Vector calculus

How do we work out (F ° ∆)G, where F and G are vector fields?

(F ° ∆)G = ((F ° ∆)G₁, (F ° ∆)G₂, (F ° ∆)G₃) So perform Fx d/dx + Fy d/dy + Fz d/dz on each element of G separately. Then G₁ → i, G₂ → j and G₃ → k. (see top equation to understand what I mean here)

Write the cross product in suffix notation.

(a × b)i = εijk aj bk

How to we write ∆ in suffix notation?

* (d/dx, d/dy, d/dz) * relabel (x, y, z) as (x₁, x₂, x₃) and then we have ∆ = (d/dx₁, d/dx₂, d/dx₃) = d/dxi

How to we write divergence in suffix notation?

* div u = ∆ ° u = dux/dx + duy/dy + duz/dz * = dui/dxi with the summation convention implying summation from i = 1 to i = 3

Pathline equations.

* dx/ dt = u * dy/dt = v * dz/dt = w

In the curvilinear coordinate system, define the 3 basis vectors ei, and the scale factors hi.

* e₁ = (1/h₁)(dr/du₁) with h₁ = dr/du₁ * e₂ = (1/h₂)(dr/du₂) with h₂ = dr/du₂ * e₃ = (1/h₃)(dr/du₃) with h₃ = dr/du₃

Give the 3 equations for grad, div and curl acting on sums of scalar or vector functions.

* ∆(f + g) = ∆(f) + ∆(g) * ∆ ° (f + g) = ∆ ° f + ∆ ° g * ∆ × (f + g) = ∆ × f + ∆ × g

Stagnation point.

A point x* where u(x*, t) = 0

Irrotational.

A vector field whose curl ∆ × u = 0 everywhere u = (x, y, z) is irrotational

Level curves.

Curves on which the scalar field ∅(x,y,z) = c

Incompressible fluids

Do not expand or contract so Dδ/Dt = 0 (acc)

The difference between the Eulerian and the Lagrangian approach in fluid dynamics.

Eulerian focuses on a point x at a time t during a fixed flow Lagrangian looks at each separate particle and its trajectory

What is 'F dot grad' written as?

F ° ∆ = (Fx, Fy, Fz) ° (d/dx, d/dy, d/dz) = Fx d/dx + Fy d/dy + Fz d/dz can act upon scalar fields, or vector fields.

Define the divergence.

Given a vector field F = (Fx, Fy, Fz) divF = ∆ ° F = dFx/dx + dFy/dy + dFz/dz result is a scalar

Give the component test for conservative vector fields.

Given a vector field F = F₁(x, y, z)i + F₂(x, y, z)j + F₃(x, y, z), the field is conservative ↔ ∆ × F = 0 ↔ dF₁/dy = dF₂/dx , dF₁/dz = dF₃/dx and dF₂/dz = dF₃/dy proof: as F is conservative, it can be written ∆∅ F = F₁(x, y, z)i + F₂(x, y, z)j + F₃(x, y, z) = d∅/dx i + d∅/dy j + d∅/dz k dF₁/dy = d/dy d∅/dx = d/dx d∅/dy = dF₂/dx others follow similarly

Give 5 rules for differentiating vector functions.

If A = Axi + ayj + Azk * dA/dt = dAx/dt + dAy/dt + dAz/dt *d(∅A)/dt = ∅dA/dt + d∅/dt A *d(A ± B)/dt = dA/dt ± dB/dt *d(A ° B)/dt = A ° dB/dt + dA/dt ° B *d(A × B)/dt = A × dB/dt + dA/dt × B

Define solenoidal.

If a vector field U is divergence free, ie div U = ∆ ° U = 0 example would be u = (0, x, 0)

Give the direction in which f increases and decreases most rapidly.

Increases most rapidly in ∆f direction Decreases most rapidly in -∆f direction

Give the general procedure for finding the tangent given an equation r(t) = x(t)i + y(t)j + z(t)k.

Parametrise r to get r(x,y,z) Differentiate to get r'(x,y,z) Given P, use r(x,y,z) to see what t P corresponds to. Insert this t into r'(x,y,z) and then: Insert into q(w) = r(t₀) + wr'(t₀)

Summation convention.

Repeated suffix j implies that j must be summed from 1 to 3, so we don't need to write the ∑.

Define a conservative vector field.

The line integral around any closed path C is zero. ie the line integral is path independent

Line integral.

