Vector Calculus

What are the unit vectors?

i = (1,0,0)^T, j = (0,1,0)^T, k = (0,0,1)^T

What is the Euclidean norm?

|x| := √(x²+y²+z²) and r := |x|

A position vector is…

… either x or r depending on if its fixed or moving

If Ω is a bounded, open set we write…

… ∂Ω for the boundary of Ω (=Ω⁻/Ω)

A map F:R³→R³ is called…

…. a vector field.

A function f:R³→R is called…

… a scalar field.

When we need to split F into components, we write…

… F = (F_i) = (F₁,F₂,F₃)^T

What are the cylindrical coordinates?

x = (ρcosθ, ρsinθ, z)^T, for 0≤ρ<∞, 0≤θ<2π and -∞<z<∞

What are the spherical coordinates?

x = (rcosθsinφ, rsinθsinφ, rcosφ)^T, for 0<r<∞, 0≤θ<2π and 0≤φ<π

If Ω⊂R² and f is a scalar function on Ω, then ∫_Ω f dA = ?

∫_Ω f(x) dA(x) = ∫∫_Ω f(x,y) dxdy, with A standing for ‘area’

Let Ω⊂R² be a bounded region. A partition, 𝒫, of Ω is…

… a finite collection of subregions Ω = ∪_i=1^N A_i s.t. A_i are pairwise disjoint.

Let Ω⊂R² be a bounded region. For each subregion, A_i, let δA_i = area(A_i) and choose a sample point x_i∈A_i.

The corresponding Riemann sum is…

… S(𝒫,f) = ∑_i=1^N f(x_i) δA_i

Let Ω⊂R² be a bounded region.

We say that f:Ω→R is Riemann integrable over Ω if…

… the limit ∫∫_Ω f(x) dA(x) := lim_{‖𝒫‖→0} S(𝒫,f) exists and is independent of both the choice of partitions, 𝒫, and the choice of sample points, x_i.

‖𝒫‖ denotes the maximum diameter of the subregions in 𝒫.

What are the two basic properties of double integrals?

1. Linearity: if λ and µ are constants and f and g are scalar fields, then ∫_Ω (λf+µg) dA = λ∫_Ω f dA + µ∫_Ω g dA

2. Additivity: if Ω = Ω₁∪Ω₂ and Ω₁∩Ω₂ = ∅, then ∫_Ω f dA = ∫_Ω₁ f dA + ∫_Ω₂ f dA

[Fubini’s theorem] Assume that f is a sufficiently smooth scalar function on Ω⊂R².

If Ω := {(x,y) : a≤x≤b, c≤y≤d} then ∫_Ω f(x) dA = ?

∫_a^b (∫_c^d f(x,y) dy) dx = ∫_c^d (∫_a^b f(x,y) dx) dy

[Fubini’s theorem] Assume that f is a sufficiently smooth scalar function on Ω⊂R².

If Ω := {(x,y) : a≤x≤b, g₁(x)≤y≤g₂(x)} then ∫_Ω f(x) dA = ?

∫_a^b (∫_g₁(x)^g₂(x) f(x,y) dy) dx

[Fubini’s theorem] Assume that f is a sufficiently smooth scalar function on Ω⊂R².

If Ω := {(x,y) : h₁(y)≤x≤h₂(y), c≤y≤d} then ∫_Ω f(x) dA = ?

∫_c^d (∫_h₁(y)^h₂(y) f(x,y) dx) dy

If the transformation x = x(u,v), y = y(u,v) maps ∫ in the (u,v)-plane to R in the (x,y)-plane, then ∫∫_R f(x,y) dxdy =?

∫∫_S f(x(u,v),y(u,v)) |∂(x,y)/∂(u,v)| dudv where the Jacobian determined ∂(x,y)/∂(u,v) is defined by (∂x/∂u)(∂y/∂v) - (∂y/∂u)(∂x/∂v)

In polar coordinates, the Jacobian is…

… r

If Ω⊂R² and F = (F₁,F₂,F₃)^T is a vector function on Ω, then ∫_Ω F dA = ?

(∫_Ω F₁ dA)i + (∫_Ω F₂ dA)j + (∫_Ω F₃ dA)k

If Ω⊂R³, ∫_Ω f dV = ∫_Ω f(x) dV(x) = ?

∫∫∫_Ω f(x,y,z) dxdydz with the V standing for volume

The properties and results for triple integrals are…

… analogous to the double-integral case

For cylindrical coordinates, the Jacobian comes out as…

… ρ

For spherical coordinates, the Jacobian comes out as…

… r²sinφ

Let Ω⊂R³ and let f:Ω→R be a scalar field.

