2

Vector calculus

2.1

Vector algebra

In this section we will be focussing on three-dimensional space R3 , since most partial differential equations used in physics pertain to this case. The most important quantity, the gradient, is a vector, so it is crucial to understand the vector algebra of R3 , with which you are well familiar already. While R3 has the usual vector space structure, it also allows for a special product to be defined, the cross product, which plays a crucial role. Notation. We will continue to use e1 , e2 and e3 for the standard unit vectors, but also x ˆ = e1 , y ˆ = e2 and ˆ z = e3 . We also use normal script to denote the norm of a vector, for example r = ||r||. We will use the summation convention exclusively throughout this section. Cross product. The cross product of two vectors u and v, denoted u × v, is the vector given by 

x ˆ  u × v = det  u1 v1

y ˆ u2 v2

 ˆ z  x + (u3 v1 − v1 u3 )ˆ y + (u1 v2 − v2 u1 )ˆ z. u3  = (u2 v3 − v3 u2 )ˆ v3

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The crucial property of the cross product is its antisymmetry: a × b = −b × a. You will also be aware that the y -component is obtained from the x-component by cyclic permutation. To formulate these rules in terms of the components of the vectors, we define Definition 2.1.1 (Levi-Civita tensor). The Levi-Civita tensor (or totally antisymmetric tensor) has the properties 1. ǫ123 = 1 2. ǫijk =

(

1

(ijk)

−1

(ijk)

even permutation of (123) odd permutation of (123)

It is easy to show (exercise), that the numerical values of ǫijk are ǫijk = 0, if any two of the indices i, j and k are the same, ǫ123 = ǫ231 = ǫ312 = 1, ǫ132 = ǫ321 = ǫ213 = −1.

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We see that the cross product can be written as (u × v)i = ǫijk uj vk .

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Note that there is a double sum on the right hand side, by the summation convention. The defintion of ǫijk guarantees the antisymmetry of the cross product. Of the properties of ǫijk , the first two are obvious from its definition, the third provides an interesting connection with the Kronecker δ .

Proposition 2.1.2 (Properties of ǫijk ). 1. If any two indices are interchanged, ǫijk changes sign, e.g. ǫijk = −ǫjik .

2. If the indices are cyclically permuted, then ǫijk is unchanged, i.e. ǫijk = ǫjki = ǫkij .

3. ǫijk ǫilm = δjl δkm − δjm δkl .

Proof. Exercise

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With the three properties of ǫijk given by Proposition 2.1.2, it is easy to derive the “Back Cab Rule” for the cross product: (A × (B × C))i = ǫijk Aj (B × C)k = ǫijk ǫklm Aj Bl Cm = ǫkij ǫklm Aj Bl Cm (using ǫijk = −ǫikj = ǫkij )   = δil δjm − δim δjl Aj Bl Cm = Aj Bi Cj − Aj Bj Ci = Bi (A · C) − Ci (A · B).

Extracting the first and last terms in the chain of equalities, this is the component version of the vector equation A × (B × C) = B(A · C) − C(A · B).

2.2

Differential vector calculus

Definition 2.2.1 (Scalar and vector fields). A scalar field on R3 is a function f : R3 → R. A vector field on R3 is a map v : R3 → R3 . The space of r-times differentiable scalar fields is C r (R3 , R). The space of r-times differentiable vector fields is C r (R3 , R3 ). Scalar and vector fields defined in R3 are of particular importance for physical applications. Examples of scalar fields are the temperature T (r), mass density ρ(r) for a fluid or gas, electric charge density q(r) and potential energy U (r) (for example, gravitational potential energy). Examples of vector fields include the velocity field v(r) of a fluid or gas, the electric and magnetic fields E(r) and B(r), and the electric current density j(r). There are three basic first-order differential operations in vector calculus. We begin with the gradient, which maps scalar to vector fields. Definition 2.2.2 (Gradient). The gradient of a scalar field f , denoted ∇f , is the vector field given by ∇f (r) =



∂f ∂f ∂f , , ∂x ∂y ∂z



(r)

The partial derivatives are the components of the gradient in Cartesian coordinates: ∇f (r) · ej = ∂j f (r).

Example 2.2.3 (Examples of the gradient). 1. ∇ tan−1

y x

=



−y x , ,0 x2 + y 2 x2 + y 2

2. [∇r]i = ∂i r = ∂i

Thus ∇r = r/r = ˆr.

p

3.



=

1 (−yˆ x + xˆ y). x2 + y 2

hri δij rj (∂i rj )rj r = √ = √ i = rj rj = √ rj rj rj rj rj rj r i

r f ′ (r) [∇f (r)]i = i , r

and thus ∇f (r) = f ′ (r)ˆr. An interpretation of the gradient. The gradient of f is perpendicular to the level surfaces of f ; the gradient points in the direction in which f changes most rapidly. Remark 2.2.4. It is very useful to think of the symbol ∇≡



∂ ∂ ∂ , , ∂x ∂y ∂z



≡ (∂1 , ∂2 , ∂2 )

in isolation - it is called the “nabla”-operator. It was introduced by William Hamilton, and named for the similarity of its shape to an Egyptian harp. It is a function (or operator) which maps the scalar function f to a vector function ∇f . In index notation, ∇ can be viewed as a vector having components ∂i : e j · ∇ = ∂j .