The line integral of a vector function F(r) over a curve C, parametrised as r(t) is ∫ F(r) ° dr = ∫ F(r(t) ° dr/dt dt = (F₁, F₂, F₃) ° (dx/dt, dy/dt, dz/dt) dt = F₁dx + F₂dy + F₃dz

Surface integral on a vector field.

This is the flux. ∫ ∫ F ° nhat dA where nhat = dr/du × dr/dv / |dr/du × dr/dv| and dA = |dr/du × dr/dv|dudv

Particle path/ pathline.

Trajectory of a fluid particle over a period of time, r(x₀,t) where x₀ is the particles initial position

Steady flow.

Velocity is independent of time ie u = u(x) = u(u(x,y,z), v(x,y,z), w(x,y,z))

What is meant by the line element?

We differentiate arc length and square the result: (ds/dt)² = dr/dt * dr/dt = (dx/dt)² + (dy/dt)² + (dz/dt)² written ds² = dx² + dy² + dz²

What are scalar and vector fields?

We think of a region, and a physical quantity in the region, eg temperature in a room, a field refers to the region and the value of the physical quantity. A scalar field is if the quantity is scalar, eg temp A vector field is if the quantity is vector, eg force

Give the vorticity of a fluid

Z = ∆ × u

Acceleration equation.

a = du/dt + (u ° ∆)u aka material derivative

Write the scalar product in suffix notation.

a ° b = a₁b₁ + a₂b₂ + a₃b₃ = ∑ ajbj summation convention allows us to remove ∑ thus a ° b = ajbj

Using suffix notation, show that a × (b × c) = (a ° c)b - (a ° b)c

a × (b × c) = εijk aj (b × c)k = εijk aj εklm bl cm = εkijεklm aj bl cm = (δilδjm - δimδjl) aj bl cm = aj δilbl δjmcm - aj δimcm δjlbl = aj bi cj -aj ci bj = bi ajcj - ci ajbj = b(a ° c) - c(a ° b)

Write a ° (b - c) + bhat ° c in suffix notation.

ajbj - ajcj + bjcj/ b

Write (a ° b)(c ° d) in suffix notation.

ajbjckdk the suffix implies which vector is dotted with which, so order doesn't matter

Streamline, and how to find the equation.

an instantaneous curve drawn in the fluid tangent to points in the velocity field ie shows direction of flow t is constant is it is at an instance of time represent the curve parametrically by r(s) = (x(s), y(s), z(s)) then find tangent by dr/ ds = (dx/ds, dy/ds, dz/ds) which is parallel to u: dr/ds = λu

Tangent vectors for r(u,v)

at a point on the surface (u₀, v₀), we can define the two tangent vectors: dr/du = (df/du, dg/du, dh/du) dr/dv = (df/dv, dg/dv, dh/dv) which define the tangent plane, and a normal vector when we take the cross product: dr/du × dr/dv

Give the equations for parametrising a circle, an ellipse, a cylinder and a sphere.

circle: general equation: x² + y² = a² parametrised: r(t) = (acost, asint) = r(x,y) ellipse: general equation: x²/a² + y²/b² = 1 parametrised: r(t) = (acost, bsint, 0) = r(x,y,z) cylinder: general equation: x² + y² = a² -1 ≤ z ≤ 1 parametrised: r(∅,z) = (acos∅, bsin∅, z) sphere: general equation: x² + y² + z² = a² parametrised: r(∅,θ) = (asin∅cosθ, asin∅sinθ, acos∅) 0 ≤ ∅ ≤ π 0 ≤ θ ≤ 2π

Define the curl.

curl u = ∆ × u = i j k d/dx d/dy d/dz ux uy uz = i(duz/dy -duy/dz) - j(duz/dx - dux/dz) + k(duy/dx - dux/dy) so result is a vector, and acts on a vector.

Define the directional derivative.

df/ds = ∆f ° a where a is the unit vector in the direction a

Define the displacement vector dr in the curvilinear system.

dr = h₁du₁e₁ + h₂du₂e₂ + h₃du₃e₃

Write the laplacian in suffix notation

d²/dxidxi

Define the gradient of a scalar field f(x,y,z).

gradf = ∆f = (df/dx, df/dy, df/dz) resulting in the vector form: (df/dx)i + (df/dy)j + (df/dz)k

Given r(t) = x(t)i + y(t)j + z(t)k, what is the length of r(t) defined by? Arc length?

l = ∫ |r'(t)²| dt a ≤ t ≤ b arc length = s(t) = ∫ |r'(T) dT t ≤ T ≤ t₀

Surface normal.

nhat = dr/du × dr/dv / |dr/du × dr/dv|

Give the parametric equation of the tangent line.

