Multivariable Calculus Flashcards

Question 1

Q

What is a scalar field?

Answer

A

By a scalar field φ on R³ we shall mean a map φ: R³ → R

Question 2

Q

What is a vector field?

Answer

A

By a vector field F on R³ we shall mean a map F: R³ → R³

Question 3

Q

A set is said to be open if ?

What is an open ball?

Answer

A

if for every x ∈ U there exists ε > 0 such that
B(x, ε) = {y ∈ Rⁿ : |y − x| < ε} ⊆ U.
and where
|y − x|² = ⁿΣᵢ₌₁|yᵢ − xᵢ|²
We refer to B(x, ε) as the open ball of radius ε,centred at x.

Question 4

Q

What is the moment of inertia in terms of a double integral, about an axis vertically through a point (x0, y0)?

Answer

A

∫∫ᵣ ρ(x, y)((x − x₀)² + (y − y₀)²) dA

Question 5

Q

Given two co-ordinates u(x, y) and v(x, y) which depend on variables x and y,
we define the Jacobian ∂(u, v)/∂(x, y) to be….

Answer

A

the determinant
| uₓ uᵧ |
| vₓ vᵧ |

Question 6

Q

What is the centre of mass of a body occupying a region R and with density ρ(r) at the point with position vector r ?

Answer

A

r(bar) = (xbar, ybar, zbar) = 1/M ∫∫∫ᵣ 𝐫ρ(𝐫) dV

all 𝐫s vectors
r under integrals should be R

Question 7

Q

What is the median of a function? Given a function f on a region R ⊆ R³

Answer

A

The median of f is the value of m that satisfies

Vol ({(x, y, z) : f(x, y, z) ≤ m}) = 1/2 Vol(R)

Question 8

Q

What does dS refer to?

Answer

A

|∂𝐫/∂u ∧ ∂𝐫/∂v| du dv

𝐫s are vectors

Question 9

Q

What does d𝐒 (vector) refer to

or written as 𝐧 dS

Answer

A

∂𝐫/∂u ∧ ∂𝐫/∂v du dv
𝐫s are vectors
𝐧 is the unit normal in the direction of ∂𝐫/∂u ∧ ∂𝐫/∂v

Question 10

Q

The surface area is [ ] of the choice of parametrization

Answer

A

independent

Question 11

Q

The surface area of r (U) is independent of the choice of parameterization
Prove it

Answer

A

Proof pg 29/30

Question 12

Q

If 𝐅 and φ are a vector field and scalar field defined on a parameterized surface
Σ = 𝐫(U), then we may define the following surface integrals:
1) ∫∫Σ 𝐅 · d𝐒

Answer

A

∫∫Σ 𝐅 · d𝐒 = ∫∫ᵤ 𝐅(𝐫(u, v)) · (∂𝐫/∂u ∧ ∂𝐫/∂v) du dv

Question 13

Q

If 𝐅 and φ are a vector field and scalar field defined on a parameterized surface
Σ = 𝐫(U), then we may define the following surface integrals:
2) ∫∫Σ 𝐅dS

Answer

A

∫∫Σ 𝐅 · dS = ∫∫ᵤ 𝐅(𝐫(u, v)) |(∂𝐫/∂u ∧ ∂𝐫/∂v)| du dv

Question 14

Q

If F and φ are a vector field and scalar field defined on a parameterized surface
Σ = 𝐫(U), then we may define the following surface integrals:
3) ∫∫Σ φ d𝐒

Answer

A

3) ∫∫Σ φ d𝐒 = ∫∫ᵤ φ(𝐫(u, v))(∂𝐫/∂u ∧ ∂𝐫/∂v) du dv

Question 15

Q

If F and φ are a vector field and scalar field defined on a parameterized surface
Σ = 𝐫(U), then we may define the following surface integrals:
4) ∫∫Σ φ dS

Answer

A

4) ∫∫Σ φ dS = ∫∫ᵤ φ(𝐫(u, v))|(∂𝐫/∂u ∧ ∂𝐫/∂v)| du dv

Question 16

Q

What is a flux integral?

