Modelling 2

Question 1

When is a function Vector Valued?

Answer 1

A function r is said to be vector-valued if it maps a number
t ∈ R to a vector r(t) ∈ Rn, where n = 2, 3, 4 …

Question 2

When is a curve Smooth?

Answer 2

Each component function can be differentiated infinitely many times

Question 3

What is a Simple Curve?

Answer 3

A curve that does not intersect itself is said to be a simple curve.

Question 4

When is a curve Closed?

Answer 4

A curve parametrised by r(t) where t ∈ [a, b] is said to be closed if r(a) = r(b).

Question 5

What is a Regular Parametrisation and when is a curve regular?

Answer 5

Consider a curve C parametrised by r(t) where t ∈ I. r is said to be a regular parametrisation if r’(t) /= 0 at all points on the curve.

A curve is said to be regular if it has a regular parametrisation.
Curves defined in segments can also be described as piecewise regular.

Question 6

What are the Differentiation Rules for Vectors?

Answer 6

I. For all λ ∈ R, d/dt [u(t) + λv(t)] = u’(t) + λv’(t)
II. Let f(t) be a differentiable real-valued function, then
d/dt [f(t)u(t)] = f(t)u’(t) + f’(t)u(t)
III. d/dt [u(t) · v(t)] = u(t) · v’(t) + u’(t) · v(t)
IV. d/dt [u(t) X v(t)] = u(t) X v’(t) + u’(t) X v(t)
V. d/dt [u (f(t))] = f’(t)u’(f(t))

To prove these properties just write out the vectors in component form and work that way.

Question 7

How are a vector and its derivative related if the vector’s length is constant?

Answer 7

Let r(t) be a vector-valued function such that |r(t)| = constant. Then r(t) and r’(t) are orthogonal.

Question 8

What is the Arc Length of a curve?

Answer 8

The arc length of the curve parametrised by r(t) between t = t0 and t = t1 is given by
∫t0,t1 |r’(t)| dt.

Question 9

What is the Arc Length Function?

Answer 9

The arc-length function is defined as
s(t) = ∫t0,t |r’(u)| du for some constant t0.

Question 10

What is the Arc Length Parametrisation?

Answer 10

The arc-length parametrisation of r(t) is denoted r(s), where s is the arc-length function.

To find it, find s(t) using the formula and solve for t, then substitute this into r(t).

Question 11

How do we calculate regular derivatives of multi-variable functions?

Answer 11

Consider the function f(x, y), then:
df/dt = ∂f/∂x df/dt + ∂f/∂y dy/dt

Draw a diagram from f to the independent variable and evaluate all paths to that variable.

Question 12

What is the Gradient of a function?

Answer 12

Let f : U ⊂ Rn -> R. The gradient of f, denoted ∇f or grad f
is defined as the vector
∇f = (∂f/∂x1, ∂f/∂x2, ….. , ∂f/∂xn)
In other words ∇f (read “grad f”) is a vector containing all the information about the rates of change of f along all the coordinate directions.

Question 13

What types of values does ∇f take?

Answer 13

The grad operator ∇ takes a scalar and returns a vector.

Question 14

What is the Directional Derivative?

Answer 14

Define f : U ⊂ R3 -> R. and let u be a unit vector. The directional derivative of f in the direction of u, denoted Duf, is defined as
Duf = limh->0 [f(x+hu) -f(x)/h]

Question 15

How are the directional derivative and gradient related?

Answer 15

Duf(x) = ∇f · u

see notes for proof

Question 16

What is the Linear Approximation for a function with one or two variables around a point?

Answer 16

For f(x) around x=a,
f(x) ≈ f(a) + f’(a)(x-a)

for f(x,y) around (a,b),
f(x) ≈ f(a,b) + ∂f/∂x (a,b) * (x-a) + ∂f/∂y (a,b) * (y-b)

it works similarly for a 3 variable function.

Question 17

What is the linear approximation for a multi-variable function?

Answer 17

The linear approximation of a function f : Rn ! R defined by f(x) near x0 is given by
f(x) ≈ f(x0) + [∂f/∂x1, ∂f/∂x2, ……, ∂f/∂xn]|x0 · (x-x0)

Question 18

What is the normal to 2 and 3 variable functions?

Answer 18

Consider the functions F : R2 -> R and G : R3 -> R.
a) The vector ∇F = (Fx, Fy) is normal to the curve F = constant.
b) The vector ∇G = (Gx, Gy, Gz) is normal to the surface G = constant.

Question 19

What is a critical point?

Answer 19

A point (a, b) is said to be a critical point of f : R2 -> R if:
∂f/∂x|(a,b) = ∂f/∂y|(a,b) = 0

Question 20

What is the Theorem for critical points?

Answer 20

Suppose f(x, y) has a critical point at (a, b). Let D = det H where
H =
(fxx fxy)
(fyx fyy),
Det(H) = fxxfyy-fxy^2
then
(a) If D>0 and fxx>0 at (a, b), then (a, b) is a local minimum point.
(b) If D>0 and fxx<0 at (a, b), then (a, b) is a local maximum point.
(c) If D < 0 at (a, b), then (a, b) is a saddle point.
(d) If D = 0 then the test is inconclusive.

Question 21

What is the direction of steepest descent?

Answer 21

The direction of steepest descent for a function f(x) is -∇f.

Question 22

What is the formula for centre of mass?

Answer 22

Consider a solid occupying a region Ω ⊂ R3 with density ρ(x, y, z) and total mass M. The coordinates of its centre of mass (also called centroid), (x~, y~, z~), are
x~ = 1/M * ∫∫∫ρx dV
similarly for y~ and z~, just substitute them in for x in the integrand.

