Scalar and vector fields What is a field in mathematics? Roughly - PowerPoint PPT Presentation

Scalar and vector fields What is a field in mathematics? Roughly speaking, a field defines how a scalar-valued or vector- valued quantity varies through space . We usually work with scalar and vector fields . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 1/17

Motivating examples A flat circular metal plate of radius 1 m is located with its centre at the origin of R 2 and heated with a blow-torch. At each point ( x, y ) of the unit disc, denote the temperature of the disc by T ( x, y ) which is a scalar-valued function . T ( x, y ) is an instance of a sca- lar field defined on a region of two-dimensional space . The above example could be extended to R 3 by replacing the disk with a ball. In this case, we’d have a scalar field T ( x, y, z ) defined over the unit ball in R 3 . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 2/17

Motivating examples (cont.) A positive point charge of Q coul is located at the origin of R 3 . The force exerted on the test charge of 1 coul at the point ( x, y, z ) is defined according to Coulomb’s law of electro- statics, viz. Q � x 2 + y 2 + z 2 . 4 πǫ 0 r 2 , r := This force is denoted by E ( x, y, z ) and it is called the electrostatic field , which is defined for all ( x, y, z ) � = 0 (or R 3 \{ 0 } ). This vector-valued function is an instance of a vector field defined everywhere in three-dimensional space R 3 except for the origin. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 3/17

Motivating examples (cont.) Suppose that some electric charge is continuously distri- buted throughout some fixed region D ⊂ R 3 . For some point ( x, y, z ) ∈ D , define a small sphere of radius 0 < ǫ ≪ 1, volume V ǫ , and en- closed (total) charge Q ǫ . If it exists, the limit ρ ( x, y, z ) = lim ǫ → 0 Q ǫ /V ǫ defines the charge density at the point ( x, y, z ). If the limit exists for each and every point ( x, y, z ) ∈ D , then we have a scalar field ρ ( x, y, z ) defined over D . Effectively, ρ ( x, y, z ) gives the quantity of charge per unit volume concentrated at ( x, y, z ) that is ρ describes the local concentration of charge at each point in the region D . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 4/17

Motivating examples (cont.) Let D = R 3 , so that the charge is spread “everywhere” in space. For an arbitrary region Ω ⊂ R 3 , the total charge enclosed within Ω must be given by � � Q = ρ ( x, y, z ) dxdydz = ρ dV. Ω Ω The above relation can be used to obtain a very important result called the continuity equation which describes the movement of charge through space. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 5/17

Motivating examples (cont.) Suppose that the charge in the fixed region D is in motion . In particular, at each point ( x, y, z ) ∈ D , the charge moves past that point with a velocity v ( x, y, z ) (which is a vector). For each ( x, y, z ) ∈ D define J ( x, y, z ) := ρ ( x, y, z ) v ( x, y, z ) , which is also a vector. This vector-valued function is a vector field defined everywhere in the region D and called the current density field . The dimensions of J are coul m 3 × m m 2 sec = amps coul sec = m 2 Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 6/17

Motivating examples (cont.) Let S be a plane with area A , and let n be the unit vector normal to S . Let us assume for simplicity that ρ ( x, y, z ) and v ( x, y, z ) are constant in space, namely ρ ( x, y, z ) = ρ, v ( x, y, z ) = v , for all ( x, y, z ) ∈ D . Then, the current density field J ( x, y, z ) is position independent and given by J = ρ v . Moreover, if n and v are collinear , then the speed of the charge is given by v = � v � = n · v . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 7/17

Motivating examples (cont.) Now, let us fix some small time interval ∆ t > 0. Since v � n , the total “volume of space” that crosses S in the time ∆ t must be Av ∆ t . Then, the total charge Q which flows across S in the time ∆ t must be equal to Q = ( Av ∆ t ) ρ, when v and n are collinear. The above relation can also be rewritten as Q = ( A ∆ t )( v · n ) ρ. Yet, what happens when v and n are not collinear? Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 8/17

Motivating examples (cont.) Suppose, for example, that v is tangential to S . Then, v is orthogonal to the unit vector n , which implies v · n = 0 . Then, the total charge Q that crosses S in the time ∆ t is Q = ( A ∆ t ) ( v · n ) ρ = 0 . � �� � 0 Intuitively, since the direction of charge movement is along S and not through it, there can be no charge crossing S . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 9/17

Motivating examples (cont.) Finally, suppose v has an ar- betrary orientation. Then, we can express v as v = v 1 + v 2 , where v 1 � n , while v 2 ⊥ n . In this case, the total charge that crosses S in the time ∆ t is given by Q = ( Av 1 ∆ t ) ρ. But v 1 is just the projection of v along n , viz. v 1 = v · n . Therefore, the total charge Q becomes Q = ( A ∆ t )( v · n ) ρ, when the charge velocity v is in a general direction. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 10/17