Answer

A

∫∫Σ F · dS

F and S bold

Question 17

Q

What is the relationship between dS and dS (one bold)?

Answer

A

dS = ndS
first S and n bold
where n is the outwards!!!! pointing normal from the body

Question 18

Q

What is the wordy definition of a solid angle?

Answer

A

The solid angle is the angle an object subtends at a point in three-dimensional
space. More precisely, half-lines from a fixed point (or observer) will either intersect with the object in question or not; those lines of sight that are blocked by the object represent a subset
of the unit sphere centred on the observer. The solid angle is the area of this subset (strictly it is the area of this subset divided by the unit of length squared to ensure the solid angle is dimensionless).

Question 19

Q

What is the unit of a solid angle?

Answer

A

The unit of solid angle is the steradian. Given that the surface area of a
sphere is 4π(radius)²
then a whole solid angle is 4π`

Question 20

Q

If Σ∗
is a surface and Σ is the subset of Σ∗
facing the unit sphere, then the solid angle Ω
subtended at O by Σ∗
equals, by definition

Answer

A

Ω = ∫∫ (eᵣ · dS)/r² = ∫∫ (r · dS)/r³
Σ Σ

First Integral:
e,S bold
Second Integral:
first r and S bold

Question 21

Q

Define a curve

Answer

A

By a curve we shall mean a piecewise smooth function γ : I → R³ defined on
an interval I of R. Notice that order on I also gives the curve γ an orientation.

Question 22

Q

What is a simple curve?

Answer

A

We say a curve γ : [a, b] → R³
is simple if γ is 1—1, with the one possible exception that γ(a) = γ(b) may be true; this means that the curve does not cross itself except possibly by its endpoints meeting

Question 23

Q

What is a closed curve?

Answer

A

We say a curve γ : [a, b] → R³ is closed if γ(a) = γ(b)

Question 24

Q

Let C be a curve in R³, parameterized by γ : [a, b] → R³ and let F be a vector field, whose domain includes C. We define the line integral of F along C as….

Answer

A

∫𝒸 F · dr = ₐ∫ᵇ F(r(t)) · r’(t) dt

Fs and rs bold

Question 25

Q

Is the line ∫𝒸 F · dr independent of choice of parametrization?

Answer

A

Yes, if orientated the same

Question 26

Q

If oriented the same, the line integral ∫𝒸 F · dr is independent of the choice
of parameterization
Prove it

Question 27

Q

If φ is a scalar field defined on a curve C with parameterization γ : [a, b] → R³, then we also define the line integral

Answer

A

∫𝒸 φ ds = ₐ∫ᵇ φ(t)|γ’(t)| dt

Question 28

Q

If F = (F1, F2, F3) is a vector field defined on the curve C then we define ∫𝒸 F ds

Answer

A

∫𝒸 F ds = (∫𝒸 F1 ds , ∫𝒸 F2 ds, ∫𝒸 F3 ds)

First F is bold

Question 29

Q

Note that if t is the unit tangent vector field along C, in the same direction as
the parameterization, then ∫𝒸 φ ds =

Answer

A

∫𝒸 φ ds = ∫𝒸 (φt) · dr

t and r bold

Question 30

Q

If C is a curve with parameterization γ : [a, b] → R³ then the arc length of
the curve is

Answer

A

∫𝒸 ds = ₐ∫ᵇ |γ’(t)| dt

Question 31

Q

Define

1) dr (r bold)
2) ds

Answer

A

dr = (dx, dy, dz)
ds = |dr|

Question 32

Q

If the vector field F represents a force
on a particle then
∫𝒸 F · dr ( F and r bold) is ???

Answer

A

The work done by the force in moving the particle along C.

Question 33

Q

If F = ∇φ and γ : [a, b] → S is any curve such that γ (a) = p, γ (b) = q then ∫γ F · dr =

F bold
r bold
p, q bold

Answer

A

∫γ F · dr = φ (q) − φ (p)
F,r,q,p bold
In particular, the integral depends only on the endpoints of the curve γ

Question 34

Q

In particular, the integral 
 ∫γ   F · dr  depends only on the endpoints of the curve γ
prove it  (not long)

Question 35

Q

Let S be an open subset of R³. A vector field F: S → R³
is said to be conservative if
F bold

Answer

A

there exists a scalar field φ: S → R such that F = ∇φ

F bold

Question 36

Q

If F = ∇φ (bold F), what is the potential?

