Scalar and vector fields

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

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 scalar 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

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 electrostatics viz. Q 4πɛ 0 r 2 r := x 2 + y 2 + z 2. 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

Suppose that some electric charge is continuously distributed throughout some fixed region D R 3. For some point (x y z) D define a small sphere of radius 0 < ɛ 1 volume V ɛ and enclosed (total) charge Q ɛ. If it exists the limit ρ(x y z) = lim ɛ 0 Q ɛ /V ɛ 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

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 The above relation can be used to obtain a very important result called the continuity equation which describes the movement of charge through space. Ω Department of ECE Fall 2014 ECE 206: Advanced Calculus 2 5

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 which is also a vector. J(x y z) := ρ(x y z) v(x y z) 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 sec = coul m 2 sec = amps m 2 Department of ECE Fall 2014 ECE 206: Advanced Calculus 2 6

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

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

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

Finally suppose v has an arbetrary 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 1

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(ρv) 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 1

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 current 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 electromagnetism. Department of ECE Fall 2014 ECE 206: Advanced Calculus 2 1

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 coordinates 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 1

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) x F j (x y z) y F k (x y z) 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) x 2 2 F i (x y z) x y 2 F j (x y z) y 2 2 F j (x y z) y z 2 F k (x y z) z 2 2 F k (x 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 1

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) x 2 2 f(x y z) x y 2 f(x y z) y 2 2 f(x y z) y z 2 f(x y z) z 2 2 f(x 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 1

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 for i = 1 2 3. 2 F i (x y z) x y 2 F i (x y z) y z 2 F i (x y z) x z = 2 F i (x y z) y x = 2 F i (x y z) z y = 2 F i (x y z) z x 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 1

Vector and scalar fields (cont.) Both vector and scalar fields can vary with time t. Such time-varying vector and scalar fields are denoted by F(x y z t) and f(x y z t) respectively. A time-varying vector field is one in which for each fixed instant t we just have a vector field which maps each (x y z) D into the vector F(x y z t) R 3. Similarly a time varying scalar field is one in which for each fixed instant t we just have a scalar field which maps each (x y z) D into the real number f(x y z t). Department of ECE Fall 2014 ECE 206: Advanced Calculus 2 1