q(w) = r(t₀) + wr'(t₀) where r(t) = x(t)i + y(t)j + z(t)k and r'(t) = (dx/dr)i + (dy/dr)j + (dz/dr)k

Define the position vector r, |r| and thus the unit vector of r

r = xi + yj + zk |r| = √ (x² + y² + z²) unit vector of r = r / |r|

Derive the acceleration equation/ material derivative/ total derivative

u(x,y,z,t) = velocity hence acceleration is du/dt using chain rule: (and pathline equations) du/dt = du/dx*dx/dt + du/dy*dy/dt + du/dz*dz/dt + du/dt*dt/dt = du/dt + u*du/dx + v*du/dy + w*du/dz = du/dt + (u ° ∆)u

What is the trace of a 3x3 matrix A in suffix notation?

we can write A as A₁₁ A₁₂ A₁₃ A₂₁ A₂₂ A₂₃ A₃₁ A₃₂ A₃₃ The trace is the sum of the diagonal elements: A₁₁ + A₂₂ + A₃₃ = ∑Ajj = Ajj (by summation convention)

Streamline equations

we have velocity u(x) = (u(x,y,z), v(x,y,z), w(x,y,z)) and tangent which is parallel: dr/ds = (dx/ds, dy/ds, dz/ds) hence * dx/ds = u(x(s),y(s),z(s)) * dy/ds = v(x(s),y(s),z(s)) * dz/ds = w(x(s),y(s),z(s)) or dx/u = dy/v = dz/w

How do we parametrise a line r = p + tm ?

we know r = (x,y), p = (px,py) and m = (mx, my) thus we can write: x = px + tmx y = py + tmy

Derive the continuity equation.

we start with mass flux = ∫ δu ° dS and we know that mass inside a volume is ∫ δdV rate of increase of mass is d/dt ∫ δdV = ∫dδ/dt dV as volume is fixed this equals mass flux ∫ dδ/dt dV = - ∫ δu ° dS = - ∫ ∆ ° δu dV by div theo thus dδ/dt + (∆ ° δu) = 0 this is the continuity equation * incompressible form is just ∆ ° u = 0 *

Define the Kronecker delta.

δij = { 1 if i = j { 0 if i ≠ j = 1 0 0 0 1 0 0 0 1

Explain why δijaj = ai

δijaj = ∑ δijaj = δi₁a₁ + δi₂a₂ + δi₃a₃ we can see when i = 1, δ₁jaj = a₁ etc so it follows that δijaj = ai

Define the Alternating tensor.

εijk = 1 if ijk = 123, 231 or 312 -1 if ijk = 321, 213 or 132 0 if any i, j, k are equal note it has 27 elements. note switching 2 suffices switches the sign.

Write the ith component of the curl in suffix notation.

εijk duk/dxj

Give the relationship between the Kronecker delta and the alternating tensor.

εijkεilm = δjlδkm - δjmδkl

Use suffix notation to prove the identity ∆ ° (∅F) = ∅ ∆ ° F + F °∆∅

∆ ° (∅F) = d∅Fi/dxi = ∅ dFi/dxi + d∅/dxi Fi = ∅∆ ° F + F ° ∆∅

Define the Laplacian.

∆ ° ∆∅ = (d/dx, d/dy, d/dz) ° (d∅/dx, d∅/dy, d∅/dz) = d²∅/dx² + d²∅/dy² + d²∅/dz² = ∆²∅

Give 4 properties of the gradient.

∆(f + g) = ∆(f) + ∆(g) ∆(cf) = c∆(f) ∆(fg) = f∆(g) + g∆(f) * ∆(f(g)) = f'(g)∆(g) *

What is the special property of a scalar field f(x,y,z), relating to ∆f ?

∆f is a vector field whose direction is perpendicular to the level surfaces of f

Calculate ∆r, where r = |r|

∆r = (dr/dx, dr/dy, dr/dz) dr/dx = 1/2 (x² + y² + z²)^-1/2 * 2x = x / √(x² + y² + z²) = x / |r| similar for dr/dy and dr/dz, thus ∆r = xi + yj + zk / |r| = r / |r| = unit vector of r

If F is conservative, we can write it as...

∆∅ where ∅ is a scalar field

Surface integral on a scalar field.

∫ ∅ dA = ∫ ∫ ∅(r(u,v)) |dr/du × dr/dv| dudv

The three alternatives to the line integral equation?

∫ ∅ ds where ds = |dr| = |dr/dt|dt ∫ ∅ dr ∫ F × dr