Then the gradient of f on Ω is defined by…

… ∇f := ∂f/∂xi + ∂f/∂yj + ∂f/∂zk

(note that ∇f is a vector field)

If f and g are differentiable scalar fields and λ and µ are constants, then…

… ∇(λf+µg) = λ∇f + µ∇g and ∇(fg) = f∇g + g∇f

[Chain Rule] If φ is differentiable scalar field then d/dt(φ(x(t))) = ?

∂φ/∂xdx/dt + ∂φ/∂ydy/dt + ∂φ/∂zdz/dt = ∇φ(x(t)) _° dx(t)/dt

Let Ω⊂R³, let f:Ω→R be a differentiable scalar field, let a be a vector in R³ and let â := a/|a|.

Then the directional derivative of f in the direction â at x₀∈Ω is defined by…

… D_af(x₀) := lim_h→0 (f(x₀+hâ) - f(x₀))/h

D_af(x₀) is the rate of change of f in the direction of â at the point x₀

D_af(x₀) = ?

∇f(x₀)_°â = |∇f(x₀)|cosθ where θ is the angle between ∇f(x₀) and â

If ∇f(x₀) = 0, then…

… D_af(x₀) = 0 for every direction a and we say that f has a stationary point

max_a∈R³\{0} |D_af(x₀)| = ?

|∇f(x₀)|

If ∇f(x₀)≠0, then the maximum is…

… attained in the direction a = ∇f

∇f is a vector perpendicular to the surface…

… f(x) = c where c is constant

A scalar field, φ, is called a scalar potential for a vector field F on Ω if…

… F = ∇φ in Ω.

The vector field is then called conservative.

If φ is a scalar potential for F, then…

… φ + c is also a scalar potential for F for every constant c∈R

A curve in R³ is…

… a set C⊂R³ with a parametrisation r:[a,b]→R³ s.t. r: [a,b] = C and r is continuous.

In component form, we write r(t) = (x(t), y(t), z(t))^T

A curve is simple if…

… it does not intersect itself

A curve is said to be closed if…

… r(a) = r(b)

The vector dr(t₀)/dt is…

… tangent to the curve at r(t₀)

The vector t := (dr(t₀)/dt)/|dr(t₀)/dt| is a…

… unit tangent to the curve

Define ds/dt = ?

|dr(t)/dt| = ((dx/dt)²+(dy/dt)²+(dz/dt)²)^½

The arc length of a simple curve from r(a) to r(t) is defined to be…

… s(t) := ∫_a^t |dr(t’)/dt| dt’ for t∈[a,b]

Letting t=s, we have |dr(s)/ds| = ?

1 ∀s

Parametrisation of the curve using s is sometimes called…

… natural parametrisation

Given a curve C⊂R³ with parametrisation r: [a,b] → R³, let -C denote…

… the curve C with the parametrisation r^∼: [-b,-a] → R³ defined by r^∼(t) = r(-t).

-C is the curve C but orientated in the opposite sense.

Let C be a suffiently nice curve with parametrisation r(t) for a≤t≤b.

The line integrals of a sufficiently smooth scalar field f along C is defined by…

… ∫_C f ds := ∫_a^b f(r(t))|dr(t)/dt| dt

ds is equal to what informally? And, if t=s, what does this result in?

ds = |dr(t)/dt|dt informally and, if t=s, then ∫_C f ds := ∫_a^b f(r(s)) ds

What are the three basic properties of line integrals of scalar fields?

1. Linearity: if λ and µ are constants and f and g are scalar fields, then ∫_C (λf+µg) ds = λ∫_C f ds + µ∫_C g ds

2. Additivity wrt curve if C = C₁∪C₂ then ∫_C f ds = ∫_C₁ f ds + ∫_C₂ f ds

3. Independence of direction: with -C, ∫_-C f ds = ∫_C f ds

Let C be a sufficiently smooth nice curve and F = (F₁,F₂,F₃)^T be a sufficiently smooth vector field.

Then ∫_C F ds = ?

(∫_C F₁ ds)i + (∫_C F₂ ds)j + (∫_C F₃ ds)k

Let C be a sufficiently nice curve with parametrisation r(t) for a≤t≤b and let F be a sufficiently smooth vector field.