Answer

A

φ is the potential for F

Question 37

Q

A subset S of R³ is said to be path connected if ….

Answer

A

for any points p, q ∈ S there exists a curve γ : [a, b] → S such that γ(a) = p and γ(b) = q.
bold q, p

Question 38

Q

Prove that
If ∇φ = 0 on a path-connected set S then φ is constant. In particular, if φ and ψ are potentials of the conservative field F, defined on S, then φ and ψ differ by a constant

Answer

A

If ∇φ = 0 then for any curve, with endpoints p, q, we have
φ (q) − φ (p) = ∫γ ∇φ · dr = 0 (bold r)
Given a fixed point p in S, any q in S is connected to p by a curve, and so φ is constant. If F = ∇φ = ∇ψ then ∇ (φ − ψ) = 0 and the result follows

Question 39

Q

Let S be an open path connected subset of R³ and let F : S → R³ be a vector
field. Then the following three statements are equivalent
(i) F is conservative
(ii) Given any two points p, q ∈ S and curve γ in S, starting at p and ending at q, then
the integral ∫γ F · dr is independent of [ ]
(iii) For any simple closed curve γ then ∫γ F · dr = [ ]

Answer

A

(i) F is conservative
(ii) Given any two points p, q ∈ S and curve γ in S, starting at p and ending at q, then
the integral ∫γ F · dr is independent of the choice of curve γ.

(iii) For any simple closed curve γ then ∫γ F · dr = 0

Question 40

Q

Define the gradient

Answer

A

Let φ: R³ → R be a scalar field. Then the gradient of φ written grad φ or ∇φ
equals
grad φ = ∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z)

Question 41

Q

Define divergence

Answer

A

Let F: R³ → R³ be a vector field with F = (F1, F2, F3). Then the divergence
of F written div F or ∇ · F equals
div F = ∇ · F = ∂F1/∂x + ∂F2/∂y + ∂F3/ ∂z

Question 42

Q

Define curl

Answer

A

Let F: R³ → R³ be a vector field with F = (F1, F2, F3). Then the curl of F written curl F or ∇ ∧ F equals
curl F = ∇ ∧ F = | i j k |
| ∂/∂x ∂/∂y ∂/∂z |
| F1 F2 F3

Question 43

Q

For any scalar field φ, what is the Laplacian?

Answer

A

div grad = ∇ · ∇

∇²φ = ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z²

Question 44

Q

Let F be a vector field on R³ and φ be a scalar field on R³
curl grad φ =
div curl F =

Answer

A

curl grad φ = 0 (bold)

div curl F = 0

Question 45

Q

(Product Rules for div and curl) Let F be a vector field on R³ and φ, ψ be scalar fields on R³. Then
div (φF) =
curl (φF) =
bold Fs

Answer

A

div (φF) = ∇φ · F + φ div F;
curl (φF) = φ curl F + ∇φ ∧ F.
Bold Fs

Question 46

Q

For vector fields F, G
(All F and G bold)
∇ (F · G) =
∇ · (F ∧ G) =

Answer

A

∇ (F · G) = (F · ∇) G + (G · ∇) F + F ∧ (∇ ∧ G) + G ∧ (∇ ∧ F)

∇ · (F ∧ G) = G · (∇ ∧ F) − F · (∇ ∧ G)

Question 47

Q

For vector fields F, G
(All F and G bold)
∇ ∧ (F ∧ G) =

∇ ∧ (∇ ∧ F) =

Answer

A

∇ ∧ (F ∧ G) = F (∇ · G) − G (∇ · F) + (G · ∇) F − (F · ∇) G

∇ ∧ (∇ ∧ F) = ∇ (∇ · F) − ∇²F

Question 48

Q

What is Fourier’s law?

Answer

A

In an isotropic medium with constant thermal conductivity k, with temperature T (x, t) at position x and at time t, Fourier’s Law states that
q = −k∇T
where q is the heat flux — that is the amount of energy that flows through a particular surface per unit area per unit time.

