Surfaces, surface area and surface integrals Our main objective here - PowerPoint PPT Presentation

Surfaces, surface area and surface integrals Our main objective here is the construction of surface integrals . Surface integrals are essential for stating in completely general terms the basic laws of electromagnetism. Also, the theorems of Gauss-Ostrogradskii and Stokes on vector calculus rely in an essential way on surface integrals. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 1/43

Parametric representation of surfaces As an example, let’s consider the surface of the north hemisphere of radius r > 0 in R 3 centred at the origin. The points of the surface are defined by: x 2 + y 2 + z 2 = r 2 , z ≥ 0 . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 2/43

Parametric representation of surfaces (cont.) Alternatively, the same surface can be traced out by the point ( x, y, f ( x, y )), when x and y vary across the domain ( x, y ) ∈ R 2 | x 2 + y 2 ≤ r 2 � ⊂ R 2 � D = xy , while f : D → R is defined as r 2 − x 2 − y 2 , � f ( x, y ) = ∀ ( x, y ) ∈ D. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 3/43

Parametric representation of surfaces (cont.) The previous example can be generalized in the following way. Suppose that f : D → R is a given continuous function defined on D ⊂ R 2 xy in the x − y plane. Then, the graph of the function f defines a surface S given by ( x, y, f ( x, y )) ∈ R 3 | ( x, y ) ∈ D � � S = . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 4/43

Parametric representation of surfaces (cont.) The graphs of functions f : D → R represent an important class of surfaces. Unfortunately, not every surface in R 3 can be represented by the graph of a function . Neither of the above surfaces can be represented by a graph of a single valued function f : D → R for some D ∈ R 2 xy . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 5/43

Parametric surfaces Definition A parametric function for a surface S is a given function or mapping Φ : D → R 3 defined on some given region D in a u − v -plane R 2 uv according to Φ ( u, v ) = ( x ( u, v ) , y ( u, v ) , z ( u, v )) = = x ( u, v ) i + y ( u, v ) j + z ( u, v ) k , ∀ ( u, v ) ∈ D. The surface S of the parametric function is the set of points in R 3 traced by Φ ( u, v ) as ( u, v ) traverses the region D . Φ ( u, v ) ∈ R 3 | ( u, v ) ∈ D � � S := . The region D is defined is called the parametric domain , the variable ( u, v ) is called the parametric variable , and the function Φ : D → R 3 is called a parametric representation of the surface. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 6/43

C 1 surfaces Suppose that the parametric function Φ is a C 1 -function, that is ∂x ( u, v ) , ∂y ( u, v ) , ∂z ( u, v ) , ∂x ( u, v ) , ∂y ( u, v ) , ∂z ( u, v ) ∂u ∂u ∂u ∂v ∂v ∂v exist and are continuous functions of ( u, v ) in D . In this case, the parametric representation Φ : D → R 3 is called a C 1 -parametric representation and the related surface S is called a C 1 -surface . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 7/43

Special case revised Suppose we are given the continuous function f : D → R , where D ⊂ R 2 xy is some region in the x − y plane. We can take the u − v plane R 2 uv (from the definition) to be iden- tical to the x − y plane R 2 xy and define Φ ( u, v ) = u i + v j + f ( u, v ) k = ( u, v, f ( u, v )) , ∀ ( u, v ) ∈ D. In this case, the scalar components of Φ ( u, v ) are given by x ( u, v ) = u, y ( u, v ) = v, z ( u, v ) = f ( u, v ) . Then it is clear that the corresponding surface S coincides with the graph of the function f : D → R . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 8/43

Example: Northern hemisphere The length of OA is the radius r of the sphere. From the right-angle triangle OAE we find OE = r cos( φ ) , OB = AE = r sin( φ ) . From the right-angle triangle OBC we find OC = OB cos( θ ) = r sin( φ ) cos( θ ) , OD = OB sin( θ ) = r sin( φ ) sin( θ ) . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 9/43

Example: Northern hemisphere (cont.) OC , OD and OE give the x , y and z coordinates of the point A in terms of the angles θ and φ so that x ( θ, φ ) = r sin( φ ) cos( θ ) , y ( θ, φ ) = r sin( φ ) sin( θ ) , z ( θ, φ ) = r cos( φ ) . If θ varies through the range 0 ≤ θ ≤ 2 π and φ varies through the range 0 ≤ φ ≤ π/ 2 then the point A having the above coordinates traverses the surface S of the northern hemisphere. In this case, the domain D is defined as D = { ( θ, φ ) | 0 ≤ θ ≤ 2 π & 0 ≤ φ ≤ π/ 2 } = [0 , 2 π ] × [0 , π/ 2] , while the parametric function Φ : D → R 3 is defined as Φ ( θ, φ ) = ( x ( θ, φ ) , y ( θ, φ ) , z ( θ, φ )) = = r sin( φ ) cos( θ ) i + r sin( φ ) sin( θ ) j + r cos( φ ) k . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 10/43

Example: Whole sphere If θ varies through the range 0 ≤ θ ≤ 2 π and φ varies through the range 0 ≤ φ ≤ π then the point A traverses the entire sphere. In this case, the domain D is given by D = { ( θ, φ ) | 0 ≤ θ ≤ 2 π & 0 ≤ φ ≤ π } = [0 , 2 π ] × [0 , π ] , while the parametric function Φ : D → R 3 is still defined as Φ ( θ, φ ) = ( x ( θ, φ ) , y ( θ, φ ) , z ( θ, φ )) = = r sin( φ ) cos( θ ) i + r sin( φ ) sin( θ ) j + r cos( φ ) k . Note that, as opposed to the case of hemisphere, the surface S of the sphere cannot be represented by the graph of a function. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 11/43