We calculate M = ∫∫∫ρ(x,y,z) dxdydz for the mass

Question 23

How do we integrate in polar coordinates?

Answer 23

We use the formula, ∫∫rdrdθ, with rdrdθ instead of dxdy.
this can be simplified to ∫r^2/2 dθ.

Question 24

What are cylindrical coordinates?

Answer 24

x = rcosθ, y = rsinθ, z = z
Volume element: dV = rdrdθdz

I.e. cylindrical coordinates are polar + z axis.

We can use this set of coordinates when the object is cylindrical, that is x^2 + y^2 appears in the equation, we can the sub in r^2 for this.

Question 25

What are spherical coordinates?

Answer 25

x = rsinφcosθ
y = rsinφsinθ
z = rcosφ
Volume element: dV = r^2 sinφ drdθdφ

We can use this set of coordinates when the object is spherical, that is x^2 + y^2 + z^2 appears in the equation, we can the sub in r^2 for this.

Question 26

How does the Area Element change under a coordinate transformation?

Answer 26

Let F : U ⊆ R2 -> R2 defined by F(u, v)=(x, y) be a bijection which represents a coordinate transformation x = x(u, v) and y = y(u, v). Then, the integral of
f : R ⊆ R2 -> R can be expressed as
∫∫R f(x, y) dx dy = ∫∫S f(u, v) |det DF(u, v)| du dv.
where S = F^-1(R) is the corresponding region in the (u, v) plane. In this above equation, the matrix DF(u, v), called the Jacobian Matrix, is defined by
DF =
(Xu Xv)
(Yu Yv)

Question 27

What are the area and volume elements under coordinate transformations?

Answer 27

dA = dx dy =|det DF(u, v)| du dv
dV = dx dy dz =|det DF(u, v, w)| du dv dw

Question 28

What is a Vector Field?

Answer 28

Functions of the form F : U ⊆ R2 -> R2. Such a function is called a Vector Field. It assigns a vector in Rn to every point in Rn.

Question 29

What is Divergence?

Answer 29

Let F : U ⊆ R3 -> R3. The divergence of F is, denoted ∇ · F or div F is defined by
div F =
(∂/∂x) (F1)
(∂/∂y) · (F2)
(∂/∂z) (F3)
= ∂F1/∂x + ∂F2/∂y + ∂F3/∂z

Question 30

What is Curl?

Answer 30

Let F : U ⊆ R3 -> R3. The of F is, denoted ∇ x F or curl F is defined by
curl F =
|i j k|
|∂/∂x ∂/∂y ∂/∂z|
|F1 F2 F3|

Question 31

What is a line Integral?

Answer 31

A line integral is of the form
∫C F · dr,
where F : Rn -> Rn is a vector field and C is a curve parametrised by r(t) with t ∈ [a, b].
In practice, we evaluate the line integral by writing it as
∫a,b F · dr/dt dt.

Question 32

What is Green’s Theorem?

Answer 32

Our first theorem reduces an area integral to a line integral. This theorem works in R2. A curve C in R2 is said to be positively oriented if it is traversed anti-clockwise.

Green’s Theorem:
Let C be a positively oriented, simple closed curve and D the region bounded by C. For any two-variable functions P, Q that have continuous partial derivatives on D, we have:
∫∫D (∂Q/∂x - ∂P/∂y) dxdy = ∮C (P dx + Q dy)
The weird notation on the RHS is simply ∫ F· dr where F = (P,Q) and r = (x, y).

Question 33

What is pointwise circulation?

Answer 33

∇ x F · n^

Question 34

What is Net Circulation?

Answer 34

It is natural to find the net circulation by summing the pointwise circulation over the entire surface S, i.e.
Net circulation = ∫∫S ∇ x F · n^ dS

Question 35

What is Stokes’ Theorem?

Answer 35

Let F(x, y, z) be a vector field. Let S be a surface with unit normal n^ and boundary curve C oriented positively. Then,
∫∫S ∇ x F · n^ dS = ∫C F · dr.

Question 36

What is the formula for a unit normal of a parametrisation of a surface?

Answer 36

Let r(u, v) be the parametrisation of a surface S. The unit normal at the point P on S is given by
n^ = ± (ru x rv)/|ru x rv|
evaluated at the point P.

Question 37

What is the formula for surface area?

Answer 37

Surface area = ∫∫Ω |ru x rv| dudv

Question 38

What are the known results about a sphere?

Answer 38

Sphere radius a:

Cartesian equation: x^2 + y^2 + z^2 = a^2

Parametrisation of points within the volume (spherical coords):
r(r, θ, φ) =
(rcosθsinφ)
(rcosθsinφ)
(rcosφ)
dV = r^2sinφ drdθdφ
Volume = 4/3 π a^2

Parametrisation of points on the surface (r=a):
r(θ, φ) =
(acosθsinφ)
(acosθsinφ)
(acosφ)
dS = a^2sinφ dθdφ
Surface area = 4 π a^2

Question 39

What is the Divergence Theorem?

Answer 39

Let F : R3 -> R3 be a differentiable vector field. Let V be a finite volume in R3 and S its (closed) surface. Then
∫∫∫V ∇ · F dV = ∫∫S F · n^ dS
where n^ is the outward-pointing unit normal to the surface S.

Question 40

When is a vector field conservative?

Answer 40

A vector field F which can be expressed as ∇f is called a conservative field
ie F = ∇f.