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MA 231 Vector Analysis

Stefan Adams

2010, revised version from 2007, update 02.12.

Contents

Preface

Health Warning: These notes give the skeleton of the course and are not a substitute for attending lectures. They are meant to make note-taking easier so that you can concentrate on the lectures. An important part in vector analysis are figures and pictures. These will not be contained in these notes. For any figure which appears on the blackboard in my lectures I leave some empty space with a reference number which coincides with the number I am using in the lectures. You can fill the diagrams and figures by your own. These notes grew out of hand written notes from Jochen Voß who gave this course 2005 and 2006. I thank him very much for letting me using his notes. Any remarks and suggestions for improvements would help to create better notes for the next year.

Stefan Adams

Motivation

What is Vector Analysis? In analysis differentiation and integration were mostly considered in one di- mension. Vector analysis generalises this to curves, surfaces and volumes in Rn, n ∈ N. As an example consider the “normal” way to calculate a one dimensional integral: You may find a primitive of a function f and use the fundamental theorem of calculus, i.e. for f = F ′^ we get

∫ (^) b

a

f (x) dx = F (b) − F (a).

The value of the integral can be determined by looking at the boundary points of the interval [a, b]. Does this also work in higher dimensions? The answer is given by Gauss’s divergence theorem.

Notation

One of the main problems in vector analysis is that there are many books with all possible different notations. During the whole course I outline alter- native notations in use. It is one of the objectives to acquaint you with the different notations and symbols. Note that most of the material originated from physics and hence many books are using notations and symbols known by people in physics.

iii

Vectors: x ∈ Rn^ with x = (x 1 ,... , xn) and with norm ‖x‖ =

x^21 + · · · + x^2 n. Alternative notations are: ~x, x, x and in dimension n = 3 ~r = (x, y, z) for the vector and |x| or r for the norm of ~x, x, x or ~r. Properties of the norm are (i)‖x‖ ≥ 0; (ii)‖λx‖ = |λ|‖x‖ for λ ∈ R; (iii)‖x + y‖ ≤ ‖x‖ + ‖y‖ for x, y ∈ Rn. Functions: f : Rm^ → Rn^ with component functions f 1 ,... , fn : Rm^ → R. Alternative notations are: f~ or f or f. Partial derivatives: Let ei, 1 ≤ i ≤ n, be the canonical basis vectors in Rn (i.e. 〈ei, ej 〉 for i, j+1,... , n). The partial derivative of a function f : Rn^ → R with respect to the i-th direction at point x ∈ Rn^ is given by

∂f (x) ∂xi

= lim h→ 0

f (x + hei) − f (x) h

, x ∈ Rn.

Alternative notation is: ∂if (x). Scalar product:∑ The scalar product (dot product) is defined as 〈x, y〉 = n i=1 xiyi^ for^ x, y^ ∈^ R n. Alternative expression is x·y. Recall that 〈x, y〉 =

‖x‖‖y‖ cos θ where θ is the angle between the two vectors x and y. Note that ‖y‖ cos θ is the component of the vector y in direction of the vector x.

iv

Remark 1.2 (a) With ϕ(x)(t) = f (x + ty) the directional derivative of f at the point x ∈ Rn^ in direction y ∈ Rn^ is given as

Dyf (x) = ϕ′ (x)(0).

(b) We have defined the directional derivative via the limit as t → 0 for the differential quotient. If one takes the limit t ↓ 0 , that is t > 0 and t → 0 , one gets the directional derivative from the right hand side. The same applies to the limit t ↑ 0 for the directional derivative from the left.

We calculate directional derivatives in the next example.

Example 1.3 Let f : R^2 → R, f (x) = x^21 + x^22 be given. Then for x, y ∈ R^2 we have

f (x + ty) = (x 1 + ty 1 )^2 + (x 2 + ty 2 )^2 = x^21 + x^22 + 2t(x 1 y 1 + x 2 y 2 ) + t^2 (y^21 + y^22 )

Hence Dyf (x) = ϕ′ (x)(0) = 2〈x, y〉.

Figure 2

Recall ϕ(x)(t) = f (x + ty), t ∈ R, x, y ∈ Rn. The function ϕ(x) : R → R is the composition of the function g : R → Rn, t 7 → x + ty, and the function f : g(R) → R (i.e. the restrition of f on g(R) ⊂ Rn), that is

ϕ(x)(t) = (f ◦ g)(t) = f (g(t)).

The chain rule gives

(f ◦ g)′(t) =

∑^ n

i=

∂if (g(t))g i′(t)

∑^ n

i=

∂if (x + ty)yi,

where gi(t) = xi + tyi, t ∈ R, 1 ≤ i ≤ n, are the component functions of g. Hence we get

Dyf (x) = ϕ′ (x)(0) =

∑^ n

i=

∂if (x)yi. (1.1)

We can write (1.1) in a shorter way with the following definition.

