Modelling 2 Flashcards
When is a function Vector Valued?
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 …
When is a curve Smooth?
Each component function can be differentiated infinitely many times
What is a Simple Curve?
A curve that does not intersect itself is said to be a simple curve.
When is a curve Closed?
A curve parametrised by r(t) where t ∈ [a, b] is said to be closed if r(a) = r(b).
What is a Regular Parametrisation and when is a curve regular?
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.
What are the Differentiation Rules for Vectors?
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.
How are a vector and its derivative related if the vector’s length is constant?
Let r(t) be a vector-valued function such that |r(t)| = constant. Then r(t) and r’(t) are orthogonal.
What is the Arc Length of a curve?
The arc length of the curve parametrised by r(t) between t = t0 and t = t1 is given by
∫t0,t1 |r’(t)| dt.
What is the Arc Length Function?
The arc-length function is defined as
s(t) = ∫t0,t |r’(u)| du for some constant t0.
What is the Arc Length Parametrisation?
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).
How do we calculate regular derivatives of multi-variable functions?
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.
What is the Gradient of a function?
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.
What types of values does ∇f take?
The grad operator ∇ takes a scalar and returns a vector.
What is the Directional Derivative?
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]
How are the directional derivative and gradient related?
Duf(x) = ∇f · u
see notes for proof
What is the Linear Approximation for a function with one or two variables around a point?
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.
What is the linear approximation for a multi-variable function?
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)
What is the normal to 2 and 3 variable functions?
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.
What is a critical point?
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
What is the Theorem for critical points?
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.
What is the direction of steepest descent?
The direction of steepest descent for a function f(x) is -∇f.
What is the formula for centre of mass?
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
How do we integrate in polar coordinates?
We use the formula, ∫∫rdrdθ, with rdrdθ instead of dxdy.
this can be simplified to ∫r^2/2 dθ.
What are cylindrical coordinates?
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.