Example: Helicoid For a “radius” r in the range 0 ≤ r ≤ 1 and an “angle” θ in the range 0 ≤ θ ≤ 2 π , define Φ ( r, θ ) = ( x ( r, θ ) , y ( r, θ ) , z ( r, θ )) = r cos( θ ) i + r sin( θ ) j + θ k . The parametric mapping Φ : D → R 3 , with D = [0 , 1] × [0 , 2 π ], represents the surface S of a helicoid . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 12/43

Tangents to a surface Suppose that we are given a C 1 -parametric function Φ : D → R 3 with D = [ a, b ] × [ c, d ], and fix some ( u 0 , v 0 ) ∈ D . Then the mapping γ v 0 : [ a, b ] → R 3 defined by γ v 0 ( u ) := Φ ( u, v 0 ) for u ∈ [ a, b ] is the parametric representation of a curve Γ v 0 . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 13/43

Tangents to a surface (cont.) The derivative of the parametric function γ v 0 ( u ) at u = u 0 is given by d γ v 0 du ( u 0 ) = ∂ Φ � ∂x ∂u ( u 0 , v 0 ) , ∂y ∂u ( u 0 , v 0 ) , ∂z � ∂u ( u 0 , v 0 ) = ∂u ( u 0 , v 0 ) . This vector is tangent to the curve Γ v 0 at the point Φ ( u 0 , v 0 ). Similarly, the mapping γ u 0 : [ c, d ] → R 3 with γ u 0 ( v ) := Φ ( u 0 , v ) and v ∈ [ c, d ] is the parametric representation of a curve Γ u 0 . In this case, the vector d γ u 0 dv ( v 0 ) = ∂ Φ � ∂x ∂v ( u 0 , v 0 ) , ∂y ∂v ( u 0 , v 0 ) , ∂z � ∂v ( u 0 , v 0 ) = ∂v ( u 0 , v 0 ) is tangent to the curve Γ u 0 at the point Φ ( u 0 , v 0 ). Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 14/43

Normal vector Define the vector cross product of the vector ∂ Φ ∂u ( u 0 , v 0 ) and the vector ∂ Φ ∂v ( u 0 , v 0 ), that is N ( u 0 , v 0 ) := ∂ Φ ∂u ( u 0 , v 0 ) × ∂ Φ ∂v ( u 0 , v 0 ) . More specifically, � � i j k � � ∂x ∂y ∂z � � N ( u 0 , v 0 ) = ∂u ( u 0 , v 0 ) ∂u ( u 0 , v 0 ) ∂u ( u 0 , v 0 ) = � � � ∂x ∂y ∂z � ∂v ( u 0 , v 0 ) ∂v ( u 0 , v 0 ) ∂v ( u 0 , v 0 ) � � � ∂y ∂u ( u 0 , v 0 ) ∂z ∂v ( u 0 , v 0 ) − ∂y ∂v ( u 0 , v 0 ) ∂z � = i ∂u ( u 0 , v 0 ) � ∂x ∂u ( u 0 , v 0 ) ∂z ∂v ( u 0 , v 0 ) − ∂x ∂v ( u 0 , v 0 ) ∂z � − j ∂u ( u 0 , v 0 ) + � ∂x ∂u ( u 0 , v 0 ) ∂y ∂v ( u 0 , v 0 ) − ∂x ∂v ( u 0 , v 0 ) ∂y � + k ∂u ( u 0 , v 0 ) . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 15/43

Normal vector (cont.) Another way to define the normal vector is N ( u 0 , v 0 ) = i ∂ ( y, z ) ∂ ( u, v )( u 0 , v 0 ) + j ∂ ( z, x ) ∂ ( u, v )( u 0 , v 0 ) + k ∂ ( x, y ) ∂ ( u, v )( u 0 , v 0 ) , where ∂y ∂y ∂ ( y, z ) � � ∂u ( u 0 , v 0 ) ∂v ( u 0 , v 0 ) � � ∂ ( u, v )( u 0 , v 0 ) = � , � ∂z ∂z � ∂u ( u 0 , v 0 ) ∂v ( u 0 , v 0 ) � ∂z ∂z ∂ ( z, x ) � � ∂u ( u 0 , v 0 ) ∂v ( u 0 , v 0 ) � � ∂ ( u, v )( u 0 , v 0 ) = � , � ∂x ∂x � ∂u ( u 0 , v 0 ) ∂v ( u 0 , v 0 ) � ∂x ∂x ∂ ( x, y ) � � ∂u ( u 0 , v 0 ) ∂v ( u 0 , v 0 ) � � ∂ ( u, v )( u 0 , v 0 ) = � . ∂y ∂y � � ∂u ( u 0 , v 0 ) ∂v ( u 0 , v 0 ) � Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 16/43

Smooth surfaces Definition A surface is called smooth at the point Φ ( u 0 , v 0 ) when N ( u 0 , v 0 ) � = 0. The surface is called smooth when it is smooth at Φ ( u, v ) , ∀ ( u, v ) ∈ D , that is N ( u, v ) � = 0 , ∀ ( u, v ) ∈ D . For a smooth surface S , one can define the unit vector N ( u, v ) n ( u, v ) := � N ( u, v ) � , ∀ ( u, v ) ∈ D, where � ∂ ( y, z ) �� ∂ ( x, y ) � 2 � 2 � 2 � ∂ ( z, x ) � N ( u, v ) � := ∂ ( u, v )( u, v ) + ∂ ( u, v )( u, v ) + ∂ ( u, v )( u, v ) . As the tangent vectors span the plane that is tangent to the sur- face at Φ ( u, v ), the unit vector n ( u, v ) is normal to the plane, & therefore also normal to the surface S at Φ ( u, v ). Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 17/43