The work integral of the vector field F along C is defined by…

… ∫_C F _° dr := ∫_a^b F(r(t)) _° dr(t)/dt dt

Finding the work integral can also be thought of as…

… adding up the component of F tangential to C in the orientation given by the parametrisation.

Given a closed curve C, the work integral is therefore called the circulation of the vector field around C.

What are the three basic properties of work integrals?

1. Linearity: if λ and µ are constants and F and G are vector fields, then ∫_C (λF+µG) _° dr = λ∫_C F _° dr + µ∫_C G _° dr

2. Additivity wrt curve if C = C₁∪C₂ then ∫_C F _° dr = ∫_C₁ F _° dr + ∫_C₂ F _° dr

3. Non-independence of direction: with -C, ∫_-C F _° dr = - ∫_C F _° dr

What is the fundamental theorem of calculus for work integrals?

Let φ be a sufficiently smooth scalar field, and let C be a sufficiently nice curve with parametrisation r(t) for a≤t≤b.

Then ∫_C ∇φ _° dr = φ(r(b)) - φ(r(a))

What is the Big Theorem on conservative forces?

The following are equivalent:

1. F is a conservative vector field on a domain Ω

2. For every sufficiently nice closed curve C⊂Ω, ∮_c F _° dr = 0

3. For any two suffiently nice curves, C₁ and C₂, that both have start point x_a and end point x_b, ∫_C₁ F _° dr = ∫_C₂ F _° dr

Define explicit representation of surfaces in R³

{x = (x,y,z)^T∈R³ : z = f(x,y)}

Define implicit representation of surfaces in R³

{x = (x,y,z)^T∈R³ : F(x) = F(x,y,z) = c}

Define parametric representation of surfaces in R³

{r ≡ r(u,v) = (x(u,v), y(u,v), z(u,v))^T for (u,v)∈D≤R²}

Given the explicit representation z = f(x,y)…

…

• an implicit representation is given with F(x,y,z) := z - f(x,y)

• a parametric representatino is given by r(x,y) = (x, y, f(x,y))^T

The equation of a plane in R³ is…

… r = r₀ + λv₁ + µv₂, where λ,µ∈R and v₁ and v₂ are two non-parallel vectors in the plane

A unit normal to the plane is…

… n̂ = (v₁xv₂)/|v₁xv₂|

Suppose that the surface, S, has parametric representation.

A unit normal vector at r(u₀,v₀)∈S is then given by…

… n̂(r(u₀,v₀)) = (∂r/∂u x ∂r/∂v)/|∂r/∂u x ∂r/∂v|

Suppose the surface, S, has implicit representation.

A unit normal vector at x∈S is given by…

… n̂(x)∈S is given by n̂(x) = ∇F(x) / |∇F(x)|

To describe a 2D surface, we must have…

… ∂r/∂u x ∂r/∂u ≠ 0

Let S be a smooth surface. At a point P∈S, choose one of the two unit normals, n̂, to be the outward-pointing unit normal vector.

The surface, S, is orientable if…

… the outward-pointing direction at P can then be continued in unique and continuous way to the entire surface S.

Given a sufficiently smooth S⊂R³, a partition 𝒫 of S is…

… a finite collection of disjoint subsurfaces S₁,…,S_N of S with surface areas δS₁,…,δS_N s.t. S = ∪_i=1^N S_i.

In each of the subsurfaces, choose an arbitrary point x_i.

Then for any scalar function, f, on S, the corresponding Riemann sum is given by…

… ∑(𝒫,f) = ∑_i=1^N f(x_i) δS_i

We say that f is integrable on S, provided that…

… the limit, lim_{‖𝒫‖→0} ∑(𝒫,f), exists and is independent both of the choice of x_i and of the particular subdivision. Here ‖𝒫‖ denotes the maximum diameter of the subregions in 𝒫.

In this case, we write ∫_S f dS = ∫_S f(x) dS(x) = lim_{‖𝒫‖→0} ∑(𝒫,f)

Let S be a sufficiently smooth surface with parametric representation.

Then ∫_S f(x) dS(x) = ?

∫_D f(r(u,v)) |∂r/∂u x ∂r/∂v| dudv

What is the surface element?

dS = |∂r/∂u x ∂r/∂v| dudv

If S is given by explicit parametrisation (i.e. z = f(x,y)), then dS = ?

√(1 + (∂f/∂x)² + (∂f/∂y)²) dxdy

Let S be a sufficiently nice orientable surface with OPUNV n and let F be a sufficiently smooth vector field.