Bold q and x

Question 49

Q

What is the divergence theorem?

Answer

A

Let R be a region of R³ with a piecewise smooth boundary ∂R, and let F be a differentiable vector field on R. Then
∫∫∫R div F dV = ∫∫∂R F · dS
where dS is oriented in the direction of the outward pointing normal from R
F and S bold

Question 50

Q

Define convex

Answer

A

We say that a region R is convex if for any p, q ∈ R the line segment connecting p and q is contained in R

Bold p, q

Question 51

Q

Let φ be a smooth scalar field defined on a region R ⊆ R³ with a piecewise
smooth boundary. Then
∫∫∫R ∇φ dV =??

Answer

A

∫∫∫R ∇φ dV = ∫∫∂R φ dS

S bold

Question 52

Q

Prove that
Let φ be a smooth scalar field defined on a region R ⊆ R³ with a piecewise
smooth boundary. Then
∫∫∫R ∇φ dV = ∫∫∂R φ dS

Answer

A

Let c be a constant vector and F = cφ
Then
div (cφ) = c · ∇φ + φ div c = c · ∇φ
as c is constant. By the Divergence Theorem .....
pg 57

Question 53

Q

What is Green’s Theorem in the plane?

Answer

A

Let D be a closed bounded region in the (x, y) plane, whose boundary C is a piecewise smooth simple closed curve, and that p(x, y), q(x, y) have continuous first-order derivatives in D. Then
∫∫D (∂p/∂x + ∂q/∂y) dx dy = ∫𝒸 (p, q) · n ds = ∫𝒸 p dy - q dx

bold n

Question 54

Q

Prove Green’s Thm in the plane

Question 55

Q

(Uniqueness of the Dirichlet Problem) Let R ⊆ R³ be a path-connected region with piecewise smooth boundary ∂R and let f be a continuous function defined on ∂R.
Suppose that φ1 and φ2 are such that
∇²φ1 = 0 = ∇²φ2 in R;
φ1 = f = φ2 on ∂R
Then ....

Answer

A

Then φ1 = φ2

in R.

Question 56

Q

Prove the Uniqueness of the Dirichlet Problem

Question 57

Q

(Uniqueness, up to a constant, of the Neumann Problem) Let R ⊆ R³
be a path-connected region with piecewise smooth boundary ∂R and let f be a continuous function
defined on ∂R. Suppose that φ1 and φ2 are such that
∇²φ1 = 0 = ∇²φ2 in R;
∂φ1/∂n = f = ∂φ2/ ∂n on ∂R
Then….

Answer

A

Then φ1 − φ2

is constant in R.

Question 58

Q

Prove

Uniqueness, up to a constant, of the Neumann Problem

Question 59

Q

(Robin (or Mixed) Boundary Problem) Let R ⊆ R³ be a path-connected region with piecewise smooth boundary ∂R and let f be a continuous function defined on ∂R.
Further let β be a real constant. Suppose that φ1 and φ2 are such that
∇²φ1 = 0 = ∇²φ2 in R;
βφ1 + ∂φ1/∂n = f = βφ2 + ∂φ2/ ∂n on ∂R
(i) If β > 0 then
(ii) If β = 0 then
(iii) If β < 0 then

Answer

A

(i) If β > 0 then φ1 = φ2
in R.
(ii) If β = 0 then φ1 − φ2
is constant in R.
(iii) If β < 0 then the solution need not be unique, even up to a constant

Question 60

Q

Prove

Robin (or Mixed) Boundary Problem

Question 61

Q

If φ : R³ → R is a continuous scalar field such that ∫∫∫R φ dV = 0
for all bounded subsets R, then φ …

Question 62

Q

If φ : R³ → R is a continuous scalar field such that ∫∫∫R φ dV = 0
for all bounded subsets R, then φ ≡ 0
Prove it

Question 63

Q

Let T (x, t) denote the temperature at position x and at time t in an isotropic
medium R with thermal conductivity k, density ρ, specific heat c with heat flow determined by
Fourier’s Law which states that q = −k∇T
where q is the heat flux
The T satisfies the [ ] equation

Answer

A

heat equation

∂T/∂t = k/ρc ∇²T.