Definition 1.4 Let the function f : Rn^ → R be differentiable. The map- ping ∇f : Rn^ → Rn, x 7 → ∇f (x) = (∂ 1 f (x),... , ∂nf (x)) is called the gra- dient mapping and the vector ∇f (x) = (∂ 1 f (x),... , ∂nf (x)) is called the gradient of f at the point x. Alternative notations are grad f, ∇f , ∇f or

∇f =

∑n i=

∂f ∂xi ei.

Note that ∇f : Rn^ → Rn^ is a “vector” of the component functions ∂if : Rn^ → R, x 7 → ∂if (x), 1 ≤ i ≤ n.

Definition 1.5 A scalar or a vector quantity is said to be a field if it is a function of the spatial position. Examples: Let D ⊂ Rm, then f : D → Rn^ is called a vector field if n > 1 , and f : D → R is called a scalar field.

Examples for vector fields are the magnetic, the electric or the velocity (vec- tor) field, whereas temperature and pressure are scalar fields. Our calculation in (1.1) shows that the directional derivative Dyf at any point x ∈ Rn^ is linear in y and we only need to know the gradient ∇f (x) in order to calculate Dyf (x)

Dyf (x) = 〈∇f (x), y〉 (1.2)

Figure 3:

Definition 2.1 Let f : D → R be a function for some domain D ⊂ Rm.

(a) The graph of f is the subset of Rm+1^ consisting of all points (x, f (x)), x ∈ D. In symbols,

graph f = {(x, f (x)) ∈ Rm+1^ : x ∈ D}.

(b) Let c ∈ R. The set

f −^1 (c) = {x ∈ D : f (x) = c}

is called the c-level set of the function f. In dimension m = 2 we speak also of a level curve and in dimension m = 3 of a level surface.

The behaviour or structure of a function is determined in part by the shape of its level sets; consequently, understanding these sets is of great help understanding the functions in question. The idea of level sets is also used in drawing contour maps, where one draws lines to represent constant altitude. Walking along such a line would mean walking on a level path. In the case of a hill rising from the x − y plane, a graph of all level curves gives us a good idea of the ’height’ function h(x, y), which represents the height of the hill at point (x, y).

Example 2.2 (a) f : R^2 → R, (x, y) 7 → f (x, y) = x^2 + y^2.

Figure 4:

(b) f : R^3 → R, (x, y, z) 7 → f (x, y, z) = z^2 − x^2 − y^2. Let c = 0: f −^1 (0) is a cone,

f (x, y, z) = 0 ⇔ r^2 := x^2 + y^2 = z^2 ⇔ r = |z|,

where r =

x^2 + y^2.

Let c > 0:

f (x, y, z) = c ⇔ r =

z^2 − c with z^2 ≥ c ⇔ z = ±

x^2 + y^2 + c.

Hence the level set is a hyperboloid of two sheets around the z axis, passing through the z axis at the points (0, 0 , ±

c).

The level set (surface) is the single-sheeted hyperboloid of revolution around the z axis, which intersects the x − y plane in the circle of radius

−c.

Figure 7: hyperboloid (single-sheet)

2.2 Vector fields, n > 1

We are going to sketch vector fields. As an example think about the velocity vector field inside a fluid or the electric and magnetic field of power currents.

Example 2.3 (a) f : R^2 → R^2 , (x, y) 7 → f (x, y) =

2 0

, see figure 8 below.

Figure 8:

(b) f : R^2 → R^2 , (x, y) 7 → f (x, y) =

x y

, see figure 9 below.

Figure 9:

Example 2.4 (a)

f : [0, 2 π] → R^3 , t 7 → f (t) =

cos t sin t t

This defines the helix seen in the figure 11 below.

Note that for c ∈ R the mapping fc : [0, 2 π] → R^3 , t 7 → fc(t) =

cos t sin t c

defines a circle line in the x − y plane shifted in z direction by c.

(b)

f : [0, 1] × [0, 2 π] → R^3 , (s, t) 7 → f (s, t) =

s cos t s sin t t

This mapping defines the helicoid seen in figure 12.

Note that for fixed parameter t we have lines within the surface of the helicoid and through any point of the helicoid there is a helix going through that point. 

Curves can be given in two ways. A parametric curve is a map ϕ : [a, b] → Rn, e.g. ϕ(t) = (cos t, sin t) ∈ R^2 for t ∈ [0, 2 π]. Curves in Rn, i.e. subsets C ⊂ Rn, can be given as the level set of some real valued function, e.g. consider the function f : R^2 → R, (x, y) 7 → f (x, y) = x^2 + y^2. The curve C is then the level set C = {(x, y) ∈ R^2 : x^2 + y^2 = 1},

which is again the circle line in x − y plane. For a parametric curve ϕ : R → Rn^ any point ϕ(t) gives the ’position’ at ’time’ t. The derivative with respect to the parameter t gives the velocity vector ϕ′(t) at time “time” t. Both are vectors in Rn^ with the following components

ϕ(t) = (ϕ 1 (t),... , ϕn(t)) ∈ Rn^ and ϕ′(t) = (ϕ′ 1 (t),... , ϕ′n(t)) ∈ Rn.