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The preceding remark suggests the defintion of another differential operator which maps vector fields to scalar fields. Namely, if a is a constant vector, x 7→ a · x maps x ∈ R3 to a scalar. If we replace the vector a by ∇, we arrive at the divergence. Definition 2.2.5 (Divergence). The divergence of a vector field v, denoted ∇ · v, is the scalar field given by ∇ · v(r) =

∂vj (r) ≡ ∂j vj . ∂rj

Example 2.2.6 (Examples of the divergence). 1. ∇ · (xyz, xyz, xyz) = (yz + zx + xy).

2. ∇ · r = ∂i ri = δii = 3.

3. ∇ · (a × r) = ∂i ǫijk aj rk = ǫijk aj δik = ǫiji aj = 0.

In three dimensions, we may also define a mappaing from a vector x ∈ R to another vector, by the mapping x 7→ a × x. This suggests the definition of the following differential operator, called the curl: Definition 2.2.7 (Curl). The curl of a vector field v, denoted ∇ × v, is the vector field given by ∇×v =



∂v ∂v ∂v ∂v ∂v ∂v3 − 2, 1 − 3, 2 − 1 ∂y ∂z ∂z ∂x ∂x ∂y



.

A more useful formula, in components, is [∇ × v]i = ǫijk ∂j vk .

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Example 2.2.8. ∇ × (y 2 , x2 , y 2 ) = (2y, 0, 2(x − y)) [∇ × r]i = ǫijk ∂j rk = ǫijk δjk = ǫijj = 0.

2.3

Second-order differential operations

The operations of grad, div and curl can be combined. Certain combinations always vanish – these are the basic null identities of vector calculus. Other combinations lead to a second-order derivative operation, the Laplacian, which is important in its own right. Schematically, grad, div and curl act as follows: grad: scalar fields → vector fields div: vector fields → scalar fields

curl: vector fields → vector fields Thus, the following combination of operations make sense: curl grad: scalar fields → vector fields div grad: scalar fields → scalar fields

grad div: vector fields → vector fields div curl: vector fields → scalar fields

curl curl: vector fields → vector fields Proposition 2.3.1 (Two null identities).

For all twice differentiable scalar fields f ∈ C 2 (R3 , R),

1. ∇ × (∇f ) = 0.

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2. ∇ · (∇ × v) = 0.

Proof.

1. We have that [∇ × (∇f )]i = ǫijk ∂j ∂k f = −ǫikj ∂j ∂k f = −ǫikj ∂k ∂j f = −[∇ × (∇f )]i .

Thus since the expression equals its own negative, it must vanish. 2. We have that ∇ · (∇ × v) = ǫijk ∂i ∂j vk ,

which must vanish for the same reason. The remaining combinations of grad, div and curl are related to a second-order differential operation called the Laplacian, which we define next. Definition 2.3.2 (Laplacian). The Laplacian of a scalar field f , denoted ∇2 f or △f , is the scalar field given by △f (r) = ∇ · ∇f (r) = ∂i2 f

The Laplacian of a vector field v, denoted △v, is the vector field given by △v = (△v1 , △v2 , △v3 ) .

Proposition 2.3.3.

For a vector field v, △v = −∇ × (∇ × v) + ∇(∇ · v).

Proof. We consider the ith component of the first term on the right: [∇ × (∇ × v)]i = ǫijk ǫklm ∂j ∂l vm .

We permutate two indices to make the product of ǫ’s to look like Proposition 2.1.2 3.: ǫijk = ǫkij . Then [∇ × (∇ × v)]i = (δil δjm − δim δjl )∂j ∂l vm = ∂i ∂m vm − ∂j ∂j vi = [∇(∇ · v) − △v]i ,

which shows that all components agree with our claim.

2.4

Curvilenear coordinate systems

The differential operators defined above were all written in Cartesian coordinates. For many practical problems (imagine describing goings-on on earth) it is better to use a description that is adapted to the symmetry of the problem under consideration. For example, we might want to take a derivative along the surface of the earth. In Cartesian coordinates, this can only be accoplished by varying all three coordinate directions at the same time. Definition 2.4.1 (Curvilinear coordinate system). Curvilinear coordinates are defined by a smooth function r : R3 → R3 which maps a point (q1 , q2 , q3 ) to a point in Cartesian space: r ≡ (x, y, z) = r(q1 , q2 , q3 ). At each point at which J(r)(q1 , q2 , q3 ) 6= 0, we define a local coordinate system with local basis vectors by q ˆ1 =

where

1 ∂r , h1 ∂q1

q ˆ2 =

1 ∂r , h2 ∂q2

∂r , hα = ∂qα

q ˆ3 =

1 ∂r , h3 ∂q3

α = 1, 2, 3

are called the metric coefficients. (In the following, we will occasionally use Greek indices to indicate that the summation convention is not applied. If all local basis vectors are mutually orthogonal, we speek of a orthogonal curvilinear coordinate system. We will also require that the system is right-handed, i.e. q ˆ1 = q ˆ2 × q ˆ3 . Remark 2.4.2. The Jacobi matrix r′ has hα q ˆα , α = 1, 2, 3 as column vectors. Thus J(r) 6= 0 implies that the local basis is nondegenerate. 23