Vector Functions of 1 Variable Flashcards
General summary of vector rules and methods used to work with them.
What are the 2 sets of symbols used to desgnate unit vectors?
e1, e2, e3 OR i, j, k.
What is the trajectory of an object r(t) as it moves?
The trajectory is path traced as r(t) moves through vector space.
Define the position vector for an object (r(t)) whose position relative to time (t) is (x(t), y(t), z(t)).
r(t) = x(t)i + y(t)j + z(t)k
(r is in bold)
What is the relationship between velocity and speed of an object of position r(t)?
v(t) = r’(t), s(t) = |r’(t)|
what is the relationship between velocity and acceleration of an object of position r(t)?
v(t) = r’(t)
a(t) = v’(t) = r’‘(t)
How do you differentiate and integrate a vector function of a scalar variable?
By components, i.e dr/dt = (dx/dt, dy/dt, dz/dt) and equivalent.
Name two rules applicable where u and w are differentiable vector functions, and where f is a differentiable scalar function.
- u + w, λu, u · w, u × w are differentiable
- (u + w)′ = u′ + w′
- (fu)′ = f′u + fu′
- (u · w)′ = u′· w + u · w′
- (u × w)′ = u′ × w + u × w′.
How do you calculate vector product (dot product) of vectors a & b where θ is the angle between them?
|a| x |b| x cos(θ)
How do you calculate vector product (cross product) of vectors a & b where θ is the angle between them?
|a| x |b| x sin(θ) x n, where n is the unit vector orthogonal to a and b.
How do you calculate the vector product of a(t) and b(t) using matrices?
find the determinant of the matrix
[ i j k ]
[ a1 a2 a3 ]
[ b1 b2 b3 ]
Why can it be important to parameterise a curve, and what is the parameter of
r(t) = x(t), y(t), z(t)?
In order to find more information from a graph, i.e w.r.t time.
The parameter of r(t) is t.
Consider r(t) = (x(t), y(t), z(t)), , t ∈ [a, b].
What does it mean for r(t) to be differentiable, smooth, closed, and simple?
r(t) is differentiable if x(t), y(t) and z(t) are all differentiable.
r(t) is smooth if it can be differentiated as many times as we wish.
r(t) is closed if r(a) = r(b).
r(t) is simple if it has no further self-intersections (it doesn’t cross itsself other than at r(a) and r(b)).
What is the formula for the arc-length of the smooth curve r : [a, b] -> R3?
s = integral between a & b of v(τ)dτ, where the scalar speed v(τ ) = ∥r˙(τ )∥.
(Must know that V(t) refers to speed, not velocity)
What is the velocity of curve C which has been parameterised by arc length?
1.
What is the unit tangent vector at t of r(t)?
T(t) = |v(t)|/v(t), where |V(t)| != 0.