The flux integral of the vector field, F, on S is defined by…

… ∫_S F _° dS := ∫_S F _° n dS

Suppose F is the velocity field of some field.

Then ∫_S F _° dS is the…

… total volume of fluid that passes through S, per unit of time.

Given a function F: Rⁿ→R^m, its Jacobian matrix, DF∈R^mxn is defined by…

… (DF(x))_ij = ∂F_i(x)/∂x_j for i=1,…,m; j=1,…,n at x∈Rⁿ

Suppose that F:Rⁿ→R^m is sufficiently nice at x∈Rⁿ.

Then for h∈Rⁿ …

… |F(x+h) - F(x) - (DF)h| / |h| → 0 as |h| → 0

For an infinitely differentiable f:R→R, by Taylor’s theorem…

… (f(x+h) - f(x) - f’(x)h) / h → 0 as h → 0

What is the relationship between DF and the derivative?

DF is an approximate generalisation of the derivative to higher dimensions.

Let Ω⊂R³ and let F: Ω→R³ be a differentiable vector field.

Then the divergence of F on Ω is defined by…

… ∇_° F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

∇_° F is…

… a scalar field and the trace of the Jacobian of F, DF

If F and G are differentiable vector field, φ is a differentiable scalar field and λ and µ are constants, then…

… ∇ _° (λF + µG) = λ∇ _° F + µ∇ _° G; and ∇ _° (φF) = (∇φ) _° F + φ(∇ _° F)

If ∇ _° F = 0, then the vector field, F, is called…

… solenoidal or incompressible

Let F: Ω→R³ be a differentiable vector field.

Then the curl of F on Ω is defined by…

… ∇ x F := (∂F₃/∂y - ∂F₂/∂z)i - (∂F₃/∂x - ∂F₁/∂z)j + (∂F₂/∂x - ∂F₁/∂y)k

∇ x F is…

… a vector field with alternative notation curlF or rotF

The components of ∇ x F are…

… the non-zero elements of (DF) - (DF)^T.

Therefore if DF is symmetric, ∇ x F = 0

If F and G are differentiable vector fields, φ is a differentiable scalar field and λ and µ are constants, then…

… ∇ x (λF + µG) = λ∇ x F + µ∇ x G; and ∇ x (φF) = (∇φ) x F + φ(∇ x F)

If ∇ x F = 0, then the vector field, F, is called…

… irrotational

What do divergence and curl actually measure?

Divergence measures expansion of a vector field, curl measures rotation

Name two important uses of divergence and curl

The Helmholtz decomposition theorem and Maxwell’s equations of electromagnetics waves

What are the two possible second derivatives of a scalar field, f?

1. ∇ _° (∇f)

2. ∇ x (∇f)

What are three possible second derivatives of a vector field, F?

1. ∇(∇ _° F)

2. ∇_° (∇ x F)

3. ∇ x (∇ x F)

For a scalar field, f, ∇²f := ?

∇ _° (∇f)

For a vector field, F, ∇²F = ?

∇²F₁i + ∇²F₂j + ∇²F₃k

We call ∇²…

… the Laplace operator or Laplacian

It is also written ∆

What is Laplace’s and Poisson’s equation?

Laplace equation: ∇²u = 0

Poisson equation: ∇²u = f

Provided that f and F are sufficiently nice enough for their second partial derivatives (of their components) to be symmetric, then what three equations are true?

1. ∇x(∇f) = 0

2. ∇_°(∇xF) = 0

3. ∇(∇_°F) - ∇x(∇xF) = ∇²F

A domain is…

… an open, connected subset of R³, Ω⊂R³

A domain is bounded if…

… there exists an R>0 s.t. Ω⊂B_R := {x : |x|<R}

i.e. B_R is the ball with radius, R, and centre 0

A domain Ω is convex if… [two points]

… given two points x₁, x₂∈Ω, the line segment {tx₁ + (1-t)x₂ : t∈(0,1)} is a subset of Ω

A domain Ω is convex if… [straight lines]

… every straight line intersects ∂Ω at two points at most.

A surface S⊂R³ is open if…

… ∀x₁,x₂∉S ∃ a curve from x₁ to x₂ which does not cross S.

(a surface is closed if it is not open)

Let Ω⊂R² be a bounded domain s.t. its boundary curve, C, is simple and sufficiently smooth.

We say C is oriented in the anticlockwise sense or traversed in the positive direction if…

… for whatever parametrisation r(t), t∈[a,b], of C we take, as t increases, r(t) moves around C in the anticlockwise direction.