If ϕ′(t) 6 = 0, then ϕ′(t) is a tangent vector of the curve. The tangent line Tϕ(t) at a point ϕ(t) is given by

Tϕ(t) : R → Rn, λ 7 → Tϕ(t)(λ) = ϕ(t) + λϕ′(t).

This is a straight line through the point ϕ(t) in direction of ϕ′(t).

Figure 11: helix

Definition 2.5 A vector x ∈ Rn^ is orthogonal to a parametric curve ϕ : R → Rn^ at the point ϕ(t) if 〈x, ϕ′(t)〉 = 0, i.e. if it is orthogonal to the tangent line.

Lemma 2.6 Let f : Rn^ → R be differentiable and a ∈ Rn. Then

∇f (a) ⊥ {x ∈ Rn^ : f (x) = f (a)} =: L(f (a)).

Proof. Let ϕ : R → L(f (a)) be a differentiable parametric curve in the surface L(f (a)) with ϕ(0) = a. We apply the chain rule (see Proposi- tion 2.8 below) and the notion of differentiability in Definition 2.7. Note that Df (x 0 ) ∈ Lin(Rn, R), x 0 ∈ Rn, is a linear mapping given by the (1 × n)- matrix (^) ( ∂f ∂x 1 (x^0 )^

∂f ∂x 2 (x^0 )^...^

∂f ∂xn (x^0 )

and that Dϕ(t) ∈ Lin(R, Rn) is a linear mapping given by the (n × 1)-matrix

      ϕ′ 1 (t) ϕ′ 2 (t) · · ϕ′ n(t)

The chain rule gives for the derivative of the composition f ◦ ϕ with respect to t at t = 0 as

D(f ◦ ϕ)(t = 0) = Df (ϕ(0)) ◦ Dϕ(0),

where the ◦-operation on the right hand side is the matrix product which in this case is the corresponding scalar product in Rn. With that we get

d dt

f (ϕ(t))

t=0 =^ 〈∇f^ (ϕ(0)), ϕ

= 〈∇f (a), ϕ′(0)〉.

This implies ∇f (a) ⊥ ϕ′(0) and ϕ′(0) is tangent vector, i.e. it is in L(f (a)). 2 Recall the notion of (total) differentiability in the following definition. It generalises the notion of differentiability for real-valued functions defined on the real line. Roughly speaking, the existence of the differential quotient is equivalent to a linear approximation (tangent line) to that function.

Definition 2.7 (Differentiability) Let D ⊂ Rn^ be a domain and f : D → Rm, m ≥ 1 , f = (f 1 ,... , fm). We say that f is differentiable at x 0 ∈ D if the partial derivatives of f exist at x 0 and if there exists a linear mapping L : Rn^ → Rm^ with

lim x→x 0

‖f (x) − f (x 0 ) − L(x − x 0 )‖ ‖x − x 0 ‖

where the linear mapping L is given by the so-called (m×n)− Jacobi matrix at the point x 0 , i.e.

L = Df (x 0 ) =

∂f 1 ∂x 1 (x^0 )^

∂f 1 ∂x 2 (x^0 )^...^

∂f 1 ∂xn (x^0 ) ∂f 2 ∂x 1 (x^0 )^

∂f 2 ∂x 2 (x^0 )^...^...

............ ......... ∂f ∂xmn (x 0 )

The matrix Df (x 0 ) is said to be the (total) derivative of f at x 0. We say that f is differemtiable if it is differentiable at every point of its domain D. In that case the derivative Df of f is mapping

Df : D → Lin(Rn, Rm),

where Lin(Rn, Rm) is the space of linear mappings from Rn^ to Rm^ isomorphic to the space of real (n × m)− matrices.

Proposition 2.8 (Chain rule) Pick n, p, q ∈ N. Let U ⊂ Rn^ and V ⊂ Rp be open subset sets and consider mappings f : U → Rp^ and g : V → Rq. Let x 0 ∈ U be such that f (x 0 ) ∈ V. Suppose f is differentiable at x 0 and g at f (x 0 ). Then g ◦ f : U → Rq, the composition of g and f , is differentiable at x 0 , and we have

D(g ◦ f )(x 0 ) = Dg(f (x 0 )) ◦ Df (x 0 ) ∈ Lin(Rn, Rq),

where the ◦-operation on the right hand side means composition of the linear mappings which corresponds to the matrix product of the matrices describing the linear mappings.

Notation 2.9 Let a curve C ⊂ Rn^ be given as a level set of some function or via some equation. A parametric curve γ : [a, b] → Rn^ is called a parametri- sation (resp C^1 parametrisation if γ is C^1 , that is, γ is differentiable and both, γ and γ′^ are continuous) of the curve C if γ([a, b]) = C. A given curve can have several parametrisations.