z = f ( x , y ) , with f ( x , y ) good enough on an open set Ω ⊆ R 2 , 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
The Basics The tangent and normal vectors of a surface Parameterized Surfaces 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 � ∂ x ∂ u , ∂ y ∂ u , ∂ z = ∂ x ∂ u i + ∂ y ∂ u j + ∂ z � � S u ( u , v ) = ∂ u k , ∂ u and � ∂ x ∂ v , ∂ y ∂ v , ∂ z � = ∂ x ∂ v i + ∂ y ∂ v j + ∂ z � S v ( u , v ) = ∂ v k . ∂ v http://rscosan.com/docencia.html Chapter 6: Surface Integrals
The Basics The tangent and normal vectors of a surface Parameterized Surfaces 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, � S u and � S v , are in the tangent plane of the surface at the point P . And a normal vector of the surface at the point is the vectorial product � S u and � S v , i.e. � � i j k � � � � N ( u , v ) = ( � � S u × � � S v )( u , v ) = S u ( u , v ) . � � � � � � S v ( 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¨ oebius strip. http://goo.gl/TrQh3 http://rscosan.com/docencia.html Chapter 6: Surface Integrals
The Basics The tangent and normal vectors of a surface Parameterized Surfaces 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 R 3 , with ( u , v ) ∈ Ω ⊆ R 2 . The area of such surface is defined by �� � � A ( S ) = N ( u , v ) � dudv . Ω If the parameterised surface is of the form: S ( x , y ) = ( x , y , f ( x , y )) , with ( x , y ) ∈ D ⊆ R 2 , then S u ( x , y ) = ( 1 , 0 , f x ) , S y ( x , y ) = ( 1 , 0 , f y ) , so �� � A ( S ) = 1 + f 2 x + f 2 y dxdy . D Here f x and f y are the respective partials derivatives with respect to x and y . http://rscosan.com/docencia.html Chapter 6: Surface Integrals
The Basics The tangent and normal vectors of a surface Parameterized Surfaces 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 R 3 to R , we define the integral of the function g over S as �� �� g ( S ( u , v )) � � g dS = N ( u , v ) � dudv . S Ω 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 �� �� � g dS = g ( S ( x , y , f ( x , y ))) 1 + f 2 x + f 2 y dxdy . S D http://rscosan.com/docencia.html Chapter 6: Surface Integrals
The Basics The tangent and normal vectors of a surface Parameterized Surfaces 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 other) 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
The Basics The tangent and normal vectors of a surface Parameterized Surfaces Area of a parameterised surface Surface integral. Interpretation Surface integral of vectorial field Surface integral of vectorial field Let � F ( x , y , z ) = ( F 1 ( x , y , z ) , F 2 ( x , y , z ) , F 3 ( 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: �� �� � F � d � F ( S ( u , v )) � � � S = N ( u , v ) dudv . S ٠Remark: 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
Recommend
More recommend
