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 R2 and heated with a blow-torch. At each point (x, y) of the unit disc, denote the temperature

• f 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 R3 by replacing the disk with a ball. In this case, we’d have a scalar field T(x, y, z) defined

• ver the unit ball in R3.

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 R3. 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 4πǫ0r2 , r :=

• x2 + y2 + z2.

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 R3\{0}). This vector-valued function is an instance of a vector field defined everywhere in three-dimensional space R3 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 ⊂ R3. For some point (x, y, z) ∈ D, define a small sphere of radius 0 < ǫ ≪ 1, volume Vǫ, and en- closed (total) charge Qǫ. If it exists, the limit ρ(x, y, z) = limǫ→0 Qǫ/Vǫ 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

• f 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 = R3, so that the charge is spread “everywhere” in space. For an arbitrary region Ω ⊂ R3, the total charge enclosed within Ω must be given by Q =

• Ω

ρ(x, y, z) dxdydz =

• Ω

ρ dV. 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 m3 × m sec = coul m2 sec = amps m2

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. 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 = v1 + v2, where v1 n, while v2 ⊥ n. In this case, the total charge that crosses S in the time ∆t is given by Q = (Av1∆t)ρ. But v1 is just the projection of v along n, viz. v1 = 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(ρv) · n = A(J · n). Conclusion The total current I through the surface S is the product of the area A

• f 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 ∈ R3, called the domain

• f the vector field, together with a function or mapping F : D → R3

which assigns to each point (x, y, z) ∈ D the vector F(x, y, z) ∈ R3. The vector field F(x, y, z) can be defined in terms of its scalar components F1(x, y, z), F2(x, y, z) and F3(x, y, z) along the co-

• rdinates x, y, z, respectively.

Given the standard i, j, k axes, one has F(x, y, z) = F1(x, y, z)i + F2(x, y, z)j + F3(x, y, z)k. Definition A scalar field comprises a specified region D ∈ R3, called the domain

• f 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 → R3 is called a C1-vector field when, for each i = 1, 2, 3, the partial derivatives ∂Fi(x, y, z) ∂x , ∂Fj(x, y, z) ∂y , ∂Fk(x, y, z) ∂z all exist and are continuous functions of (x, y, z) ∈ D. Definition A vector field F : D → R3 is called a C2-vector field when F is a C1 vector field and, for each i = 1, 2, 3, the partial derivatives ∂2Fi(x, y, z) ∂x2 , ∂2Fj(x, y, z) ∂y2 , ∂2Fk(x, y, z) ∂z2 ∂2Fi(x, y, z) ∂x∂y , ∂2Fj(x, y, z) ∂y∂z , ∂2Fk(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 14/17

Vector and scalar fields (cont.)

Definition A scalar field f : D → R is called a C1-scalar field when the partial derivatives ∂f(x, y, z) ∂x , ∂f(x, y, z) ∂y , ∂f(x, y, z) ∂z all exist and are continuous functions of (x, y, z) ∈ D. Definition A scalar field f : D → R is called a C2-scalar field when f is a C1 scalar field and the partial derivatives ∂2f(x, y, z) ∂x2 , ∂2f(x, y, z) ∂y2 , ∂2f(x, y, z) ∂z2 ∂2f(x, y, z) ∂x∂y , ∂2f(x, y, z) ∂y∂z , ∂2f(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 15/17

Vector and scalar fields (cont.)

A standard result from elementary calculus says that, when F : D → R3 is a C2-vector field, then we always have ∂2Fi(x, y, z) ∂x∂y = ∂2Fi(x, y, z) ∂y∂x , ∂2Fi(x, y, z) ∂y∂z = ∂2Fi(x, y, z) ∂z∂y , ∂2Fi(x, y, z) ∂x∂z = ∂2Fi(x, y, 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

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) ∈ R3. 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 17/17