Question 64

Q

Let T (x, t) denote the temperature at position x and at time t in an isotropic
medium R with thermal conductivity k, density ρ, specific heat c with heat flow determined by
Fourier’s Law which states that
q = −k∇T
where q is the heat flux. Then T satisfies the heat equation

Prove it

Answer 56

A

Let Σ be a smooth oriented surface in R³whose
boundary is the curve ∂Σ. Let F be a smooth vector field defined on Σ ∪ ∂Σ. Then
∫∂Σ F · dr = ∫∫Σ curlF · dS

Answer 57

A

then v = w

Answer 58

A

We have c·(v − w) = 0 for all c and thus for all basis vectors. Thus all the components
of v, w are equal, hence so are the vectors

Answer 59

A

∫∫Σ ∇ψ ∧ dS = - ∫∂Σ ψ dr

S and r bold

Answer 60

A

Not long

pg 70

Answer 61

A

every simple, closed curve
C can be continuously deformed to a single point.
Specifically, if r : [0, 1] → R is a simple closed curve beginning and ending in p then there exists a map H : [0, 1] × [0, 1] → R such that H(t, 1) = r(t) and H(t, 0) = p

Answer 62

A

Then F is conservative, i.e. there exists a

potential φ on R such that F = ∇φ.

Answer 63

A

(i) F is [conservative]
(ii) Given any two points p, q ∈ S and curve γ in S, starting at p and ending at q, then
the integral ∫𝒸 F (r) · dr is independent of [the choice of curve C]
(iii) For any simple closed curve C then ∫𝒸 F (r) · dr = [0]

Answer 64

A

𝐟 = −(GM/r³)𝐫 = −(GM/r²)𝐞ᵣ = ∇(GM/r) .

Answer 65

A

𝐅 = -(GMm/r²)𝐞ᵣ

Answer 66

A

conservative

φ = GM/r

Answer 67

A

curl 𝐟 = 0
div 𝐟 = ∇²φ = 1/r²(d/dr(r²(dφ/dr))) = 1/r²(d/dr(r²(−GM/r²))) = 0
div 𝐟 = 0

Answer 68

A

1/m ∫ʳ∞ 𝐅·d𝐫 = ∫ʳ∞ 𝐟 ·d𝐫 = [φ(r)−φ(∞)] = φ(r)

Answer 69

A

−mφ(𝐫).

Answer 70

A

φ(p) = ∫∫∫ᵣ Gρ(𝐫)/|p−𝐫| dV

𝐟(p) = -∫∫∫ᵣ Gρ(𝐫)(p−𝐫)/|p−𝐫|³ dV

Answer 71

A

∇·(𝐫/r³ )= 4πδ(r)

∫∫∫ᵣ φ(𝐫)δ(𝐫−a) dV = φ(a).

Answer 72

A

∇²φ = −4πGρ(𝐫)

Proof non-examinable

Answer 73

A

∫∫∂ᵣ 𝐟 ·d𝐒 = −4πGM.

This result is equivalent to Poisson’s equation

Answer 74

A

Poisson’s equation gives us that ∇²φ = −4πGρ(𝐫) where ρ(𝐫) is the density of the matter. Applying the Divergence Theorem we have
∫∫∂ᵣ 𝐟 ·d𝐒 = ∫∫∫ᵣ ∇·𝐟 dV = ∫∫∫ᵣ ∇²φ dV = −4πG∫∫∫ᵣ ρ dV = −4πGM.
Conversely suppose that we know ∫∫∂ᵣ 𝐟 ·d𝐒 = −4πGM
for any bounded region R.

Then ∫∫∫ᵣ ∇²φ dV = −4πG∫∫∫ᵣρ dV
and so ∫∫∫ᵣ ∇²φ+ 4πGρ dV = 0 for any bounded region R.
Hence (at least if ∇²φ and ρ are piecewise continuous) we have ∇²φ + 4πGρ ≡0

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Multivariable Calculus Flashcards

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