Motivating examples (cont.) The quantity I = Q/ ∆ t is the total current passing through the surface S , and it can be expressed as I = Q/ ∆ t = A ( n · v ) ρ = A ( ρ v ) · n = A ( J · n ) . Conclusion The total current I through the surface S is the product of the area A of S and the inner product J · n of the current density J with the unit normal n to S . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 11/17

Motivating examples (cont.) Now, let us allow v and ρ (and, hence, J ) to vary in space. Then, for an infinitesimally small surface dS with infinitesimally small area dA , J ( x, y, z ) is effectively constant as ( x, y, z ) varies through dS . Then, the infinitesimal current passing through the infinitesimal surface dS with unit normal n ( x, y, z ) is given by dI = ( J ( x, y, z ) · n ( x, y, z )) dA. Thus, knowing the vector field J ( x, y, z ) we can calculate the cur- rent dI flowing across a small planar surface dS with area dA and unit normal vector n ( x, y, z ) at a point ( x, y, z ) ∈ dS . Later, we shall see that charge and current density are absolutely essential to the formulation of Maxwell’s equations of electromag- netism. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 12/17

Vector and scalar fields Definition A vector field comprises a specified region D ∈ R 3 , called the domain of the vector field , together with a function or mapping F : D → R 3 which assigns to each point ( x, y, z ) ∈ D the vector F ( x, y, z ) ∈ R 3 . The vector field F ( x, y, z ) can be defined in terms of its scalar components F 1 ( x, y, z ), F 2 ( x, y, z ) and F 3 ( x, y, z ) along the co- ordinates x , y , z , respectively. Given the standard i , j , k axes, one has F ( x, y, z ) = F 1 ( x, y, z ) i + F 2 ( x, y, z ) j + F 3 ( x, y, z ) k . Definition A scalar field comprises a specified region D ∈ R 3 , called the domain of the scalar field , together with a function or mapping f : D → R which assigns to each point ( x, y, z ) ∈ D the real number f ( x, y, z ). Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 13/17

Vector and scalar fields (cont.) Definition A vector field F : D → R 3 is called a C 1 -vector field when, for each i = 1 , 2 , 3, the partial derivatives ∂F i ( x, y, z ) , ∂F j ( x, y, z ) , ∂F k ( x, y, z ) ∂x ∂y ∂z all exist and are continuous functions of ( x, y, z ) ∈ D . Definition A vector field F : D → R 3 is called a C 2 -vector field when F is a C 1 vector field and, for each i = 1 , 2 , 3, the partial derivatives ∂ 2 F i ( x, y, z ) , ∂ 2 F j ( x, y, z ) , ∂ 2 F k ( x, y, z ) ∂x 2 ∂y 2 ∂z 2 ∂ 2 F i ( x, y, z ) , ∂ 2 F j ( x, y, z ) , ∂ 2 F k ( x, y, z ) ∂x∂y ∂y∂z ∂x∂z all exist and are continuous functions of ( x, y, z ) ∈ D . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 14/17

Vector and scalar fields (cont.) Definition A scalar field f : D → R is called a C 1 -scalar field when the partial derivatives ∂f ( x, y, z ) , ∂f ( x, y, z ) , ∂f ( x, y, z ) ∂x ∂y ∂z all exist and are continuous functions of ( x, y, z ) ∈ D . Definition A scalar field f : D → R is called a C 2 -scalar field when f is a C 1 scalar field and the partial derivatives ∂ 2 f ( x, y, z ) , ∂ 2 f ( x, y, z ) , ∂ 2 f ( x, y, z ) ∂x 2 ∂y 2 ∂z 2 ∂ 2 f ( x, y, z ) , ∂ 2 f ( x, y, z ) , ∂ 2 f ( x, y, z ) ∂x∂y ∂y∂z ∂x∂z all exist and are continuous functions of ( x, y, z ) ∈ D . Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 15/17

Vector and scalar fields (cont.) A standard result from elementary calculus says that, when F : D → R 3 is a C 2 -vector field, then we always have ∂ 2 F i ( x, y, z ) = ∂ 2 F i ( x, y, z ) , ∂x∂y ∂y∂x ∂ 2 F i ( x, y, z ) = ∂ 2 F i ( x, y, z ) , ∂y∂z ∂z∂y ∂ 2 F i ( x, y, z ) = ∂ 2 F i ( x, y, z ) , ∂x∂z ∂z∂x for i = 1 , 2 , 3. The same rules of exchangeability of the order of differentiation apply to scalar vector fields. Department of ECE, Fall 2014 ECE 206: Advanced Calculus